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%%BeginResource: procset RangerDict 1 1
/RangerDict 150 dict def RangerDict begin /bd{bind def}bind def /ld{load def}bd
/p/closepath ld /cc/concat ld /v/curveto ld /gr/grestore ld /gs/gsave ld
/l/lineto ld /m/moveto ld /n/newpath ld /r/rlineto ld /sc/scale ld /sd/setdash
ld /g/setgray ld /lc/setlinecap ld /lj/setlinejoin ld /lw/setlinewidth ld
/c/setrgbcolor ld /sp/showpage ld /s/stroke ld /tf/transform ld /tr/translate
ld /xd{exch def}bd /MGX_Overprint false def /min{2 copy gt{exch} if pop}bd
/arct where {pop} {/arct{arcto pop pop pop pop}bd} ifelse /make_ro_array {
/packedarray where { pop packedarray }{array astore readonly } ifelse } bd
/findcmykcustomcolor where { pop }{ /findcmykcustomcolor { 5 make_ro_array }
def } ifelse /setcustomcolor where { pop } { /setcustomcolor { exch aload pop
pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } def } ifelse
/setcmykcolor where {pop /k/setcmykcolor ld} {/k{4 1 roll 3 {3 index add 1 min
1 exch sub 3 1 roll} repeat c pop}bd} ifelse /x {/tint xd findcmykcustomcolor
MGX_Overprint setoverprint tint 1 exch sub setcustomcolor} def /setoverprint
where {pop}{/setoverprint {/MGX_Overprint xd} def} ifelse /currentoverprint
where {pop}{/currentoverprint{MGX_Overprint}def} ifelse /op{setoverprint}def
/_mx matrix def /smx{/_mx _mx currentmatrix def}bd /rmx{_mx setmatrix}bd /_ctm
matrix def /sctm{/_ctm _ctm currentmatrix def /_res 72 72 matrix defaultmatrix
dtransform _ctm idtransform pop abs def}bd /rctm{_ctm setmatrix}bd /el{smx tr
sc 1 eq {0 -1 m} if 0 0 1 -90 270 arc rmx}bd /ea{smx tr sc 1 eq {arcn} {arc}
ifelse 1 eq {0 0 l} if rmx}bd /rt{1 eq {m} {l} ifelse 2 copy 0 r 0 exch r neg 0
r neg 0 exch r}bd /rr{smx tr /_wy exch 3 index div def /_wx exch 3 index div
def sc _wx 2 div 0 3 -1 roll 1 eq {m} {l} ifelse 0 0 0 _wy 0.5 arct 0 _wy _wx
_wy 0.5 arct _wx _wy _wx 0 0.5 arct _wx 0 0 0 0.5 arct _wx 2 div 0 l rmx}bd
/clp{/_ct xd {{gs _ct 1 eq {eoclip} {clip} ifelse}stopped {gr currentflat 1 add
setflat} {exit} ifelse} loop}bd /horz{_y1 _w 2 div add _w _y2 {dup _x1 exch m
_x2 exch l s} for}bd /vert{_x1 _w 2 div add _w _x2 {dup _y1 m _y2 l s} for}bd
/fdiag{_x1 _w _x2 _y2 _y1 sub add{_y2 m _y2 _y1 sub neg dup r s} for}bd
/bdiag{_x1 _y2 _y1 sub sub _w _x2{_y2 m _y2 _y1 sub dup neg r s} for}bd
/hf{/_ht xd clp rctm /_w _res 12 div def pathbbox _w div cvi 1 add _w mul /_y2
xd _w div cvi 1 add _w mul /_x2 xd _w div cvi _w mul /_y1 xd _w div cvi _w mul
/_x1 xd 1 lw [] 0 sd n _ht 0 eq {horz} if _ht 1 eq {vert} if _ht 2 eq {fdiag}
if _ht 3 eq {bdiag} if _ht 4 eq {horz vert} if _ht 5 eq {fdiag bdiag} if gr}bd
/bf{/_im xd clp rctm /_w _res 12 div round def pathbbox _w div cvi 1 add _w mul
/_y2 xd _w div cvi 1 add _w mul /_x2 xd _w div cvi _w mul /_y1 xd _w div cvi _w
mul /_x1 xd /_wx _x2 _x1 sub def /_wy _y2 _y1 sub def /_swx _wx _w div cvi 8
mul def /_swy _wy _w div cvi 8 mul def /_ic 0 def _x1 _y1 tr _wx _wy sc _swx
_swy false [_swx 0 0 _swy 0 0] /_swx _swx 8 div def {_im _ic _swx div cvi 8 mod
1 getinterval /_ic _ic 1 add def} imagemask gr}bd /readimg{currentfile _imgstr
readstring pop}bd /readimgx{currentfile _imgstr readhexstring pop}bd /i{gs tr
sc /_imgstr exch string def [4 index 0 0 6 index 0 0] exch 1 eq {{readimg}}
{{readimgx}} ifelse image gr}bd /im{gs tr sc /_imgstr exch string def 1 eq
{true} {false} ifelse [3 index 0 0 5 index 0 0] {readimgx} imagemask gr}bd
userdict /RangerImage 40 dict dup begin put /_image systemdict /image get def
/_setgray systemdict /setgray get def /_currentgray systemdict /currentgray get
def /_settransfer systemdict /settransfer get def /_currenttransfer systemdict
/currenttransfer get def /clr? /deviceinfo where {pop deviceinfo /Colors known
{deviceinfo /Colors get exec 1 gt}{false}ifelse }{ /statusdict where { pop
statusdict /processcolors known { statusdict /processcolors get exec 1 gt }{
false }ifelse }{ false }ifelse }ifelse def /blank 0 _currenttransfer exec 1
_currenttransfer exec eq def /negative blank{ 0 _currenttransfer exec 0.5 lt }{
0 _currenttransfer exec 1 _currenttransfer exec gt }ifelse def /inverted?
negative def /l2? /languagelevel where{pop languagelevel 2 ge}{false} ifelse
def /4= { 4 index eq 8 1 roll 4 index eq 8 1 roll 4 index eq 8 1 roll 4 index
eq 8 1 roll pop pop pop pop and and and }def clr? {/band 0 def}{/band 5
def}ifelse /setcmykcolor where { pop 1 0 0 0 setcmykcolor _currentgray 1 exch
sub 0 1 0 0 setcmykcolor _currentgray 1 exch sub 0 0 1 0 setcmykcolor
_currentgray 1 exch sub 0 0 0 1 setcmykcolor _currentgray 1 exch sub 4 {4 copy}
repeat 1 0 0 0 4= {/band 1 store} if 0 1 0 0 4= {/band 2 store} if 0 0 1 0 4=
{/band 3 store} if 0 0 0 1 4= {/band 4 store} if 0 0 0 0 4= {/band 6 store} if
} if blank {/band 6 store} if l2? {band 0 eq {/DeviceCMYK}{/DeviceGray} ifelse
setcolorspace } if /readdata {currentfile exch readhexstring pop} def /sepimage
l2? { /image ld }{ { begin Width Height BitsPerComponent ImageMatrix Decode
length 2 eq {/DataSource load image} if Decode length 6 eq {DataSource 0 get
DataSource 1 get DataSource 2 get true 3 colorimage }if Decode length 8 eq {
DataSource 0 get DataSource 1 get DataSource 2 get DataSource 3 get true 4
colorimage }if end }def }ifelse /_image2 l2? { /_image ld }{ { begin Width
Height BitsPerComponent ImageMatrix /DataSource load _image end } def }ifelse
/doimage { band 0 eq band 5 eq or { sepimage } if band 1 ge band 4 le and { {1
exch sub dummy exec} dup 3 _currenttransfer put _settransfer _image2 } if band
6 eq { negative {{pop 0}}{{pop 1} }ifelse _settransfer _image2 }if } def /ic {
gsave translate scale /scanlen xd /ncomp xd /IsBin xd /bpcomp xd /rows xd /cols
xd /picstr1 cols string def /picstr2 cols string def /picstr3 cols string def
/picstr4 cols string def /picstr5 cols string def 12 dict begin /ImageType 1
def /Width cols def /Height rows def /ImageMatrix [cols 0 0 rows 0 0] def
/BitsPerComponent bpcomp def band 0 eq { /Decode [0 1 0 1 0 1 0 1] def } {
/Decode [0 1] def } ifelse band 0 eq { /MultipleDataSources true def
/DataSource [ { picstr1 readdata } { picstr2 readdata } { picstr3 readdata } {
picstr4 readdata picstr5 readdata pop } ] def } if band 1 eq { /DataSource {
picstr1 readdata picstr2 readdata pop picstr3 readdata pop picstr4 readdata pop
picstr5 readdata pop } def } if band 2 eq { /DataSource { picstr1 readdata pop
picstr2 readdata picstr3 readdata pop picstr4 readdata pop picstr5 readdata pop
} def } if band 3 eq { /DataSource { picstr1 readdata pop picstr2 readdata pop
picstr3 readdata picstr4 readdata pop picstr5 readdata pop } def } if band 4 eq
{ /DataSource { picstr1 readdata pop picstr2 readdata pop picstr3 readdata pop
picstr4 readdata picstr5 readdata pop } def } if band 5 ge { /DataSource {
picstr1 readdata pop picstr2 readdata pop picstr3 readdata pop picstr4 readdata
pop picstr5 readdata } def } if currentdict end doimage grestore } def
currentdict readonly pop end /bullets{{/bullet}repeat} bd
/WE[/grave/acute/circumflex/tilde/macron
/breve/dotaccent/dieresis/ring/cedilla/hungarumlaut/ogonek/caron/dotlessi 18
bullets StandardEncoding 32 95 getinterval aload pop 3 bullets
/quotesinglbase/florin/quotedblbase/ellipsis/dagger/daggerdbl/circumflex
/perthousand/Scaron/guilsinglleft/OE 4 bullets/quoteleft/quoteright
/quotedblleft/quotedblright/bullet/endash/emdash/tilde/trademark/scaron
/guilsinglright/oe 2 bullets /Ydieresis/space/exclamdown/cent/sterling
/currency/yen/brokenbar/section/dieresis/copyright/ordfeminine
/guillemotleft/logicalnot/hyphen/registered/macron/degree/plusminus
/twosuperior/threesuperior/acute/mu/paragraph/periodcentered/cedilla
/onesuperior/ordmasculine/guillemotright/onequarter/onehalf/threequarters
/questiondown/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/AE/Ccedilla
/Egrave/Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis
/Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/multiply/Oslash
/Ugrave/Uacute/Ucircumflex/Udieresis/Yacute/Thorn/germandbls/agrave/aacute
/acircumflex/atilde/adieresis/aring/ae/ccedilla/egrave/eacute/ecircumflex
/edieresis/igrave/iacute/icircumflex/idieresis/eth/ntilde/ograve/oacute
/ocircumflex/otilde/odieresis/divide/oslash/ugrave/uacute/ucircumflex
/udieresis/yacute/thorn/ydieresis]def WE dup 39/quotesingle put 96/grave put
/selectfont where {pop} {/selectfont {exch findfont exch dup type /arraytype eq
{makefont} {scalefont} ifelse setfont}bd} ifelse /_mxe matrix def /_mxf matrix
def /we {/_fn xd FontDirectory _fn known {pop} {findfont dup maxlength dict
/_fd xd {exch dup /FID ne {exch _fd 3 1 roll put} {pop pop} ifelse} forall _fd
dup /FontName _fn put /Encoding WE put _fn _fd definefont pop} ifelse _fn} bd
/fnt {/_ys xd /_xs xd /_asc xd /_itl exch dup 0 ne {10.0 div sin} if def [1 0
_itl 1 0 0] [_xs 0 0 _ys neg 0 _asc] _mxf concatmatrix selectfont currentfont
dup dup /FontInfo known {/FontInfo get} if /UnderlinePosition known not {pop
/Courier findfont} if /FontInfo get /UnderlinePosition get 1000 div 0 exch _mxf
tf /_ulpos xd pop 0 .3 _mxf tf /_sopos xd pop currentfont dup dup /FontInfo
known {/FontInfo get} if /UnderlineThickness known not {pop /Courier findfont}
if /FontInfo get /UnderlineThickness get 1000 div _ys mul /_ulsiz xd } bd /to
{m show n} bd /tp {m false charpath} bd /ul {1 eq {/_pos _ulpos def} {/_pos
_sopos def} ifelse 1 eq {gs [] 0 sd 0 lc _ulsiz lw _pos add m 0 r s gr} {_pos
add _ulsiz 2 div add m 0 _ulsiz neg r 0 r 0 _ulsiz r p} ifelse} bd end
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11420 11382 m -26 13 r -22 17 r -28 56 r -15 75 r 5 84 r 45 175 r 40 76 r 51 60
r 112 -55 r 73 -80 r 39 -91 r 6 -95 r -26 -90 r -55 -75 r -42 -30 r -45 -24 r
-52 -13 r p 0.000 0.000 0.000 1.000 k eofill 11420 11382 m -26 13 r -22 17 r
-28 56 r -15 75 r 5 84 r 45 175 r 40 76 r 51 60 r 112 -55 r 73 -80 r 39 -91 r 6
-95 r -26 -90 r -55 -75 r -42 -30 r -45 -24 r -52 -13 r -60 -3 r 0.000 0.000
0.000 1.000 k s 11475 14075 m -18 -42 r -7 -43 r 9 -37 r 15 -35 r 43 -66 r 43
-62 r 20 62 r 7 58 r -8 52 r -14 45 r -43 61 r -23 14 r p 0.000 0.000 0.000
1.000 k eofill 11475 14075 m -18 -42 r -7 -43 r 9 -37 r 15 -35 r 43 -66 r 43
-62 r 20 62 r 7 58 r -8 52 r -14 45 r -43 61 r -23 14 r -24 -7 r 0.000 0.000
0.000 1.000 k s 7190 8763 m -40 -5 r -28 -18 r -22 -25 r -10 -38 r 5 -40 r 19
-30 r 25 -20 r 40 -10 r 38 5 r 30 18 r 22 25 r 10 38 r -5 40 r -19 30 r -26 20
r p 0.000 0.000 0.000 1.000 k eofill 7190 8763 m -40 -5 r -28 -18 r -22 -25 r
-10 -38 r 5 -40 r 19 -30 r 25 -20 r 40 -10 r 38 5 r 30 18 r 22 25 r 10 38 r -5
40 r -19 30 r -26 20 r -39 10 r 0.000 0.000 0.000 1.000 k s 10744 9037 m -39 -7
r -28 -17 r -22 -26 r -11 -40 r 6 -39 r 17 -30 r 27 -20 r 40 -11 r 35 6 r 28 17
r 22 28 r 11 39 r -6 40 r -19 28 r -26 22 r p 0.000 0.000 0.000 1.000 k eofill
10744 9037 m -39 -7 r -28 -17 r -22 -26 r -11 -40 r 6 -39 r 17 -30 r 27 -20 r
40 -11 r 35 6 r 28 17 r 22 28 r 11 39 r -6 40 r -19 28 r -26 22 r -35 10 r
0.000 0.000 0.000 1.000 k s 10479 9325 m -40 -7 r -29 -13 r -21 -27 r -10 -35 r
5 -31 r 18 -29 r 27 -21 r 38 -10 r 37 5 r 28 18 r 18 27 r 9 31 r -5 35 r -14 29
r -26 16 r p 0.000 0.000 0.000 1.000 k eofill 10479 9325 m -40 -7 r -29 -13 r
-21 -27 r -10 -35 r 5 -31 r 18 -29 r 27 -21 r 38 -10 r 37 5 r 28 18 r 18 27 r 9
31 r -5 35 r -14 29 r -26 16 r -35 12 r 0.000 0.000 0.000 1.000 k s 10209 9685
m -29 -3 r -25 -14 r -21 -50 r 16 -53 r 50 -23 r 52 16 r 17 24 r 6 26 r -3 29 r
-13 25 r -24 16 r p 0.000 0.000 0.000 1.000 k eofill 10209 9685 m -29 -3 r -25
-14 r -21 -50 r 16 -53 r 50 -23 r 52 16 r 17 24 r 6 26 r -3 29 r -13 25 r -24
16 r -26 7 r 0.000 0.000 0.000 1.000 k s 11459 8133 m -62 15 r -47 34 r -20 48
r 0 57 r 19 50 r 41 41 r 69 24 r 86 -7 r -8 -62 r -18 -61 r p 0.000 0.000 0.000
1.000 k eofill 11459 8133 m -62 15 r -47 34 r -20 48 r 0 57 r 19 50 r 41 41 r
69 24 r 86 -7 r -8 -62 r -18 -61 r -60 -139 r 0.000 0.000 0.000 1.000 k s 8965
8963 m -46 15 r -47 -1 r -87 -2 r 30 57 r 49 28 r 93 30 r p 0.000 0.000 0.000
1.000 k eofill 8965 8963 m -46 15 r -47 -1 r -87 -2 r 30 57 r 49 28 r 93 30 r 8
-127 r 0.000 0.000 0.000 1.000 k s 9435 9175 m 30 42 r 34 30 r 95 45 r -44 78 r
-66 90 r -47 -132 r -12 -70 r -1 -28 r p 0.000 0.000 0.000 1.000 k eofill 9435
9175 m 30 42 r 34 30 r 95 45 r -44 78 r -66 90 r -47 -132 r -12 -70 r -1 -28 r
11 -55 r 0.000 0.000 0.000 1.000 k s 11482 11353 m 328 12 r 0 63 r -33 37 r
-100 33 r p 0.000 0.000 0.000 1.000 k eofill 11482 11353 m 328 12 r 0 63 r -33
37 r -100 33 r -195 -145 r 0.000 0.000 0.000 1.000 k s 10480 6508 m -608 1172 r
1333 lw 2 lc 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740
3462.126 5983.465 ] cc s rctm 10165 6142 m -643 1498 r 0.000 0.000 0.000 1.000
k [0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 9534 6400
m -372 1192 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740
3462.126 5983.465 ] cc s rctm 9162 6068 m -398 1547 r 0.000 0.000 0.000 1.000 k
[0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 8629 6425 m
-180 1178 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740
3462.126 5983.465 ] cc s rctm 8164 6168 m -105 1522 r 0.000 0.000 0.000 1.000 k
[0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 7679 6577 m
38 1133 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740
3462.126 5983.465 ] cc s rctm 7182 6450 m 190 1423 r 0.000 0.000 0.000 1.000 k
[0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 6795 6950 m
254 1070 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740
3462.126 5983.465 ] cc s rctm 6290 6828 m 469 1394 r 0.000 0.000 0.000 1.000 k
[0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 6004 7430 m
450 1040 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740
3462.126 5983.465 ] cc s rctm 5494 7460 m 686 1185 r 0.000 0.000 0.000 1.000 k
[0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 5300 8057 m
604 833 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740
3462.126 5983.465 ] cc s rctm 4832 8187 m 870 1008 r 0.000 0.000 0.000 1.000 k
[0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 4735 8805 m
759 730 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740
3462.126 5983.465 ] cc s rctm 4254 8992 m 1063 810 r 0.000 0.000 0.000 1.000 k
[0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 4290 9608 m
857 527 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740
3462.126 5983.465 ] cc s rctm 3830 9868 m 1212 605 r 0.000 0.000 0.000 1.000 k
[0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 3994 10498 m
925 294 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740
3462.126 5983.465 ] cc s rctm 3600 10873 m 1310 275 r 0.000 0.000 0.000 1.000 k
[0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 3855 11442 m
954 106 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740
3462.126 5983.465 ] cc s rctm 3565 11877 m 1265 25 r 0.000 0.000 0.000 1.000 k
[0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 3880 12378 m
974 -90 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740
3462.126 5983.465 ] cc s rctm 3662 12912 m 1283 -275 r 0.000 0.000 0.000 1.000
k [0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 4102 13307
m 953 -282 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740
3462.126 5983.465 ] cc s rctm 3974 13890 m 1203 -562 r 0.000 0.000 0.000 1.000
k [0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 4444 14155
m 816 -500 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740
3462.126 5983.465 ] cc s rctm 4440 14738 m 999 -746 r 0.000 0.000 0.000 1.000 k
[0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 4902 15033 m
760 -751 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740
3462.126 5983.465 ] cc s rctm 10497 8212 m -395 -119 r -378 -83 r -370 -48 r
-357 -19 r -347 14 r -328 40 r -315 66 r -302 94 r -290 116 r -270 144 r -255
156 r -241 185 r -220 195 r -205 217 r -190 228 r -167 244 r -150 253 r -130
270 r -113 275 r -90 280 r -69 285 r -48 295 r -25 293 r -7 295 r 19 295 r 41
294 r 65 293 r 89 285 r 115 277 r 140 271 r 161 264 r 194 251 r 225 282 r 116
158 r 110 175 r 85 185 r 60 200 r 19 110 r 6 110 r -21 235 r 0.000 0.000 0.000
1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm
10140 7798 m -165 0 r -150 20 r -65 27 r -56 37 r -49 45 r -40 58 r 0.000 0.000
0.000 1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s
rctm 5849 13808 m -55 130 r -42 144 r -15 146 r 22 149 r 0.000 0.000 0.000
1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm
11369 7838 m -187 169 r -182 176 r -343 377 r -327 398 r -315 422 r -296 437 r
-292 451 r -572 907 r -580 892 r -305 428 r -313 410 r -333 383 r -349 357 r
-183 165 r -192 153 r -195 150 r -205 135 r 0.000 0.000 0.000 1.000 k [0.078740
0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 11677 8538 m -215 174
r -207 178 r -378 367 r -352 380 r -323 386 r -303 390 r -292 397 r -547 798 r
-556 784 r -289 385 r -113 145 r -193 228 r -330 363 r -357 352 r -390 333 r
-427 317 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740
3462.126 5983.465 ] cc s rctm 9742 8758 m -20 -125 r 17 -111 r 46 -87 r 70 -58
r 85 -34 r 95 -1 r 97 38 r 47 37 r 41 45 r 0.000 0.000 0.000 1.000 k [0.078740
0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 9687 8525 m -60 -17 r
-60 -8 r -97 15 r -73 40 r -50 67 r -27 80 r 5 93 r 30 97 r 65 98 r 0.000 0.000
0.000 1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s
rctm 9460 8950 m -798 860 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000
0.000000 0.078740 3462.126 5983.465 ] cc s rctm 9017 8273 m 87 -5 r 71 7 r 62
25 r 48 33 r 65 90 r 15 50 r 4 55 r -30 109 r -29 45 r -36 41 r -50 35 r -62 20
r -65 7 r -80 -7 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000
0.078740 3462.126 5983.465 ] cc s rctm 8839 8427 m -77 -75 r -47 -72 r -21 -68
r 3 -69 r 25 -60 r 40 -50 r 110 -78 r 67 -20 r 70 -3 r 63 11 r 62 27 r 50 50 r
38 63 r 27 94 r 3 113 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000
0.078740 3462.126 5983.465 ] cc s rctm 8715 8237 m -66 5 r -54 15 r -46 26 r
-37 30 r -25 37 r -22 40 r -11 92 r 18 93 r 48 80 r 75 58 r 95 30 r 0.000 0.000
0.000 1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s
rctm 8704 8687 m -272 1106 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000
0.000000 0.078740 3462.126 5983.465 ] cc s rctm 9054 8722 m -39 285 r -3 146 r
0 60 r 23 120 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740
3462.126 5983.465 ] cc s rctm 7674 8158 m -85 89 r -67 90 r -48 103 r -30 100 r
-14 103 r 4 102 r 21 98 r 34 95 r 53 92 r 63 80 r 80 73 r 94 59 r 110 45 r 120
31 r 130 10 r 146 -3 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000
0.078740 3462.126 5983.465 ] cc s rctm 7949 8418 m -60 39 r -44 46 r -36 50 r
-22 54 r -8 55 r 0 58 r 11 55 r 22 53 r 65 92 r 47 35 r 48 27 r 120 23 r 62 -8
r 70 -20 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740
3462.126 5983.465 ] cc s rctm 7885 8407 m 327 231 r 0.000 0.000 0.000 1.000 k
[0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 8137 8052 m
62 635 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740
3462.126 5983.465 ] cc s rctm 8220 8922 m 37 443 r 0.000 0.000 0.000 1.000 k
[0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 6767 8625 m
58 118 r 79 105 r 91 94 r 107 71 r 110 54 r 122 31 r 115 9 r 116 -19 r 0.000
0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc
s rctm 6652 9137 m 57 63 r 61 60 r 152 92 r 173 65 r 189 40 r 185 13 r 181 -15
r 164 -38 r 130 -64 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000
0.078740 3462.126 5983.465 ] cc s rctm 7912 9348 m 0 70 r 15 62 r 52 123 r 81
109 r 90 98 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740
3462.126 5983.465 ] cc s rctm 6622 9147 m -33 120 r -19 135 r 2 150 r 28 156 r
0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126
5983.465 ] cc s rctm 6269 9065 m 410 110 r 0.000 0.000 0.000 1.000 k [0.078740
0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 6079 9305 m 111 147 r
64 65 r 66 55 r 147 85 r 162 60 r 171 41 r 184 25 r 403 10 r 430 -6 r 453 5 r
230 16 r 232 34 r 230 48 r 230 77 r 0.000 0.000 0.000 1.000 k [0.078740
0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 5992 9425 m 42 293 r
66 252 r 44 117 r 48 103 r 118 182 r 135 150 r 150 128 r 160 105 r 175 88 r 360
135 r 362 112 r 333 118 r 155 74 r 54 28 r 81 58 r 0.000 0.000 0.000 1.000 k
[0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 6077 9778 m
122 115 r 138 94 r 152 66 r 168 50 r 175 32 r 183 17 r 390 8 r 405 -10 r 404 0
r 198 10 r 190 25 r 183 35 r 174 53 r 0.000 0.000 0.000 1.000 k [0.078740
0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 6302 10365 m 137 47 r
143 38 r 315 40 r 328 15 r 339 -5 r 335 -5 r 323 5 r 303 25 r 139 27 r 131 35 r
0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126
5983.465 ] cc s rctm 6937 10865 m 170 22 r 202 8 r 440 -12 r 441 -3 r 202 12 r
72 6 r 101 27 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740
3462.126 5983.465 ] cc s rctm 8182 11312 m -173 10 r -225 36 r -259 64 r -253
98 r -122 63 r -111 69 r -94 81 r -81 92 r -60 98 r -34 114 r -8 121 r 22 137 r
73 63 r 77 39 r 73 15 r 70 -5 r 62 -19 r 53 -38 r 45 -47 r 32 -56 r 20 -65 r 3
-67 r -15 -67 r -32 -61 r -55 -60 r -78 -47 r -105 -40 r -128 -23 r 0.000 0.000
0.000 1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s
rctm 7665 11390 m 110 107 r 94 121 r 68 122 r 32 97 r 0.000 0.000 0.000 1.000 k
[0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 7200 11528 m
115 52 r 99 67 r 83 73 r 63 87 r 47 100 r 32 115 r 11 133 r -6 152 r 0.000
0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc
s rctm 7447 11032 m -162 58 r -150 63 r -135 69 r -116 70 r -104 76 r -93 77 r
-78 80 r -65 83 r -97 169 r -53 163 r -14 163 r 22 157 r 55 138 r 83 125 r 105
100 r 127 75 r 142 44 r 153 10 r 83 -10 r 80 -20 r 165 -77 r 0.000 0.000 0.000
1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 6929
10860 m -195 110 r -175 128 r -152 152 r -125 162 r -103 175 r -75 178 r -49
185 r -20 182 r 7 176 r 37 169 r 70 156 r 103 140 r 133 122 r 167 93 r 98 42 r
105 28 r 115 20 r 124 17 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000
0.000000 0.078740 3462.126 5983.465 ] cc s rctm 6667 10655 m -245 183 r -200
175 r -158 170 r -124 162 r -93 152 r -67 146 r -46 137 r -29 132 r -28 238 r
-22 210 r -18 92 r -32 85 r -48 81 r -72 72 r 0.000 0.000 0.000 1.000 k
[0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 5500 12673 m
212 -8 r 75 -13 r 58 -24 r 94 -68 r 90 -92 r 0.000 0.000 0.000 1.000 k
[0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 5250 12192 m
185 -122 r 130 -83 r 49 -25 r 81 -34 r 0.000 0.000 0.000 1.000 k [0.078740
0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 5235 11957 m 139 -202
r 158 -165 r 167 -120 r 88 -48 r 80 -34 r 0.000 0.000 0.000 1.000 k [0.078740
0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 5824 10150 m -37 192 r
-17 95 r -1 110 r 15 125 r 48 151 r 78 177 r 129 213 r 0.000 0.000 0.000 1.000
k [0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 5815 10150
m 35 123 r 50 112 r 70 98 r 80 95 r 167 150 r 162 125 r 0.000 0.000 0.000 1.000
k [0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 5787 10708
m -168 45 r -115 55 r -90 57 r -107 62 r 0.000 0.000 0.000 1.000 k [0.078740
0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 5254 11112 m 285 455 r
0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126
5983.465 ] cc s rctm 5865 10972 m -268 141 r 145 307 r 0.000 0.000 0.000 1.000
k [0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 5779 12648
m 126 205 r 49 85 r 18 25 r 42 50 r 85 82 r 123 95 r 183 107 r 259 138 r 0.000
0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc
s rctm 5860 12825 m 15 315 r 4 168 r -4 69 r -20 131 r 0.000 0.000 0.000 1.000
k [0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 5822 13478
m 195 -16 r 125 -39 r 87 -61 r 93 -95 r 0.000 0.000 0.000 1.000 k [0.078740
0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 6497 14715 m -193 130
r -154 135 r -126 138 r -90 144 r -57 145 r -30 148 r -5 147 r 30 148 r 52 147
r 81 145 r 102 138 r 127 140 r 150 132 r 170 123 r 188 120 r 208 108 r 227 100
r 243 92 r 257 83 r 268 67 r 287 57 r 295 41 r 305 25 r 317 12 r 326 -8 r 330
-19 r 340 -45 r 340 -65 r 350 -80 r 349 -105 r 351 -128 r 350 -150 r 0.000
0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc
s rctm 6642 15038 m -125 70 r -105 80 r -90 94 r -65 95 r -45 96 r -25 109 r -5
103 r 15 110 r 33 108 r 59 107 r 71 102 r 99 100 r 120 91 r 135 90 r 158 77 r
178 73 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740
3462.126 5983.465 ] cc s rctm 6912 14667 m 48 86 r 35 132 r 10 153 r -10 80 r
-23 84 r -33 81 r -47 79 r -67 70 r -83 68 r -103 58 r -119 47 r -145 37 r -168
21 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126
5983.465 ] cc s rctm 7229 14442 m 63 115 r 43 118 r 24 128 r 10 127 r -7 127 r
-28 128 r -39 123 r -56 119 r -72 111 r -85 104 r -102 88 r -116 82 r -127 63 r
-143 48 r -155 30 r -165 10 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000
0.000000 0.078740 3462.126 5983.465 ] cc s rctm 7482 14177 m 62 106 r 50 124 r
48 135 r 32 140 r 21 150 r 9 156 r -10 154 r -25 160 r -45 153 r -67 150 r -90
143 r -113 137 r -142 123 r -168 109 r -199 95 r -228 76 r 0.000 0.000 0.000
1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 7742
13920 m 83 145 r 77 158 r 62 165 r 46 175 r 30 177 r 12 183 r 0 185 r -25 184 r
-43 185 r -69 176 r -88 175 r -112 165 r -141 159 r -164 148 r -193 133 r -222
120 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126
5983.465 ] cc s rctm 8415 14952 m -3 195 r -20 190 r -40 176 r -53 172 r -64
158 r -81 155 r -92 140 r -98 132 r -107 122 r -115 108 r -237 188 r -248 142 r
-118 50 r -120 40 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000
0.078740 3462.126 5983.465 ] cc s rctm 7534 16718 m 396 90 r 550 160 r 572 190
r 473 184 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740
3462.126 5983.465 ] cc s rctm 7832 16440 m 713 188 r 605 185 r 1275 419 r 0.000
0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc
s rctm 10850 15550 m -68 303 r -27 179 r -28 200 r -15 213 r -2 228 r 15 240 r
35 250 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740
3462.126 5983.465 ] cc s rctm 10709 16708 m 370 109 r 180 56 r 70 24 r 108 51 r
0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126
5983.465 ] cc s rctm 10735 16365 m 599 180 r 298 108 r 110 40 r 28 14 r 115 73
r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126
5983.465 ] cc s rctm 8614 14948 m -124 -5 r -115 -26 r -106 -50 r -90 -65 r -80
-85 r -62 -100 r -42 -109 r -20 -120 r 5 -123 r 29 -117 r 48 -108 r 68 -92 r 84
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102 r 42 111 r 20 122 r -5 122 r -28 113 r -49 105 r -66 95 r -85 77 r -95 60 r
-109 41 r -116 20 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000
0.078740 3462.126 5983.465 ] cc s rctm 8590 14587 m -95 -14 r -43 -16 r -38 -27
r -30 -33 r -22 -39 r -25 -91 r 13 -97 r 19 -45 r 25 -37 r 33 -28 r 42 -23 r 93
-25 r 95 13 r 40 17 r 38 26 r 30 34 r 22 41 r 25 94 r -13 95 r -17 41 r -27 37
r -33 28 r -37 24 r -95 25 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000
0.000000 0.078740 3462.126 5983.465 ] cc s rctm 8067 13545 m 38 197 r 24 65 r
36 73 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126
5983.465 ] cc s rctm 8355 13182 m 17 203 r 25 142 r 82 231 r 0.000 0.000 0.000
1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 8417
13548 m 45 -45 r 45 -35 r 50 -23 r 50 -18 r 98 -4 r 92 27 r 75 55 r 48 80 r 10
105 r -31 120 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740
3462.126 5983.465 ] cc s rctm 8704 13080 m 103 12 r 88 25 r 85 38 r 72 47 r 62
60 r 51 66 r 72 154 r 33 160 r -1 161 r -47 142 r -35 62 r -43 53 r 0.000 0.000
0.000 1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s
rctm 8810 12882 m -26 210 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000
0.000000 0.078740 3462.126 5983.465 ] cc s rctm 8969 14832 m 158 128 r 165 117
r 168 103 r 170 85 r 179 75 r 178 65 r 175 48 r 180 37 r 358 45 r 344 7 r 165
-15 r 160 -19 r 301 -61 r 267 -90 r 222 -112 r 171 -130 r 67 -75 r 52 -73 r 31
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0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126
5983.465 ] cc s rctm 12175 14383 m 77 -106 r 83 -77 r 84 -45 r 78 -3 r 17 -102
r -20 -85 r -49 -72 r -63 -70 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000
0.000000 0.078740 3462.126 5983.465 ] cc s rctm 11472 14028 m 232 -81 r 240 -74
r 125 -23 r 121 -20 r 122 2 r 118 11 r 0.000 0.000 0.000 1.000 k [0.078740
0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 12620 13070 m -75 72 r
-46 63 r -29 52 r -10 48 r 4 43 r 15 39 r 43 68 r 43 68 r 10 35 r 4 39 r -14 45
r -33 45 r -135 118 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000
0.078740 3462.126 5983.465 ] cc s rctm 8914 12497 m 211 -69 r 190 -110 r 89 -68
r 81 -77 r 139 -173 r 58 -95 r 45 -97 r 67 -201 r 23 -204 r -7 -101 r -21 -97 r
0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126
5983.465 ] cc s rctm 8612 12878 m 205 -5 r 197 -31 r 185 -55 r 168 -77 r 157
-100 r 141 -120 r 124 -133 r 108 -150 r 88 -159 r 74 -170 r 51 -173 r 34 -178 r
8 -187 r -12 -180 r -33 -180 r -60 -178 r 0.000 0.000 0.000 1.000 k [0.078740
0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 10545 10168 m 0 252 r
-16 128 r -32 129 r -52 125 r -73 130 r -103 125 r -134 126 r 0.000 0.000 0.000
1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm
10830 9843 m 47 250 r 18 244 r -13 233 r -50 223 r -45 105 r -52 110 r -68 104
r -77 106 r -93 100 r -108 102 r -260 198 r 0.000 0.000 0.000 1.000 k [0.078740
0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 11342 9330 m 5 138 r
-12 127 r -25 117 r -40 108 r -61 103 r -80 99 r -100 101 r -129 102 r 0.000
0.000 0.000 1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc
s rctm 11690 9037 m 15 240 r -8 238 r -42 230 r -73 218 r -113 204 r -155 186 r
-94 85 r -105 79 r -118 70 r -133 63 r 0.000 0.000 0.000 1.000 k [0.078740
0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s rctm 10695 11070 m 165 -70
r 194 -62 r 210 -53 r 223 -37 r 232 -21 r 223 5 r 217 30 r 195 60 r 0.000 0.000
0.000 1.000 k [0.078740 0.000000 0.000000 0.078740 3462.126 5983.465 ] cc s
rctm 10464 11802 m 116 -99 r 125 -81 r 134 -75 r 146 -57 r 152 -45 r 160 -30 r
163 -10 r 167 10 r 0.000 0.000 0.000 1.000 k [0.078740 0.000000 0.000000
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@endspecial 1668 3378 a Fx(Diplomarb)s(eit)1424 3580
y(F)-8 b(ac)m(h)m(b)s(ereic)m(h)33 b(Mathematik)1221
3782 y(T)-8 b(ec)m(hnisc)m(he)35 b(Univ)m(ersit)2100
3786 y(\177)2100 3782 y(at)d(Darmstadt)531 5365 y Fw(Betreuer:)1697
b(Prof.)31 b(Dr.)f(J.)g(Buc)m(hmann)531 5591 y(Abgab)s(edatum:)1706
b(29.)31 b(Oktob)s(er)f(1999)p eop
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1 1 bop 270 1082 a Fv(V)-19 b(orw)-6 b(ort)270 1530 y
Fw(Elliptisc)m(he)19 b(Kurv)m(en)i(erfreuen)g(sic)m(h)g(in)f(den)h
(letzten)i(Jahren)e(so)m(w)m(ohl)h(in)e(der)h(Mathematik)270
1643 y(als)41 b(auc)m(h)h(in)f(der)g(Informatik)g(immer)f(gr)1769
1647 y(\177)1769 1643 y(o\031erer)j(Beliebtheit.)e(Dies)g(mag)i(zum)e
(einen)270 1756 y(daran)27 b(liegen,)g(da\031)h(die)e(Theorie)h
(inzwisc)m(hen)e(sehr)i(umfangreic)m(h)g(ist.)g(Auf)g(der)g(anderen)270
1869 y(Seite)42 b(bieten)g(sie)g(auc)m(h)g(direkte)g(An)m(w)m(endungsm)
2057 1873 y(\177)2057 1869 y(oglic)m(hk)m(eiten)f(in)g(der)h(Zahlen)m
(theorie)270 1982 y(und)29 b(der)h(Kryptographie.)411
2097 y(Es)43 b(gibt)g(mittlerw)m(eile)f(nic)m(h)m(t)h(n)m(ur)f(F)-8
b(aktorisierungsalgorithmen)42 b(und)g(Primzahl-)270
2210 y(b)s(ew)m(eise,)35 b(die)f(auf)h(elliptisc)m(hen)e(Kurv)m(en)h(b)
s(eruhen.)g(Auc)m(h)h(b)s(ei)f(der)h(Konstruktion)f(v)m(on)270
2323 y(Kryptosystemen)g(spielen)f(sie)h(in)f(der)h(letzten)i(Zeit)e
(eine)g(b)s(eac)m(h)m(tlic)m(he)h(Rolle.)f(Bei)h(den)270
2436 y(Public)41 b(Key)h(Kryptosystemen)h(m)m(u\031)g(man)f(sic)m(h)h
(im)e(Gegensatz)k(zu)e(symmetrisc)m(hen)270 2549 y(Kryptosystemen)36
b(oft)g(die)g(Praktik)-5 b(abilit)1755 2553 y(\177)1755
2549 y(at)34 b(mit)i(einer)f(w)m(esen)m(tlic)m(h)h(sc)m(hlec)m(h)m
(teren)h(E\016-)270 2662 y(zienz)e(erk)-5 b(aufen.)35
b(Elliptisc)m(he)e(Kurv)m(en)h(heb)s(en)g(diesen)g(Nac)m(h)m(teil)i(zw)
m(ar)g(nic)m(h)m(t)f(auf,)g(man)270 2775 y(k)-5 b(ann)37
b(jedo)s(c)m(h)h(im)e(V)-8 b(ergleic)m(h)38 b(zu)g(b)s(eispielsw)m
(eise)c Fu(RSA)k Fw(b)s(ei)e(v)m(ergleic)m(h)m(barer)i(Sic)m(herheit)
270 2888 y(k)321 2892 y(\177)318 2888 y(urzere)28 b(Sc)m(hl)757
2892 y(\177)754 2888 y(ussel)f(v)m(erw)m(enden,)h(w)m(as)h(sic)m(h)f
(dann)f(b)s(ei)g(der)h(Laufzeit)h(p)s(ositiv)d(b)s(emerkbar)270
3001 y(mac)m(h)m(t.)411 3116 y(In)c(einem)g(Pro)5 b(jekt)23
b(am)g(Institut)e(f)1592 3120 y(\177)1589 3116 y(ur)h(Theoretisc)m(he)h
(Informatik)e(der)i Ft(Technischen)270 3229 y(Universit)715
3225 y(\177)712 3229 y(at)29 b(D)m(armst)-6 b(adt)25
b Fw(geh)m(t)i(es)f(darum,)f(einen)h(Pro)m(vider)f(f)2621
3233 y(\177)2618 3229 y(ur)g(An)m(w)m(endungen)g(in)270
3342 y(der)32 b(Public)e(Key)i(Kryptographie)f(zu)h(en)m(t)m(wic)m(k)m
(eln.)h(Dazu)g(geh)2483 3346 y(\177)2483 3342 y(oren)f(insb)s(esondere)
e(V)-8 b(er-)270 3455 y(sc)m(hl)423 3459 y(\177)420 3455
y(usselungsalgorithmen)23 b(und)h(digitale)h(Signaturen.)g(Im)g(V)-8
b(ordergrund)25 b(steh)m(t)i(dab)s(ei,)270 3568 y(da\031)33
b(man)f(jederzeit)g(zwisc)m(hen)g(den)f(v)m(ersc)m(hiedenen)h(V)-8
b(erfahren)33 b(ausw)2762 3572 y(\177)2762 3568 y(ahlen)d(k)-5
b(ann.)32 b(Ein)270 3681 y(w)m(eiterer)38 b(Gesic)m(h)m(tspunkt)f(ist)g
(die)g(Aktualit)1848 3685 y(\177)1848 3681 y(at,)h(so)m(wie)f(die)g
(Standardisierung)d(der)j(Al-)270 3794 y(gorithmen.)d(Gerade)h(im)f
(Bereic)m(h)h(der)f(elliptisc)m(hen)e(Kurv)m(en)h(sollen)g(hier)h(F)-8
b(ortsc)m(hritte)270 3907 y(erzielt)30 b(w)m(erden.)411
4023 y(Diese)46 b(Diplomarb)s(eit)d(b)s(esc)m(h)1458
4027 y(\177)1458 4023 y(aftigt)j(sic)m(h)f(mit)g(der)h(Implemen)m
(tierung)d(eines)i(V)-8 b(er-)270 4135 y(sc)m(hl)423
4139 y(\177)420 4135 y(usselungsalgorithm)m(us)29 b(un)m(ter)k(V)-8
b(erw)m(endung)33 b(v)m(on)g(elliptisc)m(hen)d(Kurv)m(en.)j(Als)f(Pro-)
270 4248 y(grammiersprac)m(he)i(wurde)g(JA)-10 b(V)g(A)34
b(ausgew)1806 4252 y(\177)1806 4248 y(ahlt,)h(da)g(sie)f(ein)g
(Strukturk)m(onzept)g(f)3156 4252 y(\177)3153 4248 y(ur)g(die)270
4361 y(Realisierung)19 b(kryptographisc)m(her)i(V)-8
b(erfahren)21 b(in)f(Soft)m(w)m(are)j(b)s(ereitstellt.)d(Der)i(En)m
(tsc)m(hei-)270 4474 y(dung)30 b(zugunsten)h(der)g(Arc)m(hitektur)g
(soll)f(dab)s(ei)g(die)h(E\016zienz)g(gegen)2728 4478
y(\177)2725 4474 y(ub)s(ergestellt)h(w)m(er-)270 4587
y(den.)e(So)m(w)m(eit)h(es)f(m)937 4591 y(\177)937 4587
y(oglic)m(h)g(w)m(ar,)h(wurden)e(Standards)g(mitein)m(b)s(ezogen.)411
4703 y(Die)c(Kom)m(bination)e(v)m(on)i(Theorie)e(und)g(An)m(w)m(endung)
h(w)m(ar)h(hierb)s(ei)c(aussc)m(hlaggeb)s(end)270 4816
y(f)301 4820 y(\177)298 4816 y(ur)27 b(meine)h(En)m(tsc)m(heidung,)g
(die)f(Diplomarb)s(eit)f(im)i(Bereic)m(h)h(der)f(Kryptographie)f(anzu-)
270 4929 y(fertigen.)35 b(Die)h(elliptisc)m(hen)d(Kurv)m(en)h(bilden)f
(dab)s(ei)h(einen)g(mathematisc)m(h)i(anspruc)m(hs-)270
5042 y(v)m(ollen)29 b(Bereic)m(h)h(der)g(Zahlen)m(theorie.)f(Dar)1759
5046 y(\177)1756 5042 y(ub)s(erhinaus)e(sind)h(sie)h(ab)s(er)g(auc)m(h)
i(ein)e(Gegen-)270 5154 y(stand,)e(der)f(in)g(der)g(mo)s(dernen)g
(Kryptographie)g(eingesetzt)i(w)m(erden)f(k)-5 b(ann.)26
b(Diese)h(spielt)270 5267 y(in)32 b(der)h(heutigen)g(Zeit)g(eine)g
(immer)g(wic)m(h)m(tigere)g(Rolle,)h(denn)e(gerade)i(f)2820
5271 y(\177)2817 5267 y(ur)f(das)g(inno)m(v)-5 b(a-)270
5380 y(tiv)m(e)29 b(F)-8 b(eld)28 b(des)g(E-Commerce)g(ist)g(die)f
(Kryptographie)g(sc)m(hlie\031lic)m(h)g(v)m(on)h(en)m(tsc)m(heidender)
270 5493 y(Bedeutung.)411 5609 y(An)46 b(dieser)g(Stelle)g(m)1191
5613 y(\177)1191 5609 y(oc)m(h)m(te)j(ic)m(h)d(all)f(denen)h(dank)m
(en,)h(deren)f(Un)m(terst)2946 5613 y(\177)2943 5609
y(utzung)h(zur)1807 5858 y(1)p eop
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2 2 bop 390 82 a Fw(2)390 415 y(Qualit)644 419 y(\177)644
415 y(at)30 b(dieser)f(Diplomarb)s(eit)f(b)s(eigetragen)j(hat.)g
(Zuerst)e(sei)h(mein)f(Betreuer)i Ft(Pr)n(of.)390 528
y(Buchmann)g Fw(genann)m(t,)i(der)f(auc)m(h)h(das)f(in)m(teressan)m(te)
i(Thema)e(v)m(orgesc)m(hlagen)i(hat.)f(Bei)390 641 y(Problemen)f(und)f
(F)-8 b(ragen)34 b(w)m(aren)f Ft(Markus)j(Ma)n(urer)31
b Fw(und)h Ft(Markus)j(R)m(upper)-6 b(t)32 b Fw(im-)390
754 y(mer)e(hilfsb)s(ereit)d(und)h(k)m(omp)s(eten)m(t.)k(Auc)m(h)e
Ft(Tobias)k(Hahn)c Fw(half)f(mir)f(einige)i(Male)3333
758 y(\177)3330 754 y(ub)s(er)390 867 y(tec)m(hnisc)m(he)i(Probleme)f
(hin)m(w)m(eg.)h(W)-8 b(eiterhin)30 b(dank)m(e)i(ic)m(h)g
Ft(Safua)-6 b(t)33 b(Hamd)n(y)g Fw(f)3120 871 y(\177)3117
867 y(ur)d(die)h(Er-)390 980 y(laubnis,)37 b(die)h(Abbildungen)d
(elliptisc)m(her)i(Kurv)m(en)h(aus)g(seiner)g(Diplomarb)s(eit)f(f)3272
984 y(\177)3269 980 y(ur)h(die)390 1093 y(meinige)26
b(zu)g(mo)s(di\014zieren.)f(W)-8 b(as)28 b(die)e(Arb)s(eit)f(mit)h(T)
2207 1112 y(E)2257 1093 y(X)h(b)s(etri\013t,)f(so)h(stand)g(mir)e
Ft(Stef)-8 b(an)390 1205 y(Br)519 1201 y(\177)515 1205
y(otz)29 b Fw(zur)h(Seite.)531 1318 y(Besonders)35 b(b)s(edank)m(e)f
(ic)m(h)h(mic)m(h)f(b)s(ei)g(den)g(Korrekturlesern)f
Ft(Manuel)k(Breuning)390 1431 y Fw(und)f Ft(Rainer)k(M)1030
1427 y(\177)1027 1431 y(uller)p Fw(,)c(so)m(wie)h(b)s(ei)f(meiner)g(F)
-8 b(reundin)35 b Ft(Anja)h Fw(f)2736 1435 y(\177)2733
1431 y(ur)g(ihre)g(p)s(ermanen)m(te)390 1544 y(Un)m(terst)691
1548 y(\177)688 1544 y(utzung.)2786 1757 y Ft(Markus)d(Crnjak)2485
1870 y Fw(Darmstadt,)e(Oktob)s(er)f(1999)p eop
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3 3 bop 270 1082 a Fv(Inhaltsv)-6 b(erzeic)g(hnis)270
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b(5)270 2040 y(2)84 b(Elliptisc)m(he)36 b(Kurv)m(en)2081
b(7)406 2156 y Fw(2.1)94 b(Grundlagen)59 b(.)46 b(.)g(.)f(.)h(.)g(.)f
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b(9)406 2387 y(2.3)94 b(Kryptographisc)m(h)29 b(geeignete)j(Kurv)m(en)
64 b(.)46 b(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g
(.)82 b(12)270 2596 y Fs(3)i(Der)35 b(Algorithm)m(us)2106
b(15)406 2712 y Fw(3.1)94 b(Das)32 b(ElGamal-V)-8 b(erfahren)71
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h(.)g(.)g(.)f(.)h(.)g(.)82 b(15)406 2828 y(3.2)94 b(V)-8
b(ersc)m(hl)904 2832 y(\177)901 2828 y(usselung)29 b(mit)h(elliptisc)m
(hen)e(Kurv)m(en)72 b(.)46 b(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)
h(.)g(.)82 b(17)406 2944 y(3.3)94 b(Alternativ)m(en)28
b(.)46 b(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)f(.)
h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)82
b(18)270 3152 y Fs(4)i(Die)35 b(Implemen)m(tierung)1904
b(19)406 3268 y Fw(4.1)94 b(Die)31 b(JA)-10 b(V)g(A)30
b(Cryptograph)m(y)g(Arc)m(hitecture)81 b(.)45 b(.)h(.)g(.)g(.)f(.)h(.)g
(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)82 b(19)406 3384 y(4.2)94
b(Der)31 b(Standard)e(IEEE)h(P1363)i(/)f(D8)92 b(.)45
b(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)82
b(20)615 3500 y(4.2.1)106 b(Die)31 b(Domain)f(P)m(arameter)93
b(.)46 b(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)h(.)g(.)f(.)h(.)g(.)g(.)f(.)
h(.)g(.)82 b(21)615 3616 y(4.2.2)106 b(Das)31 b(Sc)m(hl)1255
3620 y(\177)1252 3616 y(usselpaar)89 b(.)46 b(.)f(.)h(.)g(.)f(.)h(.)g
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b(22)406 3732 y(4.3)94 b(Die)31 b(C++-Klassen)m(bibliothek)c(LiDIA)60
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h(.)g(.)82 b(25)270 4288 y Fs(5)i(E\016zienz)2496 b(29)406
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5609 y(w)m(erden.)g(Die)g(erforderlic)m(he)f(Theorie)h(der)f(elliptisc)
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y(5)p eop
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1209 y(\177)2611 1205 y(origen)h(Dokumen)m(tation)390
1318 y(aufgelistet.)531 1431 y(Wie)36 b(ob)s(en)f(sc)m(hon)h(erw)1349
1435 y(\177)1349 1431 y(ahn)m(t,)g(erheb)s(en)f(die)g(Ausf)2274
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(h)2136 1661 y(\177)2136 1657 y(aftigung)f(mit)g(dem)f(Thema)h
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1883 y(sc)m(he)h(Kurv)m(en)f(b)s(ezogen)i(sind,)d(stehen)h(in)g(dem)g
(auc)m(h)h(f)2251 1887 y(\177)2248 1883 y(ur)f(Laien)g(v)m(erst)2792
1887 y(\177)2792 1883 y(andlic)m(hen)g([Bur94].)390 1996
y(Die)k(Theorie)f(elliptisc)m(her)f(Kurv)m(en)h(wird)f(sehr)h(ausf)2213
2000 y(\177)2210 1996 y(uhrlic)m(h)f(in)g([Sil86])i(und)e([Hus87)q(])i
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n 1488 2623 m 1482 2632 l gs col0 s gr
% Polyline
n 1488 2623 m 1494 2601 l gs col0 s gr
% Polyline
n 1507 2576 m 1494 2601 l gs col0 s gr
% Polyline
n 1507 2576 m 1510 2572 l gs col0 s gr
% Polyline
n 1512 2563 m 1510 2572 l gs col0 s gr
% Polyline
n 1512 2090 m 1497 2071 l 1474 2046 l gs col0 s gr
% Polyline
n 1512 2563 m 1522 2540 l gs col0 s gr
% Polyline
n 1525 2529 m 1522 2540 l gs col0 s gr
% Polyline
n 1525 2529 m 1533 2509 l gs col0 s gr
% Polyline
n 1542 2484 m 1533 2509 l gs col0 s gr
% Polyline
n 1542 2484 m 1543 2476 l gs col0 s gr
% Polyline
n 1551 2456 m 1543 2476 l gs col0 s gr
% Polyline
n 1551 2154 m 1530 2117 l 1512 2090 l gs col0 s gr
% Polyline
n 1551 2456 m 1555 2443 l gs col0 s gr
% Polyline
n 1557 2438 m 1555 2443 l gs col0 s gr
% Polyline
n 1557 2438 m 1564 2408 l gs col0 s gr
% Polyline
n 1569 2391 m 1564 2408 l gs col0 s gr
% Polyline
n 1569 2391 m 1571 2371 l gs col0 s gr
% Polyline
n 1576 2346 m 1571 2371 l gs col0 s gr
% Polyline
n 1576 2346 m 1577 2331 l gs col0 s gr
% Polyline
n 1580 2300 m 1577 2331 l gs col0 s gr
% Polyline
n 1580 2300 m 1580 2290 l gs col0 s gr
% Polyline
n 1578 2254 m 1580 2290 l gs col0 s gr
% Polyline
n 1578 2254 m 1575 2237 l gs col0 s gr
% Polyline
n 1571 2208 m 1575 2237 l gs col0 s gr
% Polyline
n 1571 2208 m 1558 2170 l gs col0 s gr
% Polyline
n 1555 2162 m 1558 2170 l gs col0 s gr
% Polyline
n 1555 2162 m 1551 2154 l gs col0 s gr
% Polyline
n 2010 2137 m 2004 2117 l 2004 2110 l gs col0 s gr
% Polyline
n 2000 2071 m 2004 2110 l gs col0 s gr
% Polyline
n 2000 2071 m 2000 2060 l gs col0 s gr
% Polyline
n 2000 2024 m 2000 2060 l gs col0 s gr
% Polyline
n 2000 2024 m 2003 2015 l gs col0 s gr
% Polyline
n 2006 1977 m 2003 2015 l gs col0 s gr
% Polyline
n 2006 1977 m 2006 1975 l gs col0 s gr
% Polyline
n 2010 1957 m 2006 1975 l gs col0 s gr
% Polyline
n 2048 2252 m 2028 2208 l 2016 2171 l gs col0 s gr
% Polyline
n 2014 2162 m 2016 2171 l gs col0 s gr
% Polyline
n 2014 2162 m 2010 2137 l gs col0 s gr
% Polyline
n 2010 1957 m 2013 1937 l gs col0 s gr
% Polyline
n 2013 1931 m 2013 1937 l gs col0 s gr
% Polyline
n 2013 1931 m 2021 1900 l gs col0 s gr
% Polyline
n 2024 1887 m 2021 1900 l gs col0 s gr
% Polyline
n 2024 1887 m 2029 1865 l gs col0 s gr
% Polyline
n 2036 1840 m 2029 1865 l gs col0 s gr
% Polyline
n 2036 1840 m 2039 1829 l gs col0 s gr
% Polyline
n 2048 1801 m 2039 1829 l gs col0 s gr
% Polyline
n 2087 2323 m 2071 2300 l 2049 2256 l gs col0 s gr
% Polyline
n 2049 2254 m 2049 2256 l gs col0 s gr
% Polyline
n 2049 2254 m 2048 2252 l gs col0 s gr
% Polyline
n 2048 1801 m 2049 1797 l gs col0 s gr
% Polyline
n 2049 1794 m 2049 1797 l gs col0 s gr
% Polyline
n 2049 1794 m 2060 1762 l gs col0 s gr
% Polyline
n 2064 1747 m 2060 1762 l gs col0 s gr
% Polyline
n 2064 1747 m 2071 1729 l gs col0 s gr
% Polyline
n 2080 1702 m 2071 1729 l gs col0 s gr
% Polyline
n 2080 1702 m 2082 1697 l gs col0 s gr
% Polyline
n 2087 1686 m 2082 1697 l gs col0 s gr
% Polyline
n 2124 2386 m 2099 2346 l 2087 2323 l gs col0 s gr
% Polyline
n 2087 1686 m 2094 1666 l gs col0 s gr
% Polyline
n 2095 1655 m 2094 1666 l gs col0 s gr
% Polyline
n 2095 1655 m 2105 1633 l gs col0 s gr
% Polyline
n 2113 1611 m 2105 1633 l gs col0 s gr
% Polyline
n 2113 1611 m 2117 1601 l gs col0 s gr
% Polyline
n 2124 1582 m 2117 1601 l gs col0 s gr
% Polyline
n 2163 2443 m 2160 2438 l 2144 2414 l gs col0 s gr
% Polyline
n 2128 2391 m 2144 2414 l gs col0 s gr
% Polyline
n 2128 2391 m 2124 2386 l gs col0 s gr
% Polyline
n 2124 1582 m 2128 1570 l gs col0 s gr
% Polyline
n 2131 1564 m 2128 1570 l gs col0 s gr
% Polyline
n 2131 1564 m 2140 1538 l gs col0 s gr
% Polyline
n 2147 1518 m 2140 1538 l gs col0 s gr
% Polyline
n 2147 1518 m 2152 1507 l gs col0 s gr
% Polyline
n 2163 1482 m 2152 1507 l gs col0 s gr
% Polyline
n 2201 2496 m 2191 2484 l 2163 2443 l gs col0 s gr
% Polyline
n 2163 1482 m 2165 1475 l gs col0 s gr
% Polyline
n 2165 1473 m 2165 1475 l gs col0 s gr
% Polyline
n 2165 1473 m 2176 1444 l gs col0 s gr
% Polyline
n 2184 1426 m 2176 1444 l gs col0 s gr
% Polyline
n 2184 1426 m 2190 1413 l gs col0 s gr
% Polyline
n 2201 1384 m 2190 1413 l gs col0 s gr
% Polyline
n 2240 2550 m 2224 2529 l 2201 2496 l gs col0 s gr
% Polyline
n 2201 1384 m 2202 1381 l gs col0 s gr
% Polyline
n 2202 1380 m 2202 1381 l gs col0 s gr
% Polyline
n 2202 1380 m 2214 1351 l gs col0 s gr
% Polyline
n 2219 1334 m 2214 1351 l gs col0 s gr
% Polyline
n 2219 1334 m 2226 1319 l gs col0 s gr
% Polyline
n 2238 1289 m 2226 1319 l gs col0 s gr
% Polyline
n 2237 1289 m 2239 1289 l gs col0 s gr
% Polyline
n 2240 1288 m 2238 1289 l gs col0 s gr
% Polyline
n 2278 2603 m 2257 2576 l 2240 2550 l gs col0 s gr
% Polyline
n 2240 1288 m 2251 1256 l gs col0 s gr
% Polyline
n 2257 1242 m 2251 1256 l gs col0 s gr
% Polyline
n 2257 1242 m 2264 1225 l gs col0 s gr
% Polyline
n 2275 1197 m 2264 1225 l gs col0 s gr
% Polyline
n 2275 1197 m 2277 1196 l gs col0 s gr
% Polyline
n 2278 1191 m 2277 1196 l gs col0 s gr
% Polyline
n 2315 2656 m 2290 2623 l 2278 2603 l gs col0 s gr
% Polyline
n 2278 1191 m 2289 1163 l gs col0 s gr
% Polyline
n 2293 1151 m 2289 1163 l gs col0 s gr
% Polyline
n 2293 1151 m 2301 1133 l gs col0 s gr
% Polyline
n 2313 1106 m 2301 1133 l gs col0 s gr
% Polyline
n 2313 1106 m 2313 1101 l gs col0 s gr
% Polyline
n 2315 1095 m 2313 1101 l gs col0 s gr
% Polyline
n 2354 2709 m 2323 2667 l 2315 2656 l gs col0 s gr
% Polyline
n 2315 1095 m 2325 1072 l gs col0 s gr
% Polyline
n 2330 1059 m 2325 1072 l gs col0 s gr
% Polyline
n 2330 1059 m 2339 1039 l gs col0 s gr
% Polyline
n 2348 1013 m 2339 1039 l gs col0 s gr
% Polyline
n 2348 1013 m 2351 1008 l gs col0 s gr
% Polyline
n 2354 1000 m 2351 1008 l gs col0 s gr
% Polyline
n 2392 2763 m 2389 2759 l 2375 2739 l gs col0 s gr
% Polyline
n 2356 2712 m 2375 2739 l gs col0 s gr
% Polyline
n 2356 2712 m 2354 2709 l gs col0 s gr
% Polyline
n 2354 1000 m 2363 978 l gs col0 s gr
% Polyline
n 2367 967 m 2363 978 l gs col0 s gr
% Polyline
n 2367 967 m 2375 945 l gs col0 s gr
% Polyline
n 2386 922 m 2375 945 l gs col0 s gr
% Polyline
n 2386 922 m 2388 915 l gs col0 s gr
% Polyline
n 2392 904 m 2388 915 l gs col0 s gr
% Polyline
n 2431 2815 m 2422 2805 l 2392 2763 l gs col0 s gr
% Polyline
n 2392 904 m 2400 886 l gs col0 s gr
% Polyline
n 2404 875 m 2400 886 l gs col0 s gr
% Polyline
n 2404 875 m 2413 853 l gs col0 s gr
% Polyline
n 2422 830 m 2413 853 l gs col0 s gr
% Polyline
n 2422 830 m 2425 822 l gs col0 s gr
% Polyline
n 2431 807 m 2425 822 l gs col0 s gr
% Polyline
n 2468 2870 m 2455 2852 l 2431 2815 l gs col0 s gr
% Polyline
n 2431 807 m 2437 791 l gs col0 s gr
% Polyline
n 2439 782 m 2437 791 l gs col0 s gr
% Polyline
n 2439 782 m 2449 759 l gs col0 s gr
% Polyline
n 2459 736 m 2449 759 l gs col0 s gr
% Polyline
n 2459 736 m 2461 729 l gs col0 s gr
% Polyline
n 2468 711 m 2461 729 l gs col0 s gr
% Polyline
n 2507 2925 m 2487 2896 l 2468 2870 l gs col0 s gr
% Polyline
n 2468 711 m 2473 696 l gs col0 s gr
% Polyline
n 2476 690 m 2473 696 l gs col0 s gr
% Polyline
n 2476 690 m 2486 667 l gs col0 s gr
% Polyline
n 2494 646 m 2486 667 l gs col0 s gr
% Polyline
n 2494 646 m 2498 634 l gs col0 s gr
% Polyline
n 2507 612 m 2498 634 l gs col0 s gr
% Polyline
n 2545 2982 m 2518 2943 l 2507 2925 l gs col0 s gr
% Polyline
n 2507 612 m 2510 603 l gs col0 s gr
% Polyline
n 2512 599 m 2510 603 l gs col0 s gr
% Polyline
n 2512 599 m 2522 571 l gs col0 s gr
% Polyline
n 2530 553 m 2522 571 l gs col0 s gr
% Polyline
n 2530 553 m 2534 540 l gs col0 s gr
% Polyline
n 2545 514 m 2534 540 l gs col0 s gr
% Polyline
n 2583 3036 m 2583 3034 l 2576 3028 l gs col0 s gr
% Polyline
n 2551 2989 m 2576 3028 l gs col0 s gr
% Polyline
n 2551 2989 m 2545 2982 l gs col0 s gr
% Polyline
n 2545 514 m 2547 509 l gs col0 s gr
% Polyline
n 2547 506 m 2547 509 l gs col0 s gr
% Polyline
n 2547 506 m 2560 478 l gs col0 s gr
% Polyline
n 2565 461 m 2560 478 l gs col0 s gr
% Polyline
n 2565 461 m 2572 447 l gs col0 s gr
% Polyline
n 2583 416 m 2572 447 l gs col0 s gr
% Polyline
n 2582 416 m 2584 416 l gs col0 s gr
% Polyline
n 2583 415 m 2583 416 l gs col0 s gr
% Polyline
n 2622 3093 m 2613 3082 l 2583 3036 l gs col0 s gr
% Polyline
n 2583 415 m 2595 384 l gs col0 s gr
% Polyline
n 2601 369 m 2595 384 l gs col0 s gr
% Polyline
n 2601 369 m 2607 351 l gs col0 s gr
% Polyline
n 2619 322 m 2607 351 l gs col0 s gr
% Polyline
n 2619 322 m 2619 321 l gs col0 s gr
% Polyline
n 2622 315 m 2619 321 l gs col0 s gr
% Polyline
n 2659 3151 m 2644 3127 l 2622 3093 l gs col0 s gr
% Polyline
n 2622 315 m 2631 290 l gs col0 s gr
% Polyline
n 2636 277 m 2631 290 l gs col0 s gr
% Polyline
n 2636 277 m 2644 258 l gs col0 s gr
% Polyline
n 2653 232 m 2644 258 l gs col0 s gr
% Polyline
n 2653 232 m 2656 226 l gs col0 s gr
% Polyline
n 2659 214 m 2656 226 l gs col0 s gr
% Polyline
n 2698 3210 m 2674 3173 l 2659 3151 l gs col0 s gr
% Polyline
n 2659 214 m 2668 194 l gs col0 s gr
% Polyline
n 2670 186 m 2668 194 l gs col0 s gr
% Polyline
n 2670 186 m 2680 164 l gs col0 s gr
% Polyline
n 2688 140 m 2680 164 l gs col0 s gr
% Polyline
n 2688 140 m 2692 131 l gs col0 s gr
% Polyline
n 2698 114 m 2692 131 l gs col0 s gr
% Polyline
n 2736 3269 m 2733 3265 l 2725 3252 l gs col0 s gr
% Polyline
n 2704 3218 m 2725 3252 l gs col0 s gr
% Polyline
n 2704 3218 m 2698 3210 l gs col0 s gr
% Polyline
n 2698 114 m 2703 100 l gs col0 s gr
% Polyline
n 2706 94 m 2703 100 l gs col0 s gr
% Polyline
n 2774 3329 m 2764 3311 l 2736 3269 l gs col0 s gr
% Polyline
n 2813 3389 m 2793 3356 l 2774 3329 l gs col0 s gr
% Polyline
n 2851 3451 m 2850 3449 l 2847 3443 l gs col0 s gr
% Polyline
n 2822 3403 m 2847 3443 l gs col0 s gr
% Polyline
n 2822 3403 m 2813 3389 l gs col0 s gr
% Polyline
n 2889 3512 m 2878 3496 l 2851 3451 l gs col0 s gr
% Polyline
n 2928 3575 m 2907 3541 l 2889 3512 l gs col0 s gr
% Polyline
n 2965 3638 m 2962 3632 l 2952 3616 l gs col0 s gr
% Polyline
n 2934 3587 m 2952 3616 l gs col0 s gr
% Polyline
n 2934 3587 m 2928 3575 l gs col0 s gr
% Polyline
n 3004 3704 m 2990 3679 l 2965 3638 l gs col0 s gr
% Polyline
n 3042 3769 m 3016 3724 l 3004 3704 l gs col0 s gr
% Polyline
n 3081 3834 m 3070 3817 l 3046 3776 l gs col0 s gr
% Polyline
n 3043 3770 m 3046 3776 l gs col0 s gr
% Polyline
n 3043 3770 m 3042 3769 l gs col0 s gr
% Polyline
n 3119 3900 m 3097 3862 l 3081 3834 l gs col0 s gr
% Polyline
n 3158 3968 m 3149 3954 l 3131 3924 l gs col0 s gr
% Polyline
n 3122 3908 m 3131 3924 l gs col0 s gr
% Polyline
n 3122 3908 m 3119 3900 l gs col0 s gr
% Polyline
n 3195 4037 m 3174 4001 l 3158 3968 l gs col0 s gr
% Polyline
n 3654 3386 m 3651 3370 l 3650 3365 l gs col0 s gr
% Polyline
n 3642 3325 m 3650 3365 l gs col0 s gr
% Polyline
n 3642 3325 m 3639 3308 l gs col0 s gr
% Polyline
n 3635 3282 m 3639 3308 l gs col0 s gr
% Polyline
n 3635 3282 m 3631 3253 l gs col0 s gr
% Polyline
n 3629 3236 m 3631 3253 l gs col0 s gr
% Polyline
n 3629 3236 m 3625 3203 l gs col0 s gr
% Polyline
n 3624 3191 m 3625 3203 l gs col0 s gr
% Polyline
n 3624 3191 m 3622 3153 l gs col0 s gr
% Polyline
n 3622 3147 m 3622 3153 l gs col0 s gr
% Polyline
n 3622 3147 m 3621 3106 l gs col0 s gr
% Polyline
n 3621 3102 m 3621 3106 l gs col0 s gr
% Polyline
n 3621 3102 m 3621 3060 l gs col0 s gr
% Polyline
n 3621 3057 m 3621 3060 l gs col0 s gr
% Polyline
n 3621 3057 m 3621 3017 l gs col0 s gr
% Polyline
n 3621 3012 m 3621 3017 l gs col0 s gr
% Polyline
n 3621 3012 m 3622 2975 l gs col0 s gr
% Polyline
n 3623 2967 m 3622 2975 l gs col0 s gr
% Polyline
n 3623 2967 m 3625 2933 l gs col0 s gr
% Polyline
n 3627 2921 m 3625 2933 l gs col0 s gr
% Polyline
n 3627 2921 m 3629 2892 l gs col0 s gr
% Polyline
n 3631 2877 m 3629 2892 l gs col0 s gr
% Polyline
n 3631 2877 m 3633 2852 l gs col0 s gr
% Polyline
n 3635 2832 m 3633 2852 l gs col0 s gr
% Polyline
n 3635 2832 m 3638 2814 l gs col0 s gr
% Polyline
n 3642 2787 m 3638 2814 l gs col0 s gr
% Polyline
n 3642 2787 m 3644 2775 l gs col0 s gr
% Polyline
n 3650 2741 m 3644 2775 l gs col0 s gr
% Polyline
n 3650 2741 m 3650 2738 l gs col0 s gr
% Polyline
n 3654 2714 m 3650 2738 l gs col0 s gr
% Polyline
n 3693 3506 m 3693 3504 l gs col0 s gr
% Polyline
n 3676 3462 m 3693 3504 l gs col0 s gr
% Polyline
n 3676 3462 m 3667 3431 l gs col0 s gr
% Polyline
n 3662 3416 m 3667 3431 l gs col0 s gr
% Polyline
n 3662 3416 m 3654 3386 l gs col0 s gr
% Polyline
n 3654 2714 m 3657 2701 l gs col0 s gr
% Polyline
n 3657 2697 m 3657 2701 l gs col0 s gr
% Polyline
n 3657 2697 m 3664 2665 l gs col0 s gr
% Polyline
n 3667 2653 m 3664 2665 l gs col0 s gr
% Polyline
n 3667 2653 m 3673 2630 l gs col0 s gr
% Polyline
n 3678 2608 m 3673 2630 l gs col0 s gr
% Polyline
n 3678 2608 m 3682 2594 l gs col0 s gr
% Polyline
n 3690 2563 m 3682 2594 l gs col0 s gr
% Polyline
n 3690 2563 m 3690 2560 l gs col0 s gr
% Polyline
n 3693 2551 m 3690 2560 l gs col0 s gr
% Polyline
n 3730 3584 m 3713 3550 l 3693 3506 l gs col0 s gr
% Polyline
n 3693 2551 m 3700 2527 l gs col0 s gr
% Polyline
n 3702 2519 m 3700 2527 l gs col0 s gr
% Polyline
n 3702 2519 m 3711 2494 l gs col0 s gr
% Polyline
n 3717 2473 m 3711 2494 l gs col0 s gr
% Polyline
n 3717 2473 m 3720 2462 l gs col0 s gr
% Polyline
n 3730 2431 m 3720 2462 l gs col0 s gr
% Polyline
n 3769 3639 m 3737 3595 l 3730 3584 l gs col0 s gr
% Polyline
n 3730 2431 m 3731 2429 l gs col0 s gr
% Polyline
n 3733 2428 m 3731 2429 l gs col0 s gr
% Polyline
n 3733 2428 m 3743 2398 l gs col0 s gr
% Polyline
n 3749 2383 m 3743 2398 l gs col0 s gr
% Polyline
n 3749 2383 m 3756 2367 l gs col0 s gr
% Polyline
n 3767 2338 m 3756 2367 l gs col0 s gr
% Polyline
n 3767 2338 m 3768 2337 l gs col0 s gr
% Polyline
n 3769 2335 m 3768 2337 l gs col0 s gr
% Polyline
n 3807 3679 m 3770 3640 l 3769 3639 l gs col0 s gr
% Polyline
n 3769 2335 m 3781 2308 l gs col0 s gr
% Polyline
n 3789 2293 m 3781 2308 l gs col0 s gr
% Polyline
n 3789 2293 m 3794 2279 l gs col0 s gr
% Polyline
n 3807 2252 m 3794 2279 l gs col0 s gr
% Polyline
n 3845 3709 m 3814 3686 l 3807 3679 l gs col0 s gr
% Polyline
n 3807 2252 m 3808 2249 l gs col0 s gr
% Polyline
n 3809 2249 m 3808 2249 l gs col0 s gr
% Polyline
n 3809 2249 m 3824 2221 l gs col0 s gr
% Polyline
n 3832 2203 m 3824 2221 l gs col0 s gr
% Polyline
n 3832 2203 m 3838 2195 l gs col0 s gr
% Polyline
n 3845 2181 m 3838 2195 l gs col0 s gr
% Polyline
n 3883 3730 m 3845 3709 l gs col0 s gr
% Polyline
n 3845 2181 m 3853 2167 l gs col0 s gr
% Polyline
n 3858 2158 m 3853 2167 l gs col0 s gr
% Polyline
n 3858 2158 m 3869 2143 l gs col0 s gr
% Polyline
n 3883 2118 m 3869 2143 l gs col0 s gr
% Polyline
n 3883 3730 m 3884 3730 l gs col0 s gr
% Polyline
n 3922 3744 m 3884 3730 l gs col0 s gr
% Polyline
n 3883 2118 m 3886 2115 l gs col0 s gr
% Polyline
n 3885 2115 m 3887 2115 l gs col0 s gr
% Polyline
n 3886 2115 m 3902 2090 l gs col0 s gr
% Polyline
n 3916 2070 m 3902 2090 l gs col0 s gr
% Polyline
n 3916 2070 m 3919 2065 l gs col0 s gr
% Polyline
n 3922 2062 m 3919 2065 l gs col0 s gr
% Polyline
n 3922 3744 m 3937 3747 l gs col0 s gr
% Polyline
n 3960 3752 m 3937 3747 l gs col0 s gr
% Polyline
n 3922 2062 m 3938 2042 l gs col0 s gr
% Polyline
n 3951 2024 m 3938 2042 l gs col0 s gr
% Polyline
n 3951 2024 m 3955 2019 l gs col0 s gr
% Polyline
n 3960 2012 m 3955 2019 l gs col0 s gr
% Polyline
n 3960 3752 m 3979 3753 l gs col0 s gr
% Polyline
n 3999 3754 m 3979 3753 l gs col0 s gr
% Polyline
n 3960 2012 m 3974 1997 l gs col0 s gr
% Polyline
n 3989 1978 m 3974 1997 l gs col0 s gr
% Polyline
n 3989 1978 m 3994 1974 l gs col0 s gr
% Polyline
n 3999 1969 m 3994 1974 l gs col0 s gr
% Polyline
n 3999 3754 m 4017 3752 l gs col0 s gr
% Polyline
n 4036 3752 m 4017 3752 l gs col0 s gr
% Polyline
n 3999 1969 m 4014 1952 l gs col0 s gr
% Polyline
n 4030 1935 m 4014 1952 l gs col0 s gr
% Polyline
n 4030 1935 m 4034 1931 l gs col0 s gr
% Polyline
n 4036 1928 m 4034 1931 l gs col0 s gr
% Polyline
n 4036 3752 m 4052 3750 l gs col0 s gr
% Polyline
n 4074 3745 m 4052 3750 l gs col0 s gr
% Polyline
n 4036 1928 m 4056 1911 l gs col0 s gr
% Polyline
n 4074 1893 m 4056 1911 l gs col0 s gr
% Polyline
n 4074 3745 m 4085 3742 l gs col0 s gr
% Polyline
n 4113 3735 m 4085 3742 l gs col0 s gr
% Polyline
n 4074 1893 m 4076 1891 l gs col0 s gr
% Polyline
n 4079 1890 m 4076 1891 l gs col0 s gr
% Polyline
n 4079 1890 m 4100 1873 l gs col0 s gr
% Polyline
n 4113 1862 m 4100 1873 l gs col0 s gr
% Polyline
n 4113 3735 m 4115 3734 l gs col0 s gr
% Polyline
n 4127 3730 m 4115 3734 l gs col0 s gr
% Polyline
n 4127 3730 m 4146 3723 l gs col0 s gr
% Polyline
n 4152 3721 m 4146 3723 l gs col0 s gr
% Polyline
n 4113 1862 m 4123 1856 l gs col0 s gr
% Polyline
n 4136 1844 m 4123 1856 l gs col0 s gr
% Polyline
n 4136 1844 m 4146 1838 l gs col0 s gr
% Polyline
n 4152 1834 m 4146 1838 l gs col0 s gr
% Polyline
n 4152 3721 m 4172 3712 l gs col0 s gr
% Polyline
n 4190 3704 m 4172 3712 l gs col0 s gr
% Polyline
n 4152 1834 m 4170 1821 l gs col0 s gr
% Polyline
n 4190 1807 m 4170 1821 l gs col0 s gr
% Polyline
n 4190 3704 m 4200 3697 l gs col0 s gr
% Polyline
n 4226 3686 m 4200 3697 l gs col0 s gr
% Polyline
n 4226 3686 m 4227 3684 l gs col0 s gr
% Polyline
n 4229 3684 m 4227 3684 l gs col0 s gr
% Polyline
n 4190 1807 m 4194 1806 l gs col0 s gr
% Polyline
n 4205 1800 m 4194 1806 l gs col0 s gr
% Polyline
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% Polyline
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n 4741 2137 m 4734 2135 l gs col0 s gr
% Polyline
n 4705 1570 m 4716 1568 l gs col0 s gr
% Polyline
n 4741 1562 m 4716 1568 l gs col0 s gr
% Polyline
n 4777 2150 m 4753 2141 l 4741 2137 l gs col0 s gr
% Polyline
n 4741 1562 m 4745 1560 l gs col0 s gr
% Polyline
n 4753 1557 m 4745 1560 l gs col0 s gr
% Polyline
n 4753 1557 m 4771 1551 l gs col0 s gr
% Polyline
n 4777 1549 m 4771 1551 l gs col0 s gr
% Polyline
n 4777 2150 m 4797 2160 l gs col0 s gr
% Polyline
n 4814 2169 m 4797 2160 l gs col0 s gr
% Polyline
n 4777 1549 m 4795 1539 l gs col0 s gr
% Polyline
n 4814 1530 m 4795 1539 l gs col0 s gr
% Polyline
n 4851 2192 m 4825 2175 l 4814 2169 l gs col0 s gr
% Polyline
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% Polyline
n 4825 1522 m 4819 1527 l gs col0 s gr
% Polyline
n 4825 1522 m 4841 1513 l gs col0 s gr
% Polyline
n 4851 1507 m 4841 1513 l gs col0 s gr
% Polyline
n 4887 2220 m 4875 2211 l 4851 2192 l gs col0 s gr
% Polyline
n 4851 1507 m 4862 1499 l gs col0 s gr
% Polyline
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% Polyline
n 4875 1489 m 4882 1483 l gs col0 s gr
% Polyline
n 4887 1480 m 4882 1483 l gs col0 s gr
% Polyline
n 4924 2250 m 4918 2245 l 4887 2220 l gs col0 s gr
% Polyline
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% Polyline
n 4918 1454 m 4902 1468 l gs col0 s gr
% Polyline
n 4918 1454 m 4921 1450 l gs col0 s gr
% Polyline
n 4924 1449 m 4921 1450 l gs col0 s gr
% Polyline
n 4960 2284 m 4955 2279 l 4924 2250 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
n 4955 1419 m 4958 1417 l gs col0 s gr
% Polyline
n 4960 1415 m 4958 1417 l gs col0 s gr
% Polyline
n 4997 2320 m 4990 2314 l 4960 2284 l gs col0 s gr
% Polyline
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% Polyline
n 4990 1385 m 4976 1399 l gs col0 s gr
% Polyline
n 4990 1385 m 4994 1381 l gs col0 s gr
% Polyline
n 4997 1378 m 4994 1381 l gs col0 s gr
% Polyline
n 5034 2358 m 5023 2348 l 4997 2320 l gs col0 s gr
% Polyline
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% Polyline
n 5023 1350 m 5011 1364 l gs col0 s gr
% Polyline
n 5023 1350 m 5029 1345 l gs col0 s gr
% Polyline
n 5034 1340 m 5029 1345 l gs col0 s gr
% Polyline
n 5071 2399 m 5055 2382 l 5034 2358 l gs col0 s gr
% Polyline
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% Polyline
n 5055 1316 m 5046 1327 l gs col0 s gr
% Polyline
n 5055 1316 m 5062 1308 l gs col0 s gr
% Polyline
n 5071 1300 m 5062 1308 l gs col0 s gr
% Polyline
n 5107 2442 m 5085 2417 l 5071 2399 l gs col0 s gr
% Polyline
n 5071 1300 m 5079 1289 l gs col0 s gr
% Polyline
n 5085 1281 m 5079 1289 l gs col0 s gr
% Polyline
n 5085 1281 m 5096 1271 l gs col0 s gr
% Polyline
n 5107 1257 m 5096 1271 l gs col0 s gr
% Polyline
n 5144 2486 m 5115 2452 l 5107 2442 l gs col0 s gr
% Polyline
n 5107 1257 m 5111 1251 l gs col0 s gr
% Polyline
n 5115 1247 m 5111 1251 l gs col0 s gr
% Polyline
n 5115 1247 m 5128 1233 l gs col0 s gr
% Polyline
n 5144 1213 m 5128 1233 l gs col0 s gr
% Polyline
n 5181 2530 m 5171 2521 l 5145 2486 l gs col0 s gr
% Polyline
n 5144 2486 m 5145 2486 l gs col0 s gr
% Polyline
n 5143 2486 m 5145 2486 l gs col0 s gr
% Polyline
n 5144 1213 m 5160 1193 l gs col0 s gr
% Polyline
n 5171 1179 m 5160 1193 l gs col0 s gr
% Polyline
n 5171 1179 m 5176 1174 l gs col0 s gr
% Polyline
n 5181 1168 m 5176 1174 l gs col0 s gr
% Polyline
n 5218 2578 m 5199 2554 l 5181 2530 l gs col0 s gr
% Polyline
n 5181 1168 m 5191 1153 l gs col0 s gr
% Polyline
n 5199 1143 m 5191 1153 l gs col0 s gr
% Polyline
n 5199 1143 m 5207 1134 l gs col0 s gr
% Polyline
n 5218 1122 m 5207 1134 l gs col0 s gr
% Polyline
n 5254 2625 m 5252 2623 l 5248 2618 l gs col0 s gr
% Polyline
n 5226 2589 m 5248 2618 l gs col0 s gr
% Polyline
n 5226 2589 m 5218 2578 l gs col0 s gr
% Polyline
n 5218 1122 m 5222 1115 l gs col0 s gr
% Polyline
n 5226 1109 m 5222 1115 l gs col0 s gr
% Polyline
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% Polyline
n 5252 1075 m 5237 1095 l gs col0 s gr
% Polyline
n 5252 1075 m 5253 1075 l gs col0 s gr
% Polyline
n 5254 1073 m 5253 1075 l gs col0 s gr
% Polyline
n 5290 2674 m 5278 2658 l 5254 2625 l gs col0 s gr
% Polyline
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% Polyline
n 5278 1042 m 5268 1055 l gs col0 s gr
% Polyline
n 5278 1042 m 5283 1034 l gs col0 s gr
% Polyline
n 5290 1024 m 5283 1034 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
n 5303 1006 m 5313 993 l gs col0 s gr
% Polyline
n 5327 975 m 5313 993 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
n 5327 975 m 5328 973 l gs col0 s gr
% Polyline
n 5329 972 m 5328 973 l gs col0 s gr
% Polyline
n 5329 972 m 5343 952 l gs col0 s gr
% Polyline
n 5353 938 m 5358 932 l gs col0 s gr
% Polyline
n 5364 923 m 5358 932 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
n 5401 871 m 5387 890 l gs col0 s gr
% Polyline
n 5437 2880 m 5426 2864 l 5405 2834 l gs col0 s gr
% Polyline
n 5402 2829 m 5405 2834 l gs col0 s gr
% Polyline
n 5402 2829 m 5401 2827 l gs col0 s gr
% Polyline
n 5401 871 m 5401 870 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
n 5426 835 m 5415 848 l gs col0 s gr
% Polyline
n 5426 835 m 5431 828 l gs col0 s gr
% Polyline
n 5437 818 m 5431 828 l gs col0 s gr
% Polyline
n 5474 2934 m 5473 2933 l 5471 2931 l gs col0 s gr
% Polyline
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% Polyline
n 5450 2899 m 5437 2880 l gs col0 s gr
% Polyline
n 5437 818 m 5445 807 l gs col0 s gr
% Polyline
n 5450 800 m 5445 807 l gs col0 s gr
% Polyline
n 5450 800 m 5459 786 l gs col0 s gr
% Polyline
n 5473 766 m 5459 786 l gs col0 s gr
% Polyline
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% Polyline
n 5510 2988 m 5496 2967 l 5474 2934 l gs col0 s gr
% Polyline
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% Polyline
n 5496 731 m 5487 745 l gs col0 s gr
% Polyline
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% Polyline
n 5510 710 m 5502 723 l gs col0 s gr
% Polyline
n 5547 3044 m 5541 3036 l 5532 3023 l gs col0 s gr
% Polyline
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% Polyline
n 5519 3002 m 5510 2988 l gs col0 s gr
% Polyline
n 5510 710 m 5516 702 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
n 5541 662 m 5530 681 l gs col0 s gr
% Polyline
n 5541 662 m 5544 659 l gs col0 s gr
% Polyline
n 5547 654 m 5544 659 l gs col0 s gr
% Polyline
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% Polyline
n 5547 654 m 5558 638 l gs col0 s gr
% Polyline
n 5563 628 m 5558 638 l gs col0 s gr
% Polyline
n 5563 628 m 5571 617 l gs col0 s gr
% Polyline
n 5583 599 m 5571 617 l gs col0 s gr
% Polyline
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% Polyline
n 5586 3106 m 5590 3111 l gs col0 s gr
% Polyline
n 5586 3106 m 5583 3101 l gs col0 s gr
% Polyline
n 5583 599 m 5585 595 l gs col0 s gr
% Polyline
n 5586 594 m 5585 595 l gs col0 s gr
% Polyline
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% Polyline
n 5609 559 m 5599 573 l gs col0 s gr
% Polyline
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% Polyline
n 5620 541 m 5613 552 l gs col0 s gr
% Polyline
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% Polyline
n 5630 3173 m 5645 3197 l gs col0 s gr
% Polyline
n 5630 3173 m 5620 3158 l gs col0 s gr
% Polyline
n 5620 541 m 5626 531 l gs col0 s gr
% Polyline
n 5630 524 m 5626 531 l gs col0 s gr
% Polyline
n 5630 524 m 5641 509 l gs col0 s gr
% Polyline
n 5652 490 m 5641 509 l gs col0 s gr
% Polyline
n 5652 490 m 5654 487 l gs col0 s gr
% Polyline
n 5657 484 m 5654 487 l gs col0 s gr
% Polyline
n 5694 3275 m 5673 3243 l 5657 3216 l gs col0 s gr
% Polyline
n 5657 484 m 5667 466 l gs col0 s gr
% Polyline
n 5673 455 m 5667 466 l gs col0 s gr
% Polyline
n 5673 455 m 5681 445 l gs col0 s gr
% Polyline
n 5694 424 m 5681 445 l gs col0 s gr
% Polyline
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% Polyline
n 5695 3277 m 5697 3280 l gs col0 s gr
% Polyline
n 5695 3277 m 5694 3275 l gs col0 s gr
% Polyline
n 5694 424 m 5695 423 l gs col0 s gr
% Polyline
n 5695 422 m 5695 423 l gs col0 s gr
% Polyline
n 5695 422 m 5708 401 l gs col0 s gr
% Polyline
n 5717 387 m 5708 401 l gs col0 s gr
% Polyline
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% Polyline
n 5730 365 m 5721 379 l gs col0 s gr
% Polyline
n 5767 3394 m 5758 3380 l 5747 3361 l gs col0 s gr
% Polyline
n 5738 3345 m 5747 3361 l gs col0 s gr
% Polyline
n 5738 3345 m 5730 3334 l gs col0 s gr
% Polyline
n 5730 365 m 5735 357 l gs col0 s gr
% Polyline
n 5738 353 m 5735 357 l gs col0 s gr
% Polyline
n 5738 353 m 5748 335 l gs col0 s gr
% Polyline
n 5758 319 m 5748 335 l gs col0 s gr
% Polyline
n 5758 319 m 5761 313 l gs col0 s gr
% Polyline
n 5767 305 m 5761 313 l gs col0 s gr
% Polyline
n 5803 3455 m 5800 3449 l 5795 3441 l gs col0 s gr
% Polyline
n 5779 3414 m 5795 3441 l gs col0 s gr
% Polyline
n 5779 3414 m 5767 3394 l gs col0 s gr
% Polyline
n 5767 305 m 5775 291 l gs col0 s gr
% Polyline
n 5779 284 m 5775 291 l gs col0 s gr
% Polyline
n 5779 284 m 5788 269 l gs col0 s gr
% Polyline
n 5800 249 m 5788 269 l gs col0 s gr
% Polyline
n 5800 249 m 5801 248 l gs col0 s gr
% Polyline
n 5803 244 m 5801 248 l gs col0 s gr
% Polyline
n 5840 3516 m 5821 3483 l 5803 3455 l gs col0 s gr
% Polyline
n 5803 244 m 5814 225 l gs col0 s gr
% Polyline
n 5821 216 m 5814 225 l gs col0 s gr
% Polyline
n 5821 216 m 5827 203 l gs col0 s gr
% Polyline
n 5840 182 m 5827 203 l gs col0 s gr
% Polyline
n 5861 3553 m 5842 3520 l gs col0 s gr
% Polyline
n 5841 3518 m 5842 3520 l gs col0 s gr
% Polyline
n 5841 3518 m 5840 3516 l gs col0 s gr
% Polyline
n 5840 182 m 5841 181 l 5854 159 l gs col0 s gr
% Polyline
n 5861 147 m 5854 159 l gs col0 s gr
% Polyline
n 5353 938 m 5343 952 l gs col0 s gr
% Polyline
0 slj
n 103 2103 m 95 2074 l 93 2067 l gs col0 s gr
% Polyline
n 86 2039 m 93 2067 l gs col0 s gr
% Polyline
n 86 2039 m 84 2028 l gs col0 s gr
% Polyline
n 79 2004 m 84 2028 l gs col0 s gr
% Polyline
n 79 2004 m 77 1991 l gs col0 s gr
% Polyline
n 74 1970 m 77 1991 l gs col0 s gr
% Polyline
n 74 1970 m 72 1953 l gs col0 s gr
% Polyline
n 70 1935 m 72 1953 l gs col0 s gr
% Polyline
n 70 1935 m 68 1917 l gs col0 s gr
% Polyline
n 67 1901 m 68 1917 l gs col0 s gr
% Polyline
n 67 1901 m 66 1880 l gs col0 s gr
% Polyline
n 66 1866 m 66 1880 l gs col0 s gr
% Polyline
n 66 1866 m 66 1846 l gs col0 s gr
% Polyline
n 66 1832 m 66 1846 l gs col0 s gr
% Polyline
n 66 1832 m 66 1812 l gs col0 s gr
% Polyline
n 67 1798 m 66 1812 l gs col0 s gr
% Polyline
n 67 1798 m 69 1778 l gs col0 s gr
% Polyline
n 70 1763 m 69 1778 l gs col0 s gr
% Polyline
n 70 1763 m 72 1746 l gs col0 s gr
% Polyline
n 74 1729 m 72 1746 l gs col0 s gr
% Polyline
n 74 1729 m 76 1715 l gs col0 s gr
% Polyline
n 79 1695 m 76 1715 l gs col0 s gr
% Polyline
n 79 1695 m 82 1682 l gs col0 s gr
% Polyline
n 86 1660 m 82 1682 l gs col0 s gr
% Polyline
n 86 1660 m 88 1652 l gs col0 s gr
% Polyline
n 95 1626 m 88 1652 l gs col0 s gr
% Polyline
n 95 1626 m 96 1622 l gs col0 s gr
% Polyline
n 103 1595 m 96 1622 l gs col0 s gr
% Polyline
n 165 2242 m 149 2211 l 142 2198 l gs col0 s gr
% Polyline
n 132 2176 m 142 2198 l gs col0 s gr
% Polyline
n 132 2176 m 122 2152 l gs col0 s gr
% Polyline
n 117 2141 m 122 2152 l gs col0 s gr
% Polyline
n 117 2141 m 105 2108 l gs col0 s gr
% Polyline
n 104 2108 m 105 2108 l gs col0 s gr
% Polyline
n 104 2108 m 103 2103 l gs col0 s gr
% Polyline
n 103 1595 m 104 1592 l gs col0 s gr
% Polyline
n 104 1591 m 104 1592 l gs col0 s gr
% Polyline
n 104 1591 m 115 1564 l gs col0 s gr
% Polyline
n 117 1557 m 115 1564 l gs col0 s gr
% Polyline
n 117 1557 m 127 1536 l gs col0 s gr
% Polyline
n 132 1522 m 127 1536 l gs col0 s gr
% Polyline
n 132 1522 m 139 1508 l gs col0 s gr
% Polyline
n 149 1489 m 139 1508 l gs col0 s gr
% Polyline
n 149 1489 m 152 1480 l gs col0 s gr
% Polyline
n 165 1457 m 152 1480 l gs col0 s gr
% Polyline
n 226 2327 m 216 2314 l 209 2304 l gs col0 s gr
% Polyline
n 190 2280 m 209 2304 l gs col0 s gr
% Polyline
n 190 2280 m 168 2247 l gs col0 s gr
% Polyline
n 167 2245 m 168 2247 l gs col0 s gr
% Polyline
n 167 2245 m 165 2242 l gs col0 s gr
% Polyline
n 165 1457 m 166 1454 l gs col0 s gr
% Polyline
n 167 1454 m 166 1454 l gs col0 s gr
% Polyline
n 167 1454 m 183 1429 l gs col0 s gr
% Polyline
n 190 1420 m 183 1429 l gs col0 s gr
% Polyline
n 190 1420 m 201 1405 l gs col0 s gr
% Polyline
n 216 1385 m 201 1405 l gs col0 s gr
% Polyline
n 216 1385 m 220 1381 l gs col0 s gr
% Polyline
n 226 1373 m 220 1381 l gs col0 s gr
% Polyline
n 288 2384 m 287 2383 l 284 2381 l gs col0 s gr
% Polyline
n 250 2348 m 284 2381 l gs col0 s gr
% Polyline
n 250 2348 m 226 2327 l gs col0 s gr
% Polyline
n 226 1373 m 241 1359 l gs col0 s gr
% Polyline
n 250 1350 m 241 1359 l gs col0 s gr
% Polyline
n 250 1350 m 264 1338 l gs col0 s gr
% Polyline
n 287 1316 m 264 1338 l gs col0 s gr
% Polyline
n 287 1316 m 288 1316 l gs col0 s gr
% Polyline
n 288 1315 m 288 1316 l gs col0 s gr
% Polyline
n 350 2423 m 340 2417 l 288 2384 l gs col0 s gr
% Polyline
n 288 1315 m 316 1297 l gs col0 s gr
% Polyline
n 340 1281 m 316 1297 l gs col0 s gr
% Polyline
n 340 1281 m 345 1278 l gs col0 s gr
% Polyline
n 350 1275 m 345 1278 l gs col0 s gr
% Polyline
n 350 2423 m 398 2444 l gs col0 s gr
% Polyline
n 412 2450 m 398 2444 l gs col0 s gr
% Polyline
n 350 1275 m 379 1264 l gs col0 s gr
% Polyline
n 412 1249 m 379 1264 l gs col0 s gr
% Polyline
n 474 2465 m 420 2452 l 412 2450 l gs col0 s gr
% Polyline
n 412 1249 m 414 1248 l gs col0 s gr
% Polyline
n 420 1247 m 414 1248 l gs col0 s gr
% Polyline
n 420 1247 m 457 1238 l gs col0 s gr
% Polyline
n 474 1235 m 457 1238 l gs col0 s gr
% Polyline
n 474 2465 m 503 2467 l gs col0 s gr
% Polyline
n 536 2471 m 503 2467 l gs col0 s gr
% Polyline
n 474 1235 m 506 1231 l gs col0 s gr
% Polyline
n 536 1227 m 506 1231 l gs col0 s gr
% Polyline
n 536 2471 m 569 2470 l gs col0 s gr
% Polyline
n 598 2469 m 569 2470 l gs col0 s gr
% Polyline
n 536 1227 m 563 1228 l gs col0 s gr
% Polyline
n 598 1229 m 563 1228 l gs col0 s gr
% Polyline
n 598 2469 m 624 2466 l gs col0 s gr
% Polyline
n 659 2461 m 624 2466 l gs col0 s gr
% Polyline
n 598 1229 m 636 1235 l gs col0 s gr
% Polyline
n 659 1238 m 636 1235 l gs col0 s gr
% Polyline
n 659 2461 m 672 2457 l gs col0 s gr
% Polyline
n 699 2452 m 672 2457 l gs col0 s gr
% Polyline
n 699 2452 m 715 2448 l gs col0 s gr
% Polyline
n 721 2446 m 715 2448 l gs col0 s gr
% Polyline
n 721 1253 m 699 1247 l 659 1238 l gs col0 s gr
% Polyline
n 721 2446 m 753 2435 l gs col0 s gr
% Polyline
n 783 2425 m 753 2435 l gs col0 s gr
% Polyline
n 721 1253 m 747 1262 l gs col0 s gr
% Polyline
n 783 1274 m 747 1262 l gs col0 s gr
% Polyline
n 783 2425 m 791 2422 l gs col0 s gr
% Polyline
n 802 2417 m 791 2422 l gs col0 s gr
% Polyline
n 802 2417 m 825 2406 l gs col0 s gr
% Polyline
n 845 2398 m 825 2406 l gs col0 s gr
% Polyline
n 845 1301 m 802 1281 l 783 1274 l gs col0 s gr
% Polyline
n 845 2398 m 859 2390 l gs col0 s gr
% Polyline
n 875 2383 m 859 2390 l gs col0 s gr
% Polyline
n 875 2383 m 891 2374 l gs col0 s gr
% Polyline
n 906 2365 m 891 2374 l gs col0 s gr
% Polyline
n 906 1333 m 875 1316 l 845 1301 l gs col0 s gr
% Polyline
n 906 2365 m 922 2357 l gs col0 s gr
% Polyline
n 937 2348 m 922 2357 l gs col0 s gr
% Polyline
n 937 2348 m 951 2339 l gs col0 s gr
% Polyline
n 969 2328 m 951 2339 l gs col0 s gr
% Polyline
n 969 1371 m 937 1350 l 906 1333 l gs col0 s gr
% Polyline
n 969 2328 m 980 2321 l gs col0 s gr
% Polyline
1 slj
n 3165 2075 m 3165 2073 l 3164 2072 l gs col0 s gr
% Polyline
0 slj
n 990 2314 m 980 2321 l gs col0 s gr
% Polyline
1 slj
n 3157 2039 m 3164 2072 l gs col0 s gr
% Polyline
0 slj
n 990 2314 m 1007 2301 l gs col0 s gr
% Polyline
n 1030 2285 m 1007 2301 l gs col0 s gr
% Polyline
n 1030 1413 m 990 1385 l 969 1371 l gs col0 s gr
% Polyline
n 1030 2285 m 1034 2282 l gs col0 s gr
% Polyline
n 1038 2280 m 1034 2282 l gs col0 s gr
% Polyline
n 1038 2280 m 1060 2261 l gs col0 s gr
% Polyline
n 1082 2245 m 1060 2261 l gs col0 s gr
% Polyline
n 1082 2245 m 1085 2242 l gs col0 s gr
% Polyline
n 1092 2236 m 1085 2242 l gs col0 s gr
% Polyline
n 1092 1463 m 1082 1454 l 1056 1434 l gs col0 s gr
% Polyline
n 1038 1420 m 1056 1434 l gs col0 s gr
% Polyline
n 1038 1420 m 1030 1413 l gs col0 s gr
% Polyline
n 1092 2236 m 1110 2220 l gs col0 s gr
% Polyline
n 1121 2211 m 1110 2220 l gs col0 s gr
% Polyline
n 1121 2211 m 1134 2199 l gs col0 s gr
% Polyline
n 1153 2180 m 1134 2199 l gs col0 s gr
% Polyline
n 1153 1519 m 1121 1489 l 1092 1463 l gs col0 s gr
% Polyline
n 1153 2180 m 1156 2178 l gs col0 s gr
% Polyline
n 1158 2176 m 1156 2178 l gs col0 s gr
% Polyline
n 1158 2176 m 1178 2156 l gs col0 s gr
% Polyline
n 1191 2141 m 1178 2156 l gs col0 s gr
% Polyline
n 1191 2141 m 1199 2132 l gs col0 s gr
% Polyline
n 1215 2114 m 1199 2132 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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1 slj
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$F2psEnd
rs
%%EndDocument
@endspecial 3052 4372 a Fq(x)2327 3632 y(y)1831 4372
y(x)1110 3632 y(y)499 5293 y Fw(Abbildung)27 b(2.2:)k(Ell.)e(Kurv)m(en)
h(in)f(W)-8 b(eierstra\031-Normalform)31 b(f)2712 5297
y(\177)2709 5293 y(ur)e(c)m(har)q(\()p Fq(K)7 b Fw(\))25
b Fo(6)p Fw(=)g(2)p Fq(;)15 b Fw(3)531 5496 y(In)35 b(der)f
(Kryptographie)g(w)m(erden)h(im)f(allgemeinen)f(zw)m(ei)j(V)-8
b(arian)m(ten)35 b(v)m(on)h(K)3250 5500 y(\177)3250 5496
y(orp)s(ern)390 5609 y(b)s(en)m(utzt.)f(Zum)e(einen)g(wird)f(b)s(ei)h
(Hardw)m(are-Implemen)m(tierungen)g(gerne)h Fn(F)3082
5623 y Fp(2)3117 5604 y Fm(n)3204 5609 y Fw(v)m(erw)m(en-)p
eop
%%Page: 9 10
9 9 bop 270 82 a Fw(2.2.)62 b(Elliptisc)m(he)28 b(Kurv)m(en)h(als)h(ab)
s(elsc)m(he)g(Grupp)s(en)1256 b(9)270 415 y(det,)34 b(w)m(as)g(sic)m(h)
f(in)f(b)s(esseren)g(Laufzeitergebnissen)g(niedersc)m(hl)2490
419 y(\177)2490 415 y(agt.)h(Andererseits)g(k)-5 b(ann)270
528 y(man)31 b(eine)f(elliptisc)m(he)f(Kurv)m(e)h(auc)m(h)1574
532 y(\177)1571 528 y(ub)s(er)g Fn(F)1831 542 y Fl(p)1908
528 y Fw(b)s(etrac)m(h)m(ten,)i(w)m(ob)s(ei)e Fq(p)g
Fw(eine)h(in)e(der)i(Regel)270 641 y(sehr)24 b(gro\031e)h(Primzahl)d
(ist.)i(Da)h(wir)e(uns)g(in)g(dieser)g(Arb)s(eit)g(n)m(ur)g(mit)h(der)f
(zw)m(eiten)i(V)-8 b(arian-)270 754 y(te)28 b(b)s(esc)m(h)591
758 y(\177)591 754 y(aftigen,)f(k)961 758 y(\177)961
754 y(onnen)f(wir)f(immer)h(v)m(on)h(Gleic)m(h)m(ung)g(\(2.2\))h
(ausgehen.)f(W)-8 b(eiterhin)26 b(sei)270 867 y(mit)f
Fq(p)g Fw(immer)g(eine)g(Primzahl)f(gr)1431 871 y(\177)1431
867 y(o\031er)j(3)f(gemein)m(t,)g(auc)m(h)g(w)m(enn)f(dies)g(nic)m(h)m
(t)h(ausdr)3119 871 y(\177)3116 867 y(uc)m(klic)m(h)270
980 y(aus)k(dem)g(T)-8 b(ext)31 b(herv)m(orgeh)m(t.)411
1095 y(Wir)k(w)m(ollen)g(no)s(c)m(h)g(kurz)g(einige)g(wic)m(h)m(tige)h
(Kennzahlen)f(v)m(on)h(elliptisc)m(hen)d(Kurv)m(en)270
1208 y(de\014nieren.)21 b(Diese)i(sind)e(insb)s(esondere)g(f)1679
1212 y(\177)1676 1208 y(ur)h(Kapitel)g(2.3)i(wic)m(h)m(tig,)f(in)e(w)m
(elc)m(hem)j(es)f(um)f(die)270 1321 y(Charakterisierung)27
b(v)m(on)j(elliptisc)m(hen)c(Kurv)m(en)j(geh)m(t,)i(die)d(sic)m(h)h(f)
2554 1325 y(\177)2551 1321 y(ur)f(die)g(Kryptographie)270
1434 y(eignen.)270 1660 y Fs(De\014nition)35 b(2)46 b
Fu(Sei)29 b(eine)g(el)5 b(liptische)30 b(Kurve)e(der)i(F)-7
b(orm)31 b Fq(y)2334 1627 y Fp(2)2398 1660 y Fw(=)25
b Fq(x)2546 1627 y Fp(3)2599 1660 y Fw(+)13 b Fq(ax)g
Fw(+)g Fq(b)28 b Fu(vor)-5 b(ge)g(geb)g(en.)270 1773
y(Ihr)g(e)34 b(Diskriminante)f(ist)f(de\014niert)i(dur)-5
b(ch)1396 1984 y Fw(\001)25 b(=)g Fo(\000)p Fw(16\(4)p
Fq(a)1882 1946 y Fp(3)1943 1984 y Fw(+)20 b(27)p Fq(b)2163
1946 y Fp(2)2203 1984 y Fw(\))p Fq(:)940 b Fw(\(2.3\))270
2208 y Fs(Satz)35 b(3)g(\(Hasse\))45 b Fu(Es)29 b(sei)g
Fq(E)35 b Fu(eine)29 b(el)5 b(liptische)30 b(Kurve)2228
2212 y(\177)2228 2208 y(ub)-5 b(er)29 b(dem)h(K)2684
2212 y(\177)2685 2208 y(orp)-5 b(er)31 b Fn(F)2971 2222
y Fl(p)3016 2208 y Fu(.)e(Die)g(A)n(n-)270 2321 y(zahl)j(der)e
(Kurvenpunkte,)g(die)h(wir)g(mit)f Fo(j)p Fq(E)5 b Fw(\()p
Fn(F)1869 2335 y Fl(p)1914 2321 y Fw(\))p Fo(j)31 b Fu(b)-5
b(ezeichnen)31 b(wol)5 b(len,)31 b(l)2785 2325 y(\177)2786
2321 y(a\031t)g(sich)f(folgen-)270 2434 y(derma\031en)k(ausdr)908
2438 y(\177)908 2434 y(ucken:)1439 2553 y Fo(j)p Fq(E)5
b Fw(\()p Fn(F)1622 2567 y Fl(p)1667 2553 y Fw(\))p Fo(j)26
b Fw(=)f Fq(p)20 b Fw(+)g(1)h Fo(\000)f Fq(t:)982 b Fw(\(2.4\))270
2728 y Fu(mit)33 b Fo(j)p Fq(t)p Fo(j)26 b(\024)e Fw(2)684
2672 y Fo(p)p 761 2672 46 4 v 761 2728 a Fq(p)o Fu(.)411
2952 y Fw(Dab)s(ei)33 b(wird)e(die)i(Gr)1145 2956 y(\177)1145
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
n 1058 3551 m 1058 3550 l gs col0 s gr
% Polyline
n 1059 3546 m 1058 3550 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
n 1078 4550 m 1059 4482 l gs col0 s gr
% Polyline
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% Polyline
n 1078 3479 m 1072 3495 l gs col0 s gr
% Polyline
n 1078 3479 m 1087 3441 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
n 1398 2890 m 1387 2901 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
n 1598 2741 m 1584 2749 l gs col0 s gr
% Polyline
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% Polyline
n 1665 5320 m 1647 5311 l gs col0 s gr
% Polyline
n 1598 2741 m 1628 2726 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
n 1736 2681 m 1725 2686 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
n 1802 2663 m 1778 2670 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
n 1802 2663 m 1836 2657 l gs col0 s gr
% Polyline
n 1868 2651 m 1836 2657 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
n 1868 2651 m 1891 2648 l gs col0 s gr
% Polyline
n 1939 2644 m 1891 2648 l gs col0 s gr
% Polyline
n 1939 5384 m 1988 5386 l gs col0 s gr
% Polyline
n 2005 5386 m 1988 5386 l gs col0 s gr
% Polyline
n 1939 2644 m 1955 2644 l gs col0 s gr
% Polyline
n 2005 2643 m 1955 2644 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
n 2005 2643 m 2022 2643 l gs col0 s gr
% Polyline
n 2072 2644 m 2022 2643 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
n 2072 2644 m 2094 2646 l gs col0 s gr
% Polyline
n 2140 2651 m 2094 2646 l gs col0 s gr
% Polyline
n 2140 5377 m 2176 5372 l gs col0 s gr
% Polyline
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% Polyline
n 2140 2651 m 2170 2655 l gs col0 s gr
% Polyline
n 2207 2662 m 2170 2655 l gs col0 s gr
% Polyline
n 2207 5366 m 2234 5361 l gs col0 s gr
% Polyline
n 2277 5352 m 2234 5361 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
n 2344 2692 m 2343 2692 l gs col0 s gr
% Polyline
n 2344 5334 m 2345 5334 l gs col0 s gr
% Polyline
n 2346 5334 m 2345 5334 l gs col0 s gr
% Polyline
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% Polyline
n 2411 5313 m 2396 5316 l gs col0 s gr
% Polyline
n 2411 2716 m 2346 2693 l 2344 2692 l gs col0 s gr
% Polyline
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% Polyline
n 2478 5288 m 2445 5297 l gs col0 s gr
% Polyline
n 2411 2716 m 2440 2727 l gs col0 s gr
% Polyline
n 2478 2741 m 2440 2727 l gs col0 s gr
% Polyline
n 2478 5288 m 2495 5279 l gs col0 s gr
% Polyline
n 2545 5261 m 2495 5279 l gs col0 s gr
% Polyline
n 2544 5261 m 2546 5261 l gs col0 s gr
% Polyline
n 2545 5260 m 2545 5261 l gs col0 s gr
% Polyline
n 2545 2767 m 2478 2741 l gs col0 s gr
% Polyline
n 2545 5260 m 2593 5240 l gs col0 s gr
% Polyline
n 2613 5229 m 2593 5240 l gs col0 s gr
% Polyline
n 2544 2767 m 2546 2767 l gs col0 s gr
% Polyline
n 2613 2800 m 2545 2767 l gs col0 s gr
% Polyline
n 2613 5229 m 2639 5216 l gs col0 s gr
% Polyline
n 2682 5196 m 2639 5216 l gs col0 s gr
% Polyline
n 2613 2800 m 2670 2825 l gs col0 s gr
% Polyline
n 6946 4481 m 6945 4478 l 6943 4476 l gs col7 0.00 shd ef gr gs col0 s gr
% Polyline
n 2682 2830 m 2670 2825 l gs col0 s gr
% Polyline
0 slc
7.500 slw
n 6614 3104 m 12283 809 l gs col7 0.00 shd ef gr gs col0 s gr
% Polyline
1 slc
30.000 slw
n 2682 5196 m 2685 5195 l gs col0 s gr
% Polyline
n 2697 5189 m 2685 5195 l gs col0 s gr
% Polyline
n 2697 5189 m 2731 5171 l gs col0 s gr
% Polyline
n 2748 5162 m 2731 5171 l gs col0 s gr
% Polyline
n 2748 2867 m 2697 2838 l 2682 2830 l gs col0 s gr
% Polyline
n 2748 5162 m 2774 5148 l gs col0 s gr
% Polyline
n 2815 5126 m 2774 5148 l gs col0 s gr
% Polyline
n 2748 2867 m 2802 2896 l gs col0 s gr
% Polyline
n 2815 2902 m 2802 2896 l gs col0 s gr
% Polyline
n 2815 5126 m 2822 5122 l gs col0 s gr
% Polyline
n 2826 5120 m 2822 5122 l gs col0 s gr
% Polyline
n 2826 5120 m 2864 5098 l gs col0 s gr
% Polyline
n 2883 5085 m 2864 5098 l gs col0 s gr
% Polyline
n 2883 2942 m 2826 2908 l 2815 2902 l gs col0 s gr
% Polyline
n 2883 5085 m 2907 5071 l gs col0 s gr
% Polyline
n 2949 5047 m 2907 5071 l gs col0 s gr
% Polyline
n 2949 5047 m 2950 5047 l gs col0 s gr
% Polyline
n 2950 5045 m 2950 5047 l gs col0 s gr
% Polyline
n 2950 2982 m 2949 2980 l 2883 2942 l gs col0 s gr
% Polyline
n 2950 5045 m 2993 5020 l gs col0 s gr
% Polyline
n 3017 5004 m 2993 5020 l gs col0 s gr
% Polyline
n 2950 2982 m 2958 2985 l gs col0 s gr
% Polyline
n 3017 3026 m 2958 2985 l gs col0 s gr
% Polyline
n 3017 5004 m 3035 4994 l gs col0 s gr
% Polyline
n 3062 4977 m 3035 4994 l gs col0 s gr
% Polyline
n 3062 4977 m 3076 4966 l gs col0 s gr
% Polyline
n 3086 4959 m 3076 4966 l gs col0 s gr
% Polyline
n 3086 3068 m 3062 3054 l 3017 3026 l gs col0 s gr
% Polyline
n 3086 4959 m 3117 4939 l gs col0 s gr
% Polyline
n 3153 4913 m 3117 4939 l gs col0 s gr
% Polyline
n 3086 3068 m 3129 3099 l gs col0 s gr
% Polyline
n 3153 3113 m 3129 3099 l gs col0 s gr
% Polyline
n 3153 4913 m 3160 4911 l gs col0 s gr
% Polyline
n 3169 4904 m 3160 4911 l gs col0 s gr
% Polyline
n 3169 4904 m 3202 4881 l gs col0 s gr
% Polyline
n 3224 4868 m 3202 4881 l gs col0 s gr
% Polyline
n 3224 3160 m 3169 3124 l 3153 3113 l gs col0 s gr
% Polyline
n 3224 4868 m 3243 4855 l gs col0 s gr
% Polyline
n 3278 4834 m 3243 4855 l gs col0 s gr
% Polyline
n 3278 4834 m 3282 4827 l gs col0 s gr
% Polyline
n 3290 4824 m 3282 4827 l gs col0 s gr
% Polyline
n 3290 3203 m 3278 3196 l 3224 3160 l gs col0 s gr
% Polyline
n 3290 4824 m 3327 4803 l gs col0 s gr
% Polyline
n 3355 4782 m 3327 4803 l gs col0 s gr
% Polyline
n 3290 3203 m 3310 3215 l gs col0 s gr
% Polyline
n 3355 3247 m 3310 3215 l gs col0 s gr
% Polyline
n 3355 4782 m 3369 4774 l gs col0 s gr
% Polyline
n 3388 4764 m 3369 4774 l gs col0 s gr
% Polyline
n 3388 4764 m 3412 4747 l gs col0 s gr
% Polyline
n 3425 4738 m 3412 4747 l gs col0 s gr
% Polyline
n 3425 3289 m 3388 3267 l 3355 3247 l gs col0 s gr
% Polyline
n 3425 4738 m 3453 4722 l gs col0 s gr
% Polyline
n 3492 4701 m 3453 4722 l gs col0 s gr
% Polyline
n 3425 3289 m 3475 3317 l gs col0 s gr
% Polyline
n 3492 3328 m 3475 3317 l gs col0 s gr
% Polyline
n 3492 4701 m 3497 4697 l gs col0 s gr
% Polyline
n 3508 4691 m 3497 4697 l gs col0 s gr
% Polyline
n 3508 4691 m 3543 4674 l gs col0 s gr
% Polyline
n 3560 4663 m 3543 4674 l gs col0 s gr
% Polyline
n 3560 3362 m 3508 3336 l 3492 3328 l gs col0 s gr
% Polyline
n 3560 4663 m 3589 4650 l gs col0 s gr
% Polyline
n 3627 4633 m 3589 4650 l gs col0 s gr
% Polyline
n 3560 3362 m 3605 3387 l gs col0 s gr
% Polyline
n 3627 3396 m 3605 3387 l gs col0 s gr
% Polyline
n 3627 4633 m 3634 4628 l gs col0 s gr
% Polyline
n 3667 4618 m 3634 4628 l gs col0 s gr
% Polyline
n 3667 4618 m 3686 4612 l gs col0 s gr
% Polyline
n 3693 4608 m 3686 4612 l gs col0 s gr
% Polyline
n 3693 3420 m 3667 3409 l 3627 3396 l gs col0 s gr
% Polyline
n 3693 4608 m 3741 4596 l gs col0 s gr
% Polyline
n 3761 4593 m 3741 4596 l gs col0 s gr
% Polyline
n 3693 3420 m 3707 3424 l gs col0 s gr
% Polyline
n 3761 3436 m 3707 3424 l gs col0 s gr
% Polyline
n 3761 4593 m 3802 4588 l gs col0 s gr
% Polyline
n 3832 4585 m 3802 4588 l gs col0 s gr
% Polyline
n 3761 3436 m 3790 3438 l gs col0 s gr
% Polyline
n 3832 3442 m 3790 3438 l gs col0 s gr
% Polyline
n 3832 4585 m 3869 4588 l gs col0 s gr
% Polyline
n 3898 4591 m 3869 4588 l gs col0 s gr
% Polyline
n 3832 3442 m 3859 3441 l gs col0 s gr
% Polyline
n 3898 3436 m 3859 3441 l gs col0 s gr
% Polyline
n 3898 4591 m 3953 4606 l gs col0 s gr
% Polyline
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% Polyline
n 946 3175 m 940 3188 l gs col0 s gr
% Polyline
n 1018 4970 m 977 4904 l 946 4855 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
n 1869 2645 m 1817 2644 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
n 2012 2663 m 1972 2656 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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n 4996 2198 m 4995 2200 l gs col0 s gr
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n 4996 2198 m 5023 2157 l gs col0 s gr
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n 5090 2054 m 5080 2071 l gs col0 s gr
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n 5090 2054 m 5108 2029 l gs col0 s gr
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% Polyline
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% Polyline
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% Polyline
n 5138 6045 m 5135 6039 l gs col0 s gr
% Polyline
n 5135 1988 m 5137 1986 l gs col0 s gr
% Polyline
n 5138 1984 m 5137 1986 l gs col0 s gr
% Polyline
n 5138 1984 m 5164 1942 l gs col0 s gr
% Polyline
n 5184 1913 m 5164 1942 l gs col0 s gr
% Polyline
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% Polyline
n 5205 1880 m 5193 1900 l gs col0 s gr
% Polyline
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% Polyline
n 5230 6187 m 5272 6255 l gs col0 s gr
% Polyline
n 5230 6187 m 5205 6150 l gs col0 s gr
% Polyline
n 5205 1880 m 5220 1857 l gs col0 s gr
% Polyline
n 5230 1842 m 5220 1857 l gs col0 s gr
% Polyline
n 5230 1842 m 5248 1813 l gs col0 s gr
% Polyline
n 5275 1771 m 5248 1813 l gs col0 s gr
% Polyline
n 5275 1771 m 5276 1769 l gs col0 s gr
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% Polyline
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% Polyline
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n 5489 1424 m 5492 1418 l gs col0 s gr
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n 5495 1415 m 5519 1372 l gs col0 s gr
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n 5631 6845 m 5622 6828 l 5609 6807 l gs col0 s gr
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n 5580 6756 m 5560 6724 l gs col0 s gr
% Polyline
n 5560 1304 m 5572 1285 l gs col0 s gr
% Polyline
n 5580 1271 m 5572 1285 l gs col0 s gr
% Polyline
n 5580 1271 m 5600 1238 l gs col0 s gr
% Polyline
n 5622 1200 m 5600 1238 l gs col0 s gr
% Polyline
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% Polyline
n 5631 1186 m 5626 1194 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
n 5664 1128 m 5652 1150 l gs col0 s gr
% Polyline
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% Polyline
n 5702 1064 m 5677 1105 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
n 5706 1059 m 5704 1061 l gs col0 s gr
% Polyline
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% Polyline
n 5746 986 m 5730 1015 l gs col0 s gr
% Polyline
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% Polyline
n 5774 940 m 5756 969 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
n 5787 7113 m 5774 7089 l gs col0 s gr
% Polyline
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% Polyline
n 5787 915 m 5783 924 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
n 5844 816 m 5834 832 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
n 5868 7256 m 5844 7214 l gs col0 s gr
% Polyline
n 5844 816 m 5860 789 l gs col0 s gr
% Polyline
n 5868 772 m 5860 789 l gs col0 s gr
% Polyline
n 5868 772 m 5885 743 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
n 5916 690 m 5911 698 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
n 8011 2645 m 7959 2644 l gs col0 s gr
% Polyline
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% Polyline
n 8081 5377 m 8057 5378 l gs col0 s gr
% Polyline
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% Polyline
n 8081 2653 m 8034 2647 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
n 8154 2663 m 8114 2656 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
n 8224 2679 m 8200 2672 l gs col0 s gr
% Polyline
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% Polyline
n 8296 5333 m 8240 5348 l gs col0 s gr
% Polyline
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% Polyline
n 8296 2695 m 8292 2695 l gs col0 s gr
% Polyline
n 8296 5333 m 8297 5333 l gs col0 s gr
% Polyline
n 8300 5333 m 8297 5333 l gs col0 s gr
% Polyline
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% Polyline
n 8366 5313 m 8351 5316 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
n 8437 2741 m 8397 2727 l gs col0 s gr
% Polyline
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% Polyline
n 8507 5261 m 8456 5279 l gs col0 s gr
% Polyline
n 8507 5261 m 8508 5261 l gs col0 s gr
% Polyline
n 8509 5260 m 8508 5261 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
n 8580 5230 m 8556 5239 l gs col0 s gr
% Polyline
n 8508 2768 m 8510 2768 l gs col0 s gr
% Polyline
n 8580 2801 m 8509 2768 l gs col0 s gr
% Polyline
n 8580 5230 m 8606 5217 l gs col0 s gr
% Polyline
n 8650 5197 m 8606 5217 l gs col0 s gr
% Polyline
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% Polyline
n 8650 2830 m 8638 2825 l gs col0 s gr
% Polyline
n 8650 5197 m 8655 5196 l gs col0 s gr
% Polyline
n 8665 5190 m 8655 5196 l gs col0 s gr
% Polyline
n 8665 5190 m 8703 5171 l gs col0 s gr
% Polyline
n 805 4481 m 804 4478 l 803 4476 l gs col0 s gr
% Polyline
n 8722 5162 m 8703 5171 l gs col0 s gr
% Polyline
7.500 slw
n 8472 5674 m 12283 544 l gs col0 s gr
% Polyline
30.000 slw
n 8722 2867 m 8665 2838 l 8650 2830 l gs col0 s gr
% Polyline
n 8722 5162 m 8749 5148 l gs col0 s gr
% Polyline
n 8792 5125 m 8749 5148 l gs col0 s gr
% Polyline
n 8722 2867 m 8780 2896 l gs col0 s gr
% Polyline
n 8792 2904 m 8780 2896 l gs col0 s gr
% Polyline
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% Polyline
n 8804 5118 m 8796 5121 l gs col0 s gr
% Polyline
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% Polyline
n 8862 5085 m 8843 5098 l gs col0 s gr
% Polyline
n 8862 2942 m 8804 2910 l 8792 2904 l gs col0 s gr
% Polyline
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% Polyline
n 8931 5047 m 8887 5072 l gs col0 s gr
% Polyline
n 8931 5047 m 8932 5047 l gs col0 s gr
% Polyline
n 8934 5045 m 8932 5047 l gs col0 s gr
% Polyline
n 8934 2983 m 8931 2981 l 8862 2942 l gs col0 s gr
% Polyline
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% Polyline
n 9004 5003 m 8977 5019 l gs col0 s gr
% Polyline
n 8934 2983 m 8940 2986 l gs col0 s gr
% Polyline
n 9004 3027 m 8940 2986 l gs col0 s gr
% Polyline
n 9004 5003 m 9021 4992 l gs col0 s gr
% Polyline
n 9049 4977 m 9021 4992 l gs col0 s gr
% Polyline
n 9049 4977 m 9065 4967 l gs col0 s gr
% Polyline
n 9076 4958 m 9065 4967 l gs col0 s gr
% Polyline
n 9076 3069 m 9049 3053 l 9004 3027 l gs col0 s gr
% Polyline
n 9076 4958 m 9109 4939 l gs col0 s gr
% Polyline
n 9147 4915 m 9109 4939 l gs col0 s gr
% Polyline
n 9076 3069 m 9122 3099 l gs col0 s gr
% Polyline
n 9147 3113 m 9122 3099 l gs col0 s gr
% Polyline
n 9147 4915 m 9152 4912 l gs col0 s gr
% Polyline
n 9163 4904 m 9152 4912 l gs col0 s gr
% Polyline
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% Polyline
n 9218 4869 m 9195 4882 l gs col0 s gr
% Polyline
n 9218 3160 m 9163 3124 l 9147 3113 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
n 9276 4834 m 9283 4828 l gs col0 s gr
% Polyline
n 9289 4825 m 9283 4828 l gs col0 s gr
% Polyline
n 9289 3204 m 9276 3196 l 9218 3160 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
n 9289 3204 m 9311 3215 l gs col0 s gr
% Polyline
n 9360 3248 m 9311 3215 l gs col0 s gr
% Polyline
n 9360 4781 m 9373 4774 l gs col0 s gr
% Polyline
n 9394 4763 m 9373 4774 l gs col0 s gr
% Polyline
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% Polyline
n 9430 4738 m 9416 4748 l gs col0 s gr
% Polyline
n 9430 3289 m 9394 3268 l 9360 3248 l gs col0 s gr
% Polyline
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% Polyline
n 9502 4701 m 9461 4722 l gs col0 s gr
% Polyline
n 9430 3289 m 9482 3318 l gs col0 s gr
% Polyline
n 9502 3330 m 9482 3318 l gs col0 s gr
% Polyline
n 9502 4701 m 9507 4697 l gs col0 s gr
% Polyline
n 9520 4690 m 9507 4697 l gs col0 s gr
% Polyline
n 9520 4690 m 9554 4672 l gs col0 s gr
% Polyline
n 9573 4663 m 9554 4672 l gs col0 s gr
% Polyline
n 9573 3364 m 9520 3336 l 9502 3330 l gs col0 s gr
% Polyline
n 9573 4663 m 9603 4650 l gs col0 s gr
% Polyline
n 9644 4635 m 9603 4650 l gs col0 s gr
% Polyline
n 9573 3364 m 9620 3386 l gs col0 s gr
% Polyline
n 9644 3396 m 9620 3386 l gs col0 s gr
% Polyline
n 9644 4635 m 9653 4630 l gs col0 s gr
% Polyline
n 9686 4619 m 9653 4630 l gs col0 s gr
% Polyline
n 9686 4619 m 9707 4612 l gs col0 s gr
% Polyline
n 9715 4608 m 9707 4612 l gs col0 s gr
% Polyline
n 9715 3420 m 9686 3408 l 9644 3396 l gs col0 s gr
% Polyline
n 9715 4608 m 9763 4598 l gs col0 s gr
% Polyline
n 9785 4594 m 9763 4598 l gs col0 s gr
% Polyline
n 9715 3420 m 9729 3423 l gs col0 s gr
% Polyline
n 9785 3436 m 9729 3423 l gs col0 s gr
% Polyline
n 9785 4594 m 9826 4590 l gs col0 s gr
% Polyline
n 9857 4586 m 9826 4590 l gs col0 s gr
% Polyline
n 9785 3436 m 9815 3438 l gs col0 s gr
% Polyline
n 9857 3442 m 9815 3438 l gs col0 s gr
% Polyline
n 9857 4586 m 9899 4590 l gs col0 s gr
% Polyline
n 9928 4592 m 9899 4590 l gs col0 s gr
% Polyline
n 9857 3442 m 9887 3440 l gs col0 s gr
% Polyline
n 9928 3436 m 9887 3440 l gs col0 s gr
% Polyline
n 9928 4592 m 9985 4606 l gs col0 s gr
% Polyline
n 10000 4609 m 9985 4606 l gs col0 s gr
% Polyline
n 9928 3436 m 9950 3430 l gs col0 s gr
% Polyline
n 10000 3420 m 9950 3430 l gs col0 s gr
% Polyline
n 10069 4638 m 10023 4619 l 10000 4609 l gs col0 s gr
% Polyline
n 10000 3420 m 10007 3416 l gs col0 s gr
% Polyline
n 10023 3408 m 10007 3416 l gs col0 s gr
% Polyline
n 10023 3408 m 10057 3396 l gs col0 s gr
% Polyline
n 10069 3392 m 10057 3396 l gs col0 s gr
% Polyline
n 10069 4638 m 10108 4658 l gs col0 s gr
% Polyline
n 10140 4676 m 10108 4658 l gs col0 s gr
% Polyline
n 10069 3392 m 10104 3372 l gs col0 s gr
% Polyline
n 10140 3354 m 10104 3372 l gs col0 s gr
% Polyline
n 10212 4724 m 10161 4690 l 10140 4676 l gs col0 s gr
% Polyline
n 10140 3354 m 10150 3346 l gs col0 s gr
% Polyline
n 10161 3336 m 10150 3346 l gs col0 s gr
% Polyline
n 10161 3336 m 10192 3318 l gs col0 s gr
% Polyline
n 10212 3307 m 10192 3318 l gs col0 s gr
% Polyline
n 10282 4781 m 10259 4763 l 10212 4724 l gs col0 s gr
% Polyline
n 10212 3307 m 10233 3288 l gs col0 s gr
% Polyline
n 10259 3268 m 10233 3288 l gs col0 s gr
% Polyline
n 10259 3268 m 10271 3256 l gs col0 s gr
% Polyline
n 10282 3250 m 10271 3256 l gs col0 s gr
% Polyline
n 10353 4845 m 10341 4834 l 10282 4781 l gs col0 s gr
% Polyline
n 10282 3250 m 10310 3224 l gs col0 s gr
% Polyline
n 10341 3196 m 10310 3224 l gs col0 s gr
% Polyline
n 10341 3196 m 10348 3188 l gs col0 s gr
% Polyline
n 10353 3186 m 10348 3188 l gs col0 s gr
% Polyline
n 10424 4915 m 10413 4904 l 10353 4845 l gs col0 s gr
% Polyline
n 10353 3186 m 10384 3153 l gs col0 s gr
% Polyline
n 10413 3124 m 10384 3153 l gs col0 s gr
% Polyline
n 10413 3124 m 10419 3119 l gs col0 s gr
% Polyline
n 10424 3114 m 10419 3119 l gs col0 s gr
% Polyline
n 10495 4990 m 10482 4977 l 10424 4915 l gs col0 s gr
% Polyline
n 10424 3114 m 10454 3082 l gs col0 s gr
% Polyline
n 10482 3053 m 10454 3082 l gs col0 s gr
% Polyline
n 10482 3053 m 10490 3045 l gs col0 s gr
% Polyline
n 10495 3039 m 10490 3045 l gs col0 s gr
% Polyline
n 10566 5068 m 10545 5047 l 10495 4990 l gs col0 s gr
% Polyline
n 10495 3039 m 10521 3008 l gs col0 s gr
% Polyline
n 10545 2981 m 10521 3008 l gs col0 s gr
% Polyline
n 10545 2981 m 10556 2971 l gs col0 s gr
% Polyline
n 10566 2959 m 10556 2971 l gs col0 s gr
% Polyline
n 10638 5153 m 10607 5118 l 10566 5068 l gs col0 s gr
% Polyline
n 10566 2959 m 10588 2933 l gs col0 s gr
% Polyline
n 10607 2910 m 10588 2933 l gs col0 s gr
% Polyline
n 10607 2910 m 10621 2894 l gs col0 s gr
% Polyline
n 10638 2876 m 10621 2894 l gs col0 s gr
% Polyline
n 10708 5240 m 10666 5190 l 10638 5153 l gs col0 s gr
% Polyline
n 10638 2876 m 10654 2855 l gs col0 s gr
% Polyline
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% Polyline
n 1162 5143 m 1144 5124 l 1091 5067 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
n 1162 2911 m 1152 2921 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
n 1443 2728 m 1403 2746 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
n 2996 3131 m 2971 3117 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
n 3490 3412 m 3467 3402 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
7.500 slw
n 472 3639 m 6115 1166 l gs col0 s gr
% Polyline
30.000 slw
n 3561 4618 m 3609 4607 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
n 3631 3451 m 3574 3439 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
n 3773 3451 m 3732 3456 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
n 3844 3436 m 3795 3446 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
n 3984 4685 m 3951 4666 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
n 4183 3213 m 4191 3205 l gs col0 s gr
% Polyline
n 4196 3203 m 4191 3205 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
n 4256 3141 m 4226 3170 l gs col0 s gr
% Polyline
n 4256 3141 m 4261 3137 l gs col0 s gr
% Polyline
n 4266 3132 m 4261 3137 l gs col0 s gr
% Polyline
n 4337 4996 m 4324 4983 l 4266 4922 l gs col0 s gr
% Polyline
n 4266 3132 m 4297 3099 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
n 4337 3057 m 4331 3063 l gs col0 s gr
% Polyline
n 4408 5075 m 4387 5054 l 4337 4996 l gs col0 s gr
% Polyline
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% Polyline
n 4387 2999 m 4363 3026 l gs col0 s gr
% Polyline
n 4387 2999 m 4397 2988 l gs col0 s gr
% Polyline
n 4408 2977 m 4397 2988 l gs col0 s gr
% Polyline
n 4478 5160 m 4449 5124 l 4408 5075 l gs col0 s gr
% Polyline
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% Polyline
n 4449 2928 m 4430 2951 l gs col0 s gr
% Polyline
n 4449 2928 m 4461 2913 l gs col0 s gr
% Polyline
n 4478 2895 m 4461 2913 l gs col0 s gr
% Polyline
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% Polyline
n 4478 2895 m 4494 2874 l gs col0 s gr
% Polyline
n 4507 2857 m 4494 2874 l gs col0 s gr
% Polyline
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% Polyline
n 4549 2808 m 4527 2835 l gs col0 s gr
% Polyline
n 4619 5337 m 4563 5267 l 4549 5247 l gs col0 s gr
% Polyline
n 4549 2808 m 4557 2795 l gs col0 s gr
% Polyline
n 4563 2787 m 4557 2795 l gs col0 s gr
% Polyline
n 4563 2787 m 4588 2756 l gs col0 s gr
% Polyline
n 4619 2717 m 4588 2756 l gs col0 s gr
% Polyline
n 4690 5429 m 4672 5409 l 4620 5338 l gs col0 s gr
% Polyline
n 4619 5338 m 4620 5338 l gs col0 s gr
% Polyline
n 4619 5338 m 4619 5337 l gs col0 s gr
% Polyline
n 4619 2717 m 4649 2676 l gs col0 s gr
% Polyline
n 4672 2646 m 4649 2676 l gs col0 s gr
% Polyline
n 4672 2646 m 4681 2636 l gs col0 s gr
% Polyline
n 4690 2623 m 4681 2636 l gs col0 s gr
% Polyline
n 4761 5526 m 4725 5479 l 4690 5429 l gs col0 s gr
% Polyline
n 4690 2623 m 4710 2594 l gs col0 s gr
% Polyline
n 4725 2574 m 4710 2594 l gs col0 s gr
% Polyline
n 4725 2574 m 4741 2554 l gs col0 s gr
% Polyline
n 4761 2529 m 4741 2554 l gs col0 s gr
% Polyline
n 4831 5624 m 4828 5619 l 4822 5610 l gs col0 s gr
% Polyline
n 4777 5549 m 4822 5610 l gs col0 s gr
% Polyline
n 4777 5549 m 4761 5526 l gs col0 s gr
% Polyline
n 4761 2529 m 4770 2514 l gs col0 s gr
% Polyline
n 4777 2502 m 4770 2514 l gs col0 s gr
% Polyline
n 4777 2502 m 4800 2472 l gs col0 s gr
% Polyline
n 4828 2432 m 4800 2472 l gs col0 s gr
% Polyline
n 4828 2432 m 4829 2432 l gs col0 s gr
% Polyline
n 4831 2428 m 4829 2432 l gs col0 s gr
% Polyline
n 4901 5724 m 4877 5692 l 4831 5624 l gs col0 s gr
% Polyline
n 4831 2428 m 4859 2390 l gs col0 s gr
% Polyline
n 4877 2363 m 4859 2390 l gs col0 s gr
% Polyline
n 4877 2363 m 4888 2348 l gs col0 s gr
% Polyline
n 4901 2328 m 4888 2348 l gs col0 s gr
% Polyline
n 4972 5828 m 4927 5763 l 4901 5724 l gs col0 s gr
% Polyline
n 4901 2328 m 4916 2306 l gs col0 s gr
% Polyline
n 4927 2290 m 4916 2306 l gs col0 s gr
% Polyline
n 4927 2290 m 4945 2264 l gs col0 s gr
% Polyline
n 4972 2225 m 4945 2264 l gs col0 s gr
% Polyline
n 5043 5932 m 5023 5903 l 4982 5843 l gs col0 s gr
% Polyline
n 4976 5834 m 4982 5843 l gs col0 s gr
% Polyline
n 4976 5834 m 4972 5828 l gs col0 s gr
% Polyline
n 4972 2225 m 4975 2222 l gs col0 s gr
% Polyline
n 4976 2220 m 4975 2222 l gs col0 s gr
% Polyline
n 4976 2220 m 5002 2180 l gs col0 s gr
% Polyline
n 5023 2149 m 5030 2137 l gs col0 s gr
% Polyline
n 5043 2119 m 5030 2137 l gs col0 s gr
% Polyline
n 5114 6040 m 5069 5975 l 5043 5932 l gs col0 s gr
% Polyline
n 5043 2119 m 5059 2094 l gs col0 s gr
% Polyline
n 5069 2077 m 5059 2094 l gs col0 s gr
% Polyline
n 5069 2077 m 5087 2052 l gs col0 s gr
% Polyline
n 5114 2012 m 5087 2052 l gs col0 s gr
% Polyline
n 5184 6150 m 5162 6117 l 5122 6055 l gs col0 s gr
% Polyline
n 5117 6046 m 5122 6055 l gs col0 s gr
% Polyline
n 5117 6046 m 5114 6040 l gs col0 s gr
% Polyline
n 5114 2012 m 5114 2010 l gs col0 s gr
% Polyline
n 5117 2008 m 5114 2010 l gs col0 s gr
% Polyline
n 5117 2008 m 5143 1966 l gs col0 s gr
% Polyline
n 5162 1938 m 5143 1966 l gs col0 s gr
% Polyline
n 5162 1938 m 5171 1923 l gs col0 s gr
% Polyline
n 5184 1904 m 5171 1923 l gs col0 s gr
% Polyline
n 5254 6261 m 5253 6258 l 5250 6254 l gs col0 s gr
% Polyline
n 5208 6188 m 5250 6254 l gs col0 s gr
% Polyline
n 5208 6188 m 5184 6150 l gs col0 s gr
% Polyline
n 5184 1904 m 5198 1881 l gs col0 s gr
% Polyline
n 5208 1866 m 5198 1881 l gs col0 s gr
% Polyline
n 5208 1866 m 5227 1837 l gs col0 s gr
% Polyline
n 5253 1796 m 5227 1837 l gs col0 s gr
% Polyline
n 5253 1796 m 5254 1794 l gs col0 s gr
% Polyline
n 5325 6372 m 5297 6328 l 5254 6261 l gs col0 s gr
% Polyline
n 5254 1794 m 5280 1752 l gs col0 s gr
% Polyline
n 5297 1724 m 5280 1752 l gs col0 s gr
% Polyline
n 5297 1724 m 5309 1708 l gs col0 s gr
% Polyline
n 5325 1680 m 5309 1708 l gs col0 s gr
% Polyline
n 5396 6487 m 5385 6470 l 5367 6444 l gs col0 s gr
% Polyline
n 5341 6400 m 5367 6444 l gs col0 s gr
% Polyline
n 5341 6400 m 5325 6372 l gs col0 s gr
% Polyline
n 5325 1680 m 5335 1664 l gs col0 s gr
% Polyline
n 5341 1653 m 5335 1664 l gs col0 s gr
% Polyline
n 5341 1653 m 5362 1620 l gs col0 s gr
% Polyline
n 5385 1582 m 5362 1620 l gs col0 s gr
% Polyline
n 5385 1582 m 5390 1576 l gs col0 s gr
% Polyline
n 5396 1565 m 5390 1576 l gs col0 s gr
% Polyline
n 5466 6604 m 5427 6542 l 5396 6487 l gs col0 s gr
% Polyline
n 5396 1565 m 5416 1532 l gs col0 s gr
% Polyline
n 5427 1512 m 5416 1532 l gs col0 s gr
% Polyline
n 5427 1512 m 5443 1489 l gs col0 s gr
% Polyline
n 5466 1450 m 5443 1489 l gs col0 s gr
% Polyline
n 5537 6721 m 5514 6684 l 5479 6625 l gs col0 s gr
% Polyline
n 5471 6614 m 5479 6625 l gs col0 s gr
% Polyline
n 5471 6614 m 5466 6604 l gs col0 s gr
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n 5466 1450 m 5469 1444 l gs col0 s gr
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n 5471 1441 m 5469 1444 l gs col0 s gr
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n 5471 1441 m 5496 1399 l gs col0 s gr
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n 5514 1370 m 5496 1399 l gs col0 s gr
% Polyline
n 5514 1370 m 5522 1355 l gs col0 s gr
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n 5537 1332 m 5522 1355 l gs col0 s gr
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n 5607 6841 m 5597 6825 l 5585 6803 l gs col0 s gr
% Polyline
n 5556 6753 m 5585 6803 l gs col0 s gr
% Polyline
n 5556 6753 m 5537 6721 l gs col0 s gr
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n 5537 1332 m 5549 1312 l gs col0 s gr
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n 5556 1299 m 5549 1312 l gs col0 s gr
% Polyline
n 5556 1299 m 5575 1266 l gs col0 s gr
% Polyline
n 5597 1228 m 5575 1266 l gs col0 s gr
% Polyline
n 5597 1228 m 5602 1223 l gs col0 s gr
% Polyline
n 5607 1213 m 5602 1223 l gs col0 s gr
% Polyline
n 5678 6961 m 5639 6896 l 5607 6841 l gs col0 s gr
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n 5607 1213 m 5628 1179 l gs col0 s gr
% Polyline
n 5639 1156 m 5628 1179 l gs col0 s gr
% Polyline
n 5639 1156 m 5653 1133 l gs col0 s gr
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n 5678 1092 m 5653 1133 l gs col0 s gr
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n 5749 7084 m 5722 7038 l 5685 6975 l gs col0 s gr
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$F2psEnd
rs
%%EndDocument
@endspecial 1426 3072 a Fq(P)2019 3878 y Fo(\000)p Fq(P)2149
2848 y Fo(O)1511 3509 y(O)23 b(\003)e(O)1116 4376 y Fw(Abbildung)27
b(2.6:)32 b(Das)f(in)m(v)m(erse)f(Elemen)m(t)411 4593
y(Wir)g(halten)g(das)g(Resultat)g(in)f(einem)h(Satz)h(fest:)270
4819 y Fs(Satz)k(4)46 b Fu(Die)35 b(Punkte)g(auf)g(einer)h(el)5
b(liptischen)36 b(Kurve)g(bilden)g(mit)g(der)g(ob)-5
b(en)36 b(b)-5 b(eschrie-)270 4932 y(b)g(enen)33 b(A)-5
b(ddition)33 b(eine)g(ab)-5 b(elsche)33 b(Grupp)-5 b(e.)411
5155 y Fw(Als)24 b(n)616 5159 y(\177)616 5155 y(ac)m(hstes)h(sollen)e
(die)g(arithmetisc)m(hen)h(F)-8 b(ormeln)24 b(f)2297
5159 y(\177)2294 5155 y(ur)f(die)g(Berec)m(hn)m(ung)i(der)f(Sum-)270
5268 y(me)d(zw)m(eier)h(Punkte)f(angegeb)s(en)h(w)m(erden.)f(Auc)m(h)g
(hier)f(gehen)i(wir)e(da)m(v)m(on)i(aus,)f(da\031)h(unsere)270
5381 y(Kurv)m(e)31 b Fq(E)i Fw(:)28 b Fq(y)745 5348 y
Fp(2)812 5381 y Fw(=)f Fq(x)962 5348 y Fp(3)1023 5381
y Fw(+)20 b Fq(ax)h Fw(+)g Fq(b)1400 5385 y Fw(\177)1398
5381 y(ub)s(er)30 b Fn(F)1659 5395 y Fl(p)1736 5381 y
Fw(de\014niert)g(ist.)i(Der)g(ausgezeic)m(hnete)h(Punkt)e
Fo(O)270 5494 y Fw(sei)f(der)g(F)-8 b(ernpunkt)29 b(im)g(Unendlic)m
(hen.)411 5609 y(Ist)j Fq(P)41 b Fw(=)28 b(\()p Fq(x)832
5623 y Fp(1)872 5609 y Fq(;)15 b(y)957 5623 y Fp(1)996
5609 y Fw(\))29 b Fo(2)f Fq(E)5 b Fw(\()p Fn(F)1306 5623
y Fl(p)1352 5609 y Fw(\),)32 b(dann)f(gilt)h Fo(\000)p
Fq(P)41 b Fw(=)28 b(\()p Fq(x)2193 5623 y Fp(1)2232 5609
y Fq(;)15 b Fo(\000)p Fq(y)2388 5623 y Fp(1)2427 5609
y Fw(\))29 b Fo(2)f Fq(E)5 b Fw(\()p Fn(F)2737 5623 y
Fl(p)2783 5609 y Fw(\).)32 b(Sei)g(w)m(eiterhin)p eop
%%Page: 12 13
12 12 bop 390 82 a Fw(12)1411 b(KAPITEL)30 b(2.)61 b(ELLIPTISCHE)28
b(KUR)-10 b(VEN)390 415 y Fq(Q)33 b Fw(=)f(\()p Fq(x)685
429 y Fp(2)725 415 y Fq(;)15 b(y)810 429 y Fp(2)850 415
y Fw(\))33 b Fo(2)f Fq(E)5 b Fw(\()p Fn(F)1169 429 y
Fl(p)1214 415 y Fw(\))35 b(mit)g Fq(P)46 b Fo(6)p Fw(=)32
b Fo(\000)p Fq(Q)p Fw(,)j(so)g(k)-5 b(ann)34 b Fq(P)j
Fw(+)23 b Fq(Q)32 b Fw(=)h(\()p Fq(x)2690 429 y Fp(3)2730
415 y Fq(;)15 b(y)2815 429 y Fp(3)2854 415 y Fw(\))35
b(durc)m(h)f(folgende)390 528 y(Sc)m(hritten)c(b)s(erec)m(hnet)g(w)m
(erden:)1239 976 y Fq(\025)83 b Fw(=)1529 735 y Fk(8)1529
817 y(>)1529 844 y(>)1529 871 y(<)1529 1035 y(>)1529
1062 y(>)1529 1089 y(:)1670 784 y Fq(y)1715 798 y Fp(2)1775
784 y Fo(\000)19 b Fq(y)1910 798 y Fp(1)p 1663 825 294
4 v 1663 908 a Fq(x)1715 922 y Fp(2)1775 908 y Fo(\000)g
Fq(x)1917 922 y Fp(1)2058 845 y Fw(:)91 b(falls)34 b
Fq(P)k Fo(6)p Fw(=)25 b Fq(Q)1661 1033 y Fw(3)p Fq(x)1758
1000 y Fp(2)1758 1058 y(1)1818 1033 y Fw(+)20 b Fq(a)p
1661 1074 297 4 v 1744 1157 a Fw(2)p Fq(y)1834 1171 y
Fp(1)2058 1095 y Fw(:)91 b(falls)34 b Fq(P)k Fw(=)25
b Fq(Q)3323 976 y Fw(\(2.5\))1200 1288 y Fq(x)1252 1302
y Fp(3)1375 1288 y Fw(=)83 b Fq(\025)1582 1251 y Fp(2)1641
1288 y Fo(\000)20 b Fq(x)1784 1302 y Fp(1)1844 1288 y
Fo(\000)g Fq(x)1987 1302 y Fp(2)3323 1288 y Fw(\(2.6\))1208
1426 y Fq(y)1253 1440 y Fp(3)1375 1426 y Fw(=)83 b Fq(\025)p
Fw(\()p Fq(x)1669 1440 y Fp(1)1729 1426 y Fo(\000)19
b Fq(x)1871 1440 y Fp(3)1911 1426 y Fw(\))h Fo(\000)g
Fq(y)2102 1440 y Fp(1)3323 1426 y Fw(\(2.7\))531 1621
y(Die)37 b(Berec)m(hn)m(ung)h(v)m(on)g(Vielfac)m(hen)e(eines)h(Punktes)
f(l)2446 1625 y(\177)2446 1621 y(a\031t)i(sic)m(h)f(b)s(esc)m
(hleunigen,)e(in-)390 1734 y(dem)h(man)f(die)g Fu(schnel)5
b(le)39 b(Multiplikation)44 b Fw(v)m(erw)m(endet.)37
b(Sie)e(en)m(tspric)m(h)m(t)h(der)f(b)s(ek)-5 b(ann)m(ten)390
1847 y Fu(schnel)5 b(len)44 b(Exp)-5 b(onentiation)52
b Fw(v)m(on)42 b(Zahlen)f(in)g(Restklassenringen.)g(Um)h(einen)g(Punkt)
390 1960 y Fq(P)49 b Fw(mit)35 b(einer)f(nat)1030 1964
y(\177)1027 1960 y(urlic)m(hen)g(Zahl)h Fq(n)g Fw(zu)g(m)m
(ultiplizieren,)d(b)s(edien)m(t)j(man)g(sic)m(h)g(dab)s(ei)f(der)390
2073 y(Bin)531 2077 y(\177)531 2073 y(ardarstellung)e(v)m(on)k
Fq(n)p Fw(.)f(Im)f(V)-8 b(ergleic)m(h)36 b(zur)e(fortlaufenden)g
(Addition)f(des)i(Punktes)390 2185 y Fq(P)13 b Fw(,)40
b(reic)m(hen)f(n)m(un)g(sc)m(hon)h(w)m(esen)m(tlic)m(h)g(w)m(eniger)f
(\(in)g(der)g(Gr)2501 2189 y(\177)2501 2185 y(o\031enordn)m(ung)h(v)m
(on)g Fq(l)r(og)s Fw(\()p Fq(n)p Fw(\)\))390 2298 y(Additionen)28
b(aus.)390 2583 y Fz(2.3)135 b(Kryptographisc)l(h)46
b(geeignete)h(Kurv)l(en)390 2786 y Fw(Es)26 b(stellt)h(sic)m(h)f(n)m
(un)g(die)f(F)-8 b(rage,)29 b(wie)d(elliptisc)m(he)f(Kurv)m(en)h(gew)
2552 2790 y(\177)2552 2786 y(ahlt)h(w)m(erden)f(m)3122
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3238 y(che)35 b Fw(elliptisc)m(he)27 b(Kurv)m(en)g(erk)m(ennen)i(und)d
(damit)i(aussortieren)g(k)-5 b(ann.)28 b(Die)h(meisten)f(die-)390
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(hen)d(Kurv)m(en)h(b)s(esonders)g(geeignet)j(bzw.)e(b)s(esonders)390
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(en)2732 4032 y(\177)2730 4028 y(ub)s(er)e(die)h(Gr)3212
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y(ab)s(er)27 b(auc)m(h)i(die)e(E\016zienz.)g(Hier)h(m)m(u\031)g(man)f
(also)h(abh)2257 4936 y(\177)2257 4932 y(angig)f(v)m(on)i(der)e(Aufgab)
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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n 1639 3708 m 1700 3686 l gs col0 s gr
% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
n 3794 2461 m 3787 2464 l gs col0 s gr
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% Polyline
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% Polyline
n 3922 2371 m 3897 2385 l gs col0 s gr
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% Polyline
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% Polyline
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% Polyline
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% Polyline
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% Polyline
n 4432 1984 m 4482 1948 l gs col0 s gr
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% Polyline
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% Polyline
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% Polyline
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n 4560 1885 m 4585 1860 l gs col0 s gr
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% Polyline
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$F2psEnd
rs
%%EndDocument
@endspecial 2928 1968 a Fq(x)2262 1231 y(y)1711 1968
y(x)601 1231 y(y)963 2902 y Fw(Abbildung)d(2.7:)32 b(Singul)1832
2906 y(\177)1832 2902 y(are)c(elliptisc)m(he)g(Kurv)m(en)411
3135 y(In)34 b(b)s(ezug)g(auf)g(kryptographisc)m(he)g(An)m(w)m
(endungen)g(ist)g(es)h(n)m(un)e(m)2760 3139 y(\177)2760
3135 y(oglic)m(h,)h(das)h Fu(Pr)-5 b(o-)270 3248 y(blem)29
b(des)h(diskr)-5 b(eten)30 b(L)-5 b(o)g(garithmus)36
b Fw(\(siehe)26 b(hierzu)g(Kapitel)f(3.1\))j(b)s(ei)d(singul)2936
3252 y(\177)2936 3248 y(aren)f(ellipti-)270 3360 y(sc)m(hen)h(Kurv)m
(en)e(in)g(den)h(un)m(tergeordneten)h(K)1833 3364 y(\177)1833
3360 y(orp)s(er)e(abzubilden.)f(Dort)j(ist)f(es)h(v)m(ergleic)m(hs-)270
3473 y(w)m(eise)32 b(einfac)m(h)h(zu)f(l)967 3477 y(\177)967
3473 y(osen,)g(da)h(man)f(sic)m(h)g(durc)m(h)f(die)h(V)-8
b(erw)m(endung)32 b(elliptisc)m(her)e(Kurv)m(en)270 3586
y(ja)h(f)405 3590 y(\177)402 3586 y(ur)e(relativ)g(kleine)g(K)1134
3590 y(\177)1134 3586 y(orp)s(er)h(en)m(tsc)m(heiden)g(k)m(onn)m(te.)
270 3807 y Fs(De\014nition)35 b(6)46 b Fu(Eine)37 b(el)5
b(liptische)38 b(Kurve)1784 3811 y(\177)1784 3807 y(ub)-5
b(er)38 b(dem)g(K)2257 3811 y(\177)2258 3807 y(orp)-5
b(er)39 b Fn(F)2551 3821 y Fl(p)2635 3807 y Fu(hei\031t)e(sup)-5
b(ersingul)3303 3811 y(\177)3304 3807 y(ar)270 3920 y(genau)33
b(dann,)g(wenn)g Fo(j)p Fq(E)5 b Fw(\()p Fn(F)1208 3934
y Fl(p)1253 3920 y Fw(\))p Fo(j)26 b(\021)f Fw(1)h(mo)s(d)e
Fq(p)p Fu(.)411 4140 y Fw(Damit)30 b(gilt)f(insb)s(esondere)f
Fq(t)d Fw(=)g(0)30 b(in)f Fn(F)1772 4154 y Fl(p)1817
4140 y Fw(,)h(w)m(as)h(nic)m(h)m(ts)e(anderes)h(hei\031t,)g(als)f
(da\031)i(diese)270 4253 y(Kurv)m(en)j(die)f(Spur)g(0)i(hab)s(en.)e
(Nun)h(sind)e(nic)m(h)m(t)j(n)m(ur)e(sup)s(ersingul)2601
4257 y(\177)2601 4253 y(are)f(Kurv)m(en)h(sondern)270
4366 y(auc)m(h)39 b(Kurv)m(en)g(mit)f(Spur)f(1)j(o)s(der)e(2)i(anf)1734
4370 y(\177)1734 4366 y(allig)d(f)1970 4370 y(\177)1967
4366 y(ur)h(div)m(erse)g(A)m(ttac)m(k)m(en.)43 b(F)-8
b(erner)39 b(gibt)g(es)270 4478 y(die)32 b(Klasse)h(der)f(sogenann)m
(ten)i Fu(Koblitz)11 b Fw(-Kurv)m(en,)34 b(die)e(ein)g(Sp)s(ezialfall)e
(der)i Fu(T)-7 b(eilk)3142 4482 y(\177)3143 4478 y(orp)i(er)10
b Fw(-)270 4591 y(Kurv)m(en)22 b(sind.)e(Ob)m(w)m(ohl)i(es)h(b)s(ei)e
(diesen)g(m)1702 4595 y(\177)1702 4591 y(oglic)m(h)h(ist,)g(eine)g
(geringf)2551 4595 y(\177)2548 4591 y(ugige)g(Besc)m(hleunigung)270
4704 y(b)s(ei)32 b(der)g(Berec)m(hn)m(ung)i(diskreter)d(Logarithmen)i
(zu)g(erreic)m(hen,)f(w)m(erden)h(sie)f(v)m(on)i(einem)270
4817 y(T)-8 b(eil)24 b(der)g(F)-8 b(ac)m(h)m(w)m(elt)27
b(w)m(eiterhin)c(als)h(geeignet)i(angesehen.)g(Eine)d(genauere)j(Besc)m
(hreibung)270 4930 y(w)339 4934 y(\177)336 4930 y(urde)e(allerdings)946
4934 y(\177)944 4930 y(ub)s(er)g(den)h(Rahmen)g(dieser)g(Arb)s(eit)g
(hinausgehen.)f(Der)i(in)m(teressierte)270 5043 y(Leser)k(sei)g(auf)g
([Hus87)q(])h(und)d([Ham98)r(])i(v)m(erwiesen.)411 5157
y(Es)21 b(gibt)f(im)g(w)m(esen)m(tlic)m(hen)h(zw)m(ei)h(Strategien,)f
(eine)g(geeignete)h(Kurv)m(e)f(zu)g(\014nden.)f(Man)270
5270 y(k)-5 b(ann)29 b(einerseits)f(aufgrund)f(der)i(Kriterien)f(v)m
(ersuc)m(hen,)h(zu)g(einer)g Fu(guten)36 b Fw(Punkteanzahl)270
5383 y(eine)41 b Fu(gute)49 b Fw(Kurv)m(e)41 b(zu)h(\014nden.)e(Diese)i
Fu(Konstruktion)j(mit)e(komplexer)i(Multiplikation)270
5496 y Fw(hat)29 b(den)g(V)-8 b(orteil,)29 b(da\031)g(man)g(sc)m(hon)g
(b)s(ei)e(der)i(Punkteanzahl)f(sagen)h(k)-5 b(ann,)29
b(ob)g(A)m(ttac)m(k)m(en)270 5609 y(auf)j(die)g(Grupp)s(enstruktur)c
(mit)k(Algorithmen)f(wie)h Fu(Pohlig-Hel)5 b(lman)p Fw(,)33
b Fu(Pol)5 b(lar)-5 b(d-)p Fq(\032)34 b Fw(o)s(der)p
eop
%%Page: 14 15
14 14 bop 390 82 a Fw(14)1411 b(KAPITEL)30 b(2.)61 b(ELLIPTISCHE)28
b(KUR)-10 b(VEN)390 415 y Fu(Index-Calculus)40 b Fw(erfolgreic)m(h)32
b(sein)g(k)1688 419 y(\177)1688 415 y(onn)m(ten.)g(Daf)2160
419 y(\177)2157 415 y(ur)g(m)m(u\031)h(man)f(dann)f(allerdings)f(no)s
(c)m(h)390 528 y(eine)37 b(Kurv)m(e)g(dazu)h(\014nden,)e(b)s(ei)g(der)h
(auc)m(h)h(die)f(Kennzahlen)f(wie)h(Diskriminan)m(te)e(und)390
641 y(Spur)28 b(stimmen.)531 754 y(Andererseits)c(b)s(esteh)m(t)h(die)f
(M)1587 758 y(\177)1587 754 y(oglic)m(hk)m(eit)h(der)f
Fu(r)-5 b(andomisierten)30 b(Konstruktion)p Fw(.)d(Man)390
867 y(gibt)f(sic)m(h)g(dab)s(ei)g(eine)g(Kurv)m(e)g(mit)g
Fu(guten)33 b Fw(Kennzahlen)26 b(v)m(or,)h(b)s(erec)m(hnet)g(die)f
(Anzahl)f(der)390 980 y(Punkte)g(und)f(testet)k(sc)m(hlie\031lic)m(h,)c
(ob)i(die)e(ob)s(engenann)m(ten)i(Algorithmen)e(erfolglos)h(blei-)390
1093 y(b)s(en.)31 b(Ist)g(das)g(der)g(F)-8 b(all,)31
b(so)h(k)-5 b(ann)31 b(diese)g(Kurv)m(e)g(in)f(kryptographisc)m(hen)g
(An)m(w)m(endungen)390 1205 y(b)s(en)m(utzt)g(w)m(erden.)531
1318 y(Hinsic)m(h)m(tlic)m(h)f(der)h(E\016zienz)g(ist)g(die)g
(Konstruktion)f(mit)h(der)h(k)m(omplexen)f(Multipli-)390
1431 y(k)-5 b(ation)37 b(dem)g(zw)m(eiten)g(V)-8 b(erfahren)37
b(v)m(orzuziehen,)g(w)m(as)g(haupts)2609 1435 y(\177)2609
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(Berec)m(hn)m(ung)i(der)e(Punktanzahl)g(b)s(ei)g(der)h(randomisierten)d
(Konstruktion)390 1657 y(liegt.)e(Eine)g(detaillierte)f(Gegen)1552
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1774 y(\177)2381 1770 y(autert)h(als)g(auc)m(h)h(die)e(E\016zienz)390
1883 y(v)m(erglic)m(hen.)p eop
%%Page: 15 16
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(Sinn,)f(die)h(not)m(w)m(endigen)h(Kenn)m(tnisse)2705
2588 y(\177)2702 2584 y(ub)s(er)f(V)-8 b(ersc)m(hl)3200
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3036 y(liptisc)m(hen)k(Kurv)m(en)h(f)1053 3040 y(\177)1050
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(w)m(erden.)270 3447 y Fz(3.1)135 b(Das)46 b(ElGamal-V)-11
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3880 y(Grupp)s(en)k(zu)h(b)s(erec)m(hnen.)g(Bev)m(or)i(das)f(V)-8
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3993 y(w)m(erden,)39 b(w)m(as)i(das)e(DLP)h(aussagt.)h(Die)e(folgenden)
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390 867 y(P)m(ohlig-Hellman,)e(P)m(ollard-)p Fq(\032)g
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y(nieren.)f(Auc)m(h)h(hier)f(gehen)i(wir)e(v)m(on)h(einer)g(b)s
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1215 y(aus)c(und)735 1219 y(\177)732 1215 y(ub)s(ernehmen)e(die)i
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1780 y(eine)e(Zahl)f(zu)h(mac)m(hen.)i(Diese)e(und)f(viele)h(andere)g
(Grundlagen)f(der)h(Kryptographie)390 1893 y(sind)28
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2623 y Fq(K)g Fw(=)25 b(\()p Fq(G;)15 b(\013;)g(a;)g(\014)5
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2364 y(\177)2898 2360 y(al)5 b(lig)32 b(gew)3240 2364
y(\177)3241 2360 y(ahl-)270 2473 y(ten)f(Zahl)i Fq(k)28
b Fo(2)d Fn(Z)854 2491 y Fh(j)p Fl(H)5 b Fh(j)987 2473
y Fu(und)32 b(der)g(Nachricht)g Fq(m)25 b Fw(=)g(\()p
Fq(m)2057 2487 y Fp(1)2097 2473 y Fq(;)15 b(m)2217 2487
y Fp(2)2256 2473 y Fw(\))32 b Fu(mit)g Fq(m)2567 2487
y Fp(1)2606 2473 y Fq(;)15 b(m)2726 2487 y Fp(2)2791
2473 y Fo(2)25 b Fn(F)2926 2487 y Fl(p)3003 2473 y Fu(de\014nier)-5
b(en)270 2586 y(wir)1392 2719 y Fq(e)1434 2733 y Fl(K)1502
2719 y Fw(\()p Fq(m;)15 b(k)s Fw(\))27 b(=)e(\()p Fq(y)1945
2733 y Fp(0)1984 2719 y Fq(;)15 b(y)2069 2733 y Fp(1)2109
2719 y Fq(;)g(y)2194 2733 y Fp(2)2233 2719 y Fw(\))935
b(\(3.7\))270 2910 y Fu(mit)1673 3043 y Fq(y)1718 3057
y Fp(0)1782 3043 y Fw(=)25 b Fq(k)s(\013)1217 b Fw(\(3.8\))1582
3255 y(\()p Fq(c)1656 3269 y Fp(1)1696 3255 y Fq(;)15
b(c)1775 3269 y Fp(2)1815 3255 y Fw(\))26 b(=)f Fq(k)s(\014)1130
b Fw(\(3.9\))1628 3466 y Fq(y)1673 3480 y Fp(1)1738 3466
y Fw(=)25 b Fq(c)1873 3480 y Fp(1)1912 3466 y Fq(m)1992
3480 y Fp(1)3158 3466 y Fw(\(3.10\))270 3657 y Fu(und)1616
3791 y Fq(y)1661 3805 y Fp(2)1725 3791 y Fw(=)g Fq(c)1860
3805 y Fp(2)1900 3791 y Fq(m)1980 3805 y Fp(2)2019 3791
y Fq(:)1114 b Fw(\(3.11\))270 3982 y Fu(F)329 3986 y(\177)329
3982 y(ur)33 b(eine)f(verschl)914 3986 y(\177)914 3982
y(usselte)i(Nachricht)f Fq(y)28 b Fw(=)d(\()p Fq(y)1876
3996 y Fp(0)1915 3982 y Fq(;)15 b(y)2000 3996 y Fp(1)2040
3982 y Fq(;)g(y)2125 3996 y Fp(2)2164 3982 y Fw(\))33
b Fu(sei)1379 4207 y Fq(d)1426 4221 y Fl(K)1495 4207
y Fw(\()p Fq(y)s Fw(\))26 b(=)e(\()p Fq(y)1814 4221 y
Fp(1)1854 4207 y Fq(c)1893 4168 y Fh(\000)p Fp(1)1893
4233 y(1)1988 4207 y Fq(;)15 b(y)2073 4221 y Fp(2)2112
4207 y Fq(c)2151 4168 y Fh(\000)p Fp(1)2151 4233 y(2)2246
4207 y Fw(\))877 b(\(3.12\))270 4431 y Fu(mit)1556 4565
y Fq(ay)1649 4579 y Fp(0)1714 4565 y Fw(=)25 b(\()p Fq(c)1884
4579 y Fp(1)1924 4565 y Fq(;)15 b(c)2003 4579 y Fp(2)2043
4565 y Fw(\))p Fq(:)1055 b Fw(\(3.13\))411 4812 y(W)-8
b(enn)32 b(man)f(b)s(eac)m(h)m(tet,)j(da\031)e(man)g(sic)m(h)f(in)f
(einer)h(additiv)m(en)g(Grupp)s(e)f(b)s(e\014ndet,)g(b)s(e-)270
4925 y(steh)m(t)45 b(der)e(Nac)m(h)m(w)m(eis)i(der)e(Korrektheit)h(des)
g(obigen)f(Kryptosystems)h(in)e(der)3174 4906 y(\177)3162
4925 y(Ub)s(er-)270 5038 y(pr)360 5042 y(\177)357 5038
y(ufung)28 b(v)m(on)j(Gleic)m(h)m(ung)f(\(3.6\))i(f)1456
5042 y(\177)1453 5038 y(ur)d Fq(m)1649 5052 y Fp(1)1719
5038 y Fw(und)f Fq(m)1980 5052 y Fp(2)2020 5038 y Fw(.)411
5157 y(Dar)564 5161 y(\177)561 5157 y(ub)s(erhinaus)c(m)m(u\031)i(no)s
(c)m(h)h(gesagt)h(w)m(erden,)e(da\031)h(in)e(der)h(Gleic)m(h)m(ung)g
(\(3.9\))i(die)e(b)s(ei-)270 5270 y(den)32 b(Ko)s(ordinaten)f(v)m(on)i
Fq(k)s(\014)5 b Fw(,)33 b(d.)15 b(h.)32 b Fq(c)1550 5284
y Fp(1)1622 5270 y Fw(und)f Fq(c)1845 5284 y Fp(2)1885
5270 y Fw(,)h(nat)2076 5274 y(\177)2073 5270 y(urlic)m(h)f(ungleic)m(h)
g(0,)i(also)g(in)e Fn(F)3135 5237 y Fh(\003)3135 5293
y Fl(p)3180 5270 y Fw(,)i(sein)270 5383 y(m)349 5387
y(\177)346 5383 y(ussen,)28 b(um)f(die)h(In)m(v)m(ertierung)f(zu)i(erm)
1697 5387 y(\177)1697 5383 y(oglic)m(hen.)f(In)g(der)f(Praxis)h(tritt)g
(der)g(nic)m(h)m(tin)m(v)m(er-)270 5496 y(tierbare)37
b(F)-8 b(all)36 b(jedo)s(c)m(h)h(b)s(ei)f(zuf)1367 5500
y(\177)1367 5496 y(allig)f(gew)1721 5500 y(\177)1721
5496 y(ahltem)j Fq(k)i Fw(mit)c(hinreic)m(hend)f(geringer)h(W)-8
b(ahr-)270 5609 y(sc)m(heinlic)m(hk)m(eit)29 b(auf)h(und)f(k)-5
b(ann)30 b(in)f(einem)g(Programm)i(abgefangen)g(w)m(erden.)p
eop
%%Page: 18 19
18 18 bop 390 82 a Fw(18)1519 b(KAPITEL)29 b(3.)62 b(DER)30
b(ALGORITHMUS)390 415 y Fz(3.3)135 b(Alternativ)l(en)390
618 y Fw(Es)32 b(wurde)f(sc)m(hon)h(im)f(v)m(orangegangenen)j(Absc)m
(hnitt)d(erw)2452 622 y(\177)2452 618 y(ahn)m(t,)i(da\031)f(auc)m(h)h
(andere)f(V)-8 b(a-)390 731 y(rian)m(ten)30 b(der)h(V)-8
b(ersc)m(hl)1148 735 y(\177)1145 731 y(usselung)29 b(mit)h(elliptisc)m
(hen)f(Kurv)m(en)h(denkbar)g(sind.)f(Diese)i(sollen)390
844 y(hier)26 b(nic)m(h)m(t)i(im)e(Detail)i(v)m(orgestellt)h(sondern)d
(n)m(ur)h(kurz)g(angespro)s(c)m(hen)g(w)m(erden.)h(F)3283
848 y(\177)3280 844 y(ur)f(die)390 957 y(genaue)g(Besc)m(hreibung)e
(wird)f(an)h(en)m(tsprec)m(hender)h(Stelle)f(auf)h(die)f(Literatur)g(v)
m(erwiesen.)531 1070 y(Zuerst)20 b(sollte)g(gesagt)i(w)m(erden,)e
(da\031)g(b)s(ei)f(dem)h(V)-8 b(erfahren)20 b(v)m(on)h(Menezes)h(und)c
(V)-8 b(ansto-)390 1183 y(ne)30 b(b)s(eide)f(Ko)s(ordinaten)h(des)1437
1187 y(\177)1437 1183 y(o\013en)m(tlic)m(hen)g(Sc)m(hl)2087
1187 y(\177)2084 1183 y(ussels)f(v)m(erw)m(endet)i(w)m(erden.)f(Diese)h
(sind)390 1296 y(k)m(einesw)m(egs)36 b(unabh)1097 1300
y(\177)1097 1296 y(angig)e(v)m(oneinander,)h(w)m(as)h(die)f(Sic)m
(herheit)f(b)s(eein)m(tr)2922 1300 y(\177)2922 1296 y(ac)m(h)m(tigen)i
(k)3333 1300 y(\177)3333 1296 y(onn-)390 1408 y(te.)j(Das)g(Problem)d
(ist)i(jedo)s(c)m(h,)g(da\031)g(b)s(ei)f(der)g(Ben)m(utzung)i(n)m(ur)e
(einer)g(Ko)s(ordinate)g(die)390 1521 y(Nac)m(hric)m(h)m(t)28
b(auf)e(die)g(vierfac)m(he)g(L)1528 1525 y(\177)1528
1521 y(ange)h(expandiert)f(w)2254 1525 y(\177)2251 1521
y(urde,)f(w)m(as)i(nic)m(h)m(t)g(sehr)f(praktik)-5 b(ab)s(el)390
1634 y(ist.)531 1747 y(Nat)682 1751 y(\177)679 1747 y(urlic)m(h)55
b(ist)h(auc)m(h)g(die)f(V)-8 b(erw)m(endung)56 b(v)m(on)g(K)2338
1751 y(\177)2338 1747 y(orp)s(ern)e(der)i(Charakteristik)f(2)390
1860 y(m)466 1864 y(\177)466 1860 y(oglic)m(h,)28 b(da)f(sie)h(v)m(or)g
(allem)f(b)s(ei)g(Hardw)m(are-Implemen)m(tierungen)f(V)-8
b(orteile)28 b(in)f(der)g(E\016-)390 1973 y(zienz)g(bieten.)f(Man)h
(denk)m(e)g(hierb)s(ei)d(z.)16 b(B.)27 b(an)f(Smartcards.)h(Eine)e
(sinn)m(v)m(olle)g(Gr)3166 1977 y(\177)3166 1973 y(o\031enord-)390
2086 y(n)m(ung)32 b(liegt)h(auc)m(h)g(hier)f(b)s(ei)f(et)m(w)m(a)k(200)
f(Bit.)f(Man)g(\014ndet)f(in)g([Ros99)q(])h(eine)f(ausf)3197
2090 y(\177)3194 2086 y(uhrlic)m(he)390 2199 y(Betrac)m(h)m(tung)f(v)m
(on)f(elliptisc)m(hen)d(Kurv)m(en)1875 2203 y(\177)1873
2199 y(ub)s(er)h Fn(F)2131 2213 y Fp(2)2167 2194 y Fm(n)2219
2199 y Fw(.)i(Dab)s(ei)f(wird)e(auc)m(h)j(auf)f(die)f(Imple-)390
2312 y(men)m(tierung)i(v)m(on)g(Kryptosystemen)g(n)1768
2316 y(\177)1768 2312 y(aher)g(eingegangen.)531 2425
y(Es)f(wurde)g(sc)m(hon)g(mehrfac)m(h)h(der)f(Asp)s(ekt)g(der)g(Ein)m
(b)s(ettung)f(der)h(Nac)m(hric)m(h)m(t)i(auf)e(die)390
2538 y(Kurv)m(e)23 b(angespro)s(c)m(hen.)g(Man)g(suc)m(h)m(t)g(hierf)
1851 2542 y(\177)1848 2538 y(ur)e(eine)h(injektiv)m(e)h(Abbildung)c(v)m
(on)k(der)g(Menge)390 2650 y(der)i(Nac)m(hric)m(h)m(ten)i(in)e(die)g
(Menge)i(der)e(Kurv)m(enpunkte.)g(Zur)f(Zeit)h(ist)h(k)m(ein)f
(deterministi-)390 2763 y(sc)m(her)k(Algorithm)m(us)d(b)s(ek)-5
b(ann)m(t,)29 b(der)e(das)h(in)f(p)s(olynomieller)e(Zeit)j(leistet.)h
(In)e([Kob94)q(])i(ist)390 2876 y(jedo)s(c)m(h)h(ein)g(probabilistisc)m
(her)c(Algorithm)m(us)j(dargestellt,)i(der)f(f)2682 2880
y(\177)2679 2876 y(ur)f(elliptisc)m(he)f(Kurv)m(en)393
2993 y(\177)390 2989 y(ub)s(er)j(b)s(eliebigen)e(endlic)m(hen)h(K)1504
2993 y(\177)1504 2989 y(orp)s(ern)g(funktioniert.)h(Im)g(F)-8
b(all)32 b Fn(F)2693 3003 y Fl(p)2771 2989 y Fw(b)s(esteh)m(t)g(eine)g
(direk-)390 3102 y(te)43 b(Bezieh)m(ung)f(zwisc)m(hen)f(der)h(Nac)m
(hric)m(h)m(t)h(und)d(der)i Fq(x)p Fw(-Ko)s(ordinate)f(des)h(zugeh)3267
3106 y(\177)3267 3102 y(origen)390 3215 y(Punktes.)p
eop
%%Page: 19 20
19 19 bop 270 1043 a Fr(Kapitel)66 b(4)270 1463 y Fv(Die)77
b(Implemen)-6 b(tierung)270 1913 y Fw(Es)41 b(soll)f(n)m(un)g(der)g
(Pro)m(vider)g(f)1351 1917 y(\177)1348 1913 y(ur)g(die)g(V)-8
b(ersc)m(hl)1921 1917 y(\177)1918 1913 y(usselung)40
b(mit)g(elliptisc)m(hen)f(Kurv)m(en)h(im)270 2026 y(einzelnen)26
b(v)m(orgestellt)h(w)m(erden.)g(Die)g(JA)-10 b(V)g(A)27
b(Cryptograph)m(y)g(Arc)m(hitecture)g(als)g(zugrun-)270
2139 y(deliegende)34 b(Struktur)f(und)h(andere)h(Asp)s(ekte)g(v)m(on)g
(JA)-10 b(V)g(A)35 b(bilden)e(dab)s(ei)g(den)i(Anfang.)270
2252 y(Bev)m(or)45 b(dann)e(der)g(Zugri\013)f(auf)h(C++-Co)s(de)g(erl)
2020 2256 y(\177)2020 2252 y(autert)g(wird,)f(widmet)h(sic)m(h)g(ein)g
(Ab-)270 2365 y(sc)m(hnitt)33 b(dem)g(Standard)f(der)h(Public)e(Key)j
(Kryptographie,)e(der)h(so)g(w)m(eit)h(wie)e(m)3160 2369
y(\177)3160 2365 y(oglic)m(h)270 2478 y(b)s(ei)38 b(der)g(En)m(t)m(wic)
m(klung)g(b)s(er)1256 2482 y(\177)1253 2478 y(uc)m(ksic)m(h)m(tigt)h
(wurde.)f(Zuletzt)h(soll)f(die)g(Struktur)f(des)i(Pro-)270
2591 y(viders)29 b(dargestellt)h(w)m(erden,)g(w)m(oran)g(sic)m(h)g
(eine)g(Besc)m(hreibung)f(der)h(Programmsc)m(hnitt-)270
2704 y(stelle)k(ansc)m(hlie\031t.)h(Die)f(in)m(terne)g(F)-8
b(unktionsw)m(eise)34 b(wird)f(innerhalb)e(des)j(Quellco)s(des)f(in)270
2817 y(den)d(Anh)612 2821 y(\177)612 2817 y(angen)g(b)s(esc)m(hrieb)s
(en.)270 3133 y Fz(4.1)135 b(Die)46 b(JA)-15 b(V)g(A)44
b(Cryptograph)l(y)h(Arc)l(hitecture)270 3346 y Fw(Die)33
b(v)m(on)g(der)f(Firma)g Fg(SUN)g Fw(en)m(t)m(wic)m(k)m(elte)i
(Programmiersprac)m(he)e(JA)-10 b(V)g(A)33 b(ist)f(dab)s(ei,)g(sic)m(h)
270 3459 y(in)d(vielen)g(Bereic)m(hen)i(der)e(Soft)m(w)m(are-En)m(t)m
(wic)m(klung)i(durc)m(hzusetzen.)g(Durc)m(h)f(die)f(Eigen-)270
3571 y(sc)m(haft)37 b(der)e(Plattform)m(unabh)1335 3575
y(\177)1335 3571 y(angigk)m(eit)g(ist)h(sie)f(f)2029
3575 y(\177)2026 3571 y(ur)g(An)m(w)m(endungen,)g(die)g(sic)m(h)g(mit)g
(In-)270 3684 y(formationssic)m(herheit)i(b)s(esc)m(h)1326
3688 y(\177)1326 3684 y(aftigen,)i(b)s(esonders)e(geeignet.)j(Dab)s(ei)
f(denkt)f(man)h(z.)15 b(B.)270 3797 y(an)37 b(eine)f(sic)m(here)910
3778 y(\177)898 3797 y(Ub)s(ermittlung)f(v)m(on)i(E-Mails)f(o)s(der)g
(an)h(eine)f(Ben)m(utzerauthen)m(ti\014k)-5 b(a-)270
3910 y(tion.)38 b(JA)-10 b(V)g(A)38 b(b)s(einhaltet)e(auc)m(h)i(als)g
(erste)g(Sprac)m(he)g(ein)2296 3914 y(\177)2293 3910
y(ub)s(erzeugendes)f(Konzept)h(f)3307 3914 y(\177)3304
3910 y(ur)270 4023 y(die)j(In)m(tegration)i(v)m(on)g
Fu(Sicherheit)p Fw(.)g(Un)m(ter)g(diesem)e(Stic)m(h)m(w)m(ort)i(v)m
(ersteh)m(t)h(man)e(so)m(w)m(ohl)270 4136 y(die)37 b(Sic)m(herheit)f
(in)g(b)s(ezug)i(auf)f(die)g(v)m(erw)m(endete)i(Plattform)e(als)g(auc)m
(h)h(die)f(in)f(die)h(An-)270 4249 y(w)m(endung)25 b(selbst)h
(eingebaute)g(Sic)m(herheit.)f(Mit)h(dem)g(zw)m(eiten)h(Asp)s(ekt)f(b)s
(esc)m(h)3000 4253 y(\177)3000 4249 y(aftigt)h(sic)m(h)270
4362 y(diese)j(Arb)s(eit.)f(Mehr)1049 4366 y(\177)1046
4362 y(ub)s(er)g(Plattformsic)m(herheit)g(\014ndet)h(man)g(in)f([Oak98)
q(].)411 4480 y(Die)38 b(hier)e(v)m(erw)m(endete)j(JA)-10
b(V)g(A-Klassen)m(bibliothek,)35 b(das)i Fu(JA)-9 b(V)g(A)37
b(Development)j(Kit)270 4593 y(\(JDK\))59 b(1.2)p Fw(,)i(stellt)f(ein)f
(F)-8 b(ramew)m(ork)61 b(f)1762 4597 y(\177)1759 4593
y(ur)e(kryptographisc)m(he)g(An)m(w)m(endungen)g(zur)270
4706 y(V)-8 b(erf)437 4710 y(\177)434 4706 y(ugung.)34
b(Dieses)g(gliedert)f(sic)m(h)h(in)e(die)h Fu(JA)-9 b(V)g(A)34
b(Crypto)-5 b(gr)g(aphy)40 b(A)n(r)-5 b(chite)g(ctur)g(e)36
b(\(JCA\))270 4819 y Fw(und)30 b(die)g Fu(JA)-9 b(V)g(A)31
b(Crypto)-5 b(gr)g(aphy)37 b(Extension)d(\(JCE\))p Fw(.)d(In)g(der)f
(JCA)h(b)s(e\014nden)e(sic)m(h)h(Klas-)270 4932 y(sen,)41
b(die)f(sic)m(h)g(auf)h(die)f(Sc)m(hl)1300 4936 y(\177)1297
4932 y(ussel)f(und)g(deren)h(Erzeugung,)h(auf)g(digitale)e(Signaturen)
270 5044 y(o)s(der)23 b(auc)m(h)h(Pseudozufallszahlen)d(b)s(eziehen.)i
(Bei)g(der)g(JCE)g(handelt)f(es)i(sic)m(h)f(im)g(w)m(esen)m(t-)270
5157 y(lic)m(hen)31 b(um)g(die)g(Sc)m(hnittstelle)g(f)1389
5161 y(\177)1386 5157 y(ur)f(V)-8 b(ersc)m(hl)1792 5161
y(\177)1789 5157 y(usselungsalgorithmen.)30 b(Die)i(un)m(tersc)m
(hiedli-)270 5270 y(c)m(hen)23 b(Anf)619 5274 y(\177)619
5270 y(ange)f(der)g(P)m(ak)m(etnamen,)j Ff(java.security)18
b Fw(b)s(ei)k(der)f(JCA)h(und)f Ff(javax.crypto)270 5383
y Fw(b)s(ei)h(der)g(JCE,)h(deuten)g(sc)m(hon)g(auf)g(eine)g
(Besonderheit)g(dieser)f(Arc)m(hitektur)g(hin.)g(Die)h(Re-)270
5496 y(gierung)29 b(der)g(USA)h(hat)g(den)f(Exp)s(ort)g(v)m(on)h(V)-8
b(ersc)m(hl)2067 5500 y(\177)2064 5496 y(usselungssoft)m(w)m(are)30
b(b)s(esc)m(hr)3045 5500 y(\177)3045 5496 y(ankt,)f(so-)270
5609 y(da\031)24 b(man)f(auf)g(JCE-Implemen)m(tierungen)d(v)m(on)k
(anderer)f(Seite)g(zur)2599 5613 y(\177)2596 5609 y(uc)m(kgreifen)g(m)m
(u\031.)g(Sol-)1785 5858 y(19)p eop
%%Page: 20 21
20 20 bop 390 82 a Fw(20)1313 b(KAPITEL)30 b(4.)61 b(DIE)30
b(IMPLEMENTIER)m(UNG)390 415 y(c)m(he)i(existieren)e(z.)15
b(B.)32 b(v)m(on)f(der)g(T)-8 b(ec)m(hnisc)m(hen)30 b(Univ)m(ersit)2360
419 y(\177)2360 415 y(at)h(Graz)h(in)2814 396 y(\177)2801
415 y(Osterreic)m(h)f(\(IAIK-)390 528 y(Implemen)m(tierung\),)j(w)m
(elc)m(he)j(in)d(dieser)h(Diplomarb)s(eit)f(auc)m(h)i(v)m(erw)m(endet)g
(wurde.)f(Alle)390 641 y(diese)g(Klassen)g(w)m(erden)g(als)g
Fu(Konzeptklassen)45 b Fw(b)s(ezeic)m(hnet.)36 b(Wir)f(w)m(erden)g(im)g
(w)m(eiteren)390 754 y(V)-8 b(erlauf)30 b(darauf)g(zur)1122
758 y(\177)1119 754 y(uc)m(kk)m(ommen.)531 873 y(Die)37
b(Aufgab)s(e)g(der)f(JCA)h(ist,)f(die)g(standardisierte)g(V)-8
b(erw)m(endung)36 b(v)m(on)h(kryptogra-)390 986 y(phisc)m(hen)c(V)-8
b(erfahren)34 b(zu)g(erm)1464 990 y(\177)1464 986 y(oglic)m(hen.)g
(Diese)h(Implemen)m(tierungen)d(w)m(erden)i(in)g(soge-)390
1099 y(nann)m(ten)39 b Fu(Pr)-5 b(ovidern)47 b Fw(zusammengefa\031t.)40
b(Dar)2056 1103 y(\177)2053 1099 y(ub)s(erhinaus)c(bietet)j(die)f(JCA)h
(auc)m(h)g(die)390 1211 y(M)473 1215 y(\177)473 1211
y(oglic)m(hk)m(eit,)28 b(eigene)g(Implemen)m(tierungen)d(in)h(F)-8
b(orm)28 b(eines)f(solc)m(hen)g(Pro)m(viders)g(zu)g(en)m(t-)390
1324 y(wic)m(k)m(eln.)20 b(Das)h(k)945 1328 y(\177)945
1324 y(onnen)f Fu(digitale)k(Signatur)-5 b(en)29 b Fw(o)s(der)20
b(V)-8 b(ersc)m(hl)2448 1328 y(\177)2445 1324 y(usselungsalgorithmen,)
17 b(ab)s(er)390 1437 y(auc)m(h)33 b(andere)g(V)-8 b(erfahren)33
b(sein,)f(die)g(mit)h(der)f(Kryptographie)g(zusammenh)3089
1441 y(\177)3089 1437 y(angen,)h(wie)390 1550 y(z.)16
b(B.)31 b Fu(Pseudozfal)5 b(lszahlengener)-5 b(ator)g(en)p
Fw(.)531 1669 y(Bei)23 b(den)f(Konzeptklassen)g(in)f(den)h(ob)s
(engenann)m(ten)g(P)m(ak)m(eten)j(gibt)d(es)h(n)m(un)e(no)s(c)m(h)i
(ein-)390 1782 y(mal)30 b(eine)g(Zw)m(eiteilung.)f(Auf)h(der)g(Seite)g
(des)h(Ben)m(utzers)g(v)m(on)g(kryptographisc)m(her)f(Soft-)390
1895 y(w)m(are)k(b)s(e\014nden)e(sic)m(h)h(die)g(Klassen)g(des)h
Fu(Applic)-5 b(ation)37 b(Pr)-5 b(o)g(gr)g(amming)39
b(Interfac)-5 b(e)36 b(\(API\))p Fw(.)390 2008 y(Sie)20
b(v)m(ereinheitlic)m(hen)f(die)h(V)-8 b(erw)m(endung)20
b(v)m(ersc)m(hiedener)h(Implemen)m(tierungen.)e(Um)h(ab)s(er)390
2121 y(auc)m(h)26 b(eigene)g(Pro)m(vider)e(m)1312 2125
y(\177)1309 2121 y(uhelos)h(in)m(tegrieren)f(zu)i(k)2195
2125 y(\177)2195 2121 y(onnen,)f(greift)g(man)g(auf)g(die)g(Klassen)390
2234 y(des)j Fu(Servic)-5 b(e)31 b(Pr)-5 b(ovider)32
b(Interfac)-5 b(e)31 b(\(SPI\))h Fw(zur)1989 2238 y(\177)1986
2234 y(uc)m(k.)d(In)f(ihnen)e(wird)h(der)h(Rahmen)g(f)3283
2238 y(\177)3280 2234 y(ur)f(die)390 2346 y(Implemen)m(tierungen)h(v)m
(orgegeb)s(en.)531 2465 y(Will)41 b(man)h(z.)16 b(B.)43
b(einen)e(V)-8 b(ersc)m(hl)1708 2469 y(\177)1705 2465
y(usselungsalgorithm)m(us)40 b(implemen)m(tieren,)h(so)h(b)s(e-)390
2578 y(erbt)64 b(man)h(die)f(Klasse)g Ff(javax.crypto.CipherSpi)58
b Fw(mitsam)m(t)65 b(ihrer)e Fu(abstr)-5 b(akten)390
2691 y Fw(Metho)s(den.)52 b(Nac)m(hdem)h(diese)e(implemen)m(tiert)f(w)m
(orden)h(sind,)g(k)-5 b(ann)51 b(man)g(sic)m(h)h(un-)390
2804 y(ter)45 b(Angab)s(e)f(des)g(en)m(tsprec)m(henden)h(Algorithm)m
(usnamens)d(eine)i(Instanz)h(der)e(Klasse)390 2917 y
Ff(javax.crypto.Cipher)18 b Fw(erzeugen)24 b(lassen,)f(die)g(die)f
(Implemen)m(tierung)f(an)m(bietet.)3324 2898 y(\177)3313
2917 y(Ub)s(er)390 3030 y(die)43 b(Sc)m(hnittstelle)h(v)m(on)g
Ff(javax.crypto.Cipher)c Fw(ist)j(diese)h(v)m(erf)2749
3034 y(\177)2746 3030 y(ugbar.)h(F)3106 3034 y(\177)3103
3030 y(ur)e(die)h(ge-)390 3143 y(nauere)37 b(Besc)m(hreibung)g(wird)e
(auf)i(Kapitel)f(4.4,)j(den)d(in)g(den)h(Anh)2779 3147
y(\177)2779 3143 y(angen)g(en)m(thaltenen)390 3256 y(Sourceco)s(de)30
b(und)f(auf)h(das)g(Buc)m(h)h([Kn)m(u98])g(v)m(erwiesen.)531
3375 y(Will)52 b(man)g(auf)h(b)s(ereits)f(implemen)m(tierte)g
(Algorithmen)g(zur)2802 3379 y(\177)2799 3375 y(uc)m(kgreifen,)h(so)h
(ist)390 3487 y(dies)44 b(eb)s(enso)h(m)974 3491 y(\177)974
3487 y(oglic)m(h.)g(Die)h(Firma)e Fg(SUN)h Fw(und)f(auc)m(h)i(andere)f
(Hersteller)f(bieten)h(ei-)390 3600 y(gene)d(Pro)m(vider)d(an.)i(Die)g
(g)1371 3604 y(\177)1371 3600 y(angigsten)h(V)-8 b(erfahren)40
b(sind)f(darin)g(v)m(orhanden,)i(man)f(v)m(er-)390 3713
y(traut)29 b(dab)s(ei)f(jedo)s(c)m(h)h(auf)f(die)g(k)m(orrekte)i
(Implemen)m(tierung.)e(F)2590 3717 y(\177)2587 3713 y(ur)g(die)g(V)-8
b(erw)m(endung)29 b(die-)390 3826 y(ser)42 b(Algorithmen)e(m)m(u\031)i
(man)f(lediglic)m(h)f(den)h(en)m(tsprec)m(henden)g(Pro)m(vider)g(der)g
(Klasse)390 3939 y Ff(java.security.Security)30 b Fw(b)s(ek)-5
b(ann)m(tgeb)s(en)35 b(und)f(k)-5 b(ann)35 b(dann)f(die)h(Implemen)m
(tierun-)390 4052 y(gen)c(in)e(der)g(gleic)m(hen)h(W)-8
b(eise)31 b(b)s(en)m(utzen,)g(wie)e(dies)h(mit)f(selbst)h(en)m(t)m(wic)
m(k)m(elten)h(Pro)m(vidern)390 4165 y(der)f(F)-8 b(all)30
b(ist.)390 4485 y Fz(4.2)135 b(Der)46 b(Standard)e(IEEE)i(P1363)g(/)f
(D8)390 4700 y Fw(Der)e(v)m(ollst)791 4704 y(\177)791
4700 y(andige)e(Titel)g(der)h(Ric)m(h)m(tlinie,)f(die)g(v)m(on)i(dem)f
Ft(Institute)j(of)h(Electri-)390 4813 y(cal)30 b(and)g(Electr)n(onics)f
(Engineers,)h(Inc.)g(\(IEEE\))d Fw(v)m(er)2562 4817 y(\177)2562
4813 y(o\013en)m(tlic)m(h)m(t)h(wurde,)e(lautet)390 4926
y("Standard)38 b(Sp)s(eci\014cations)f(for)h(Public)f(Key)h
(Cryptograph)m(y\\.)h(T)-8 b(eile)38 b(daraus)g(fanden)390
5038 y(in)i(dieser)h(Diplomarb)s(eit)e(V)-8 b(erw)m(endung,)42
b(darum)e(soll)h(an)g(dieser)g(Stelle)g(kurz)g(darauf)390
5151 y(eingegangen)31 b(w)m(erden.)f(F)1298 5155 y(\177)1295
5151 y(ur)g(die)f(Details)i(wird)d(direkt)h(auf)i([ISD98])g(v)m
(erwiesen.)531 5270 y(Der)44 b(Standard)e(b)s(ehandelt)f(div)m(erse)h
(Gesic)m(h)m(tspunkte)i(der)e(Public)f(Key)i(Krypto-)390
5383 y(graphie.)f(Dies)g(sind)e(in)h(erster)i(Linie)d(Sc)m(hl)1960
5387 y(\177)1957 5383 y(usselgenerierungs-,)g(Signatur-)h(und)g(V)-8
b(er-)390 5496 y(sc)m(hl)543 5500 y(\177)540 5496 y(usselungsv)m
(erfahren.)41 b(F)-8 b(erner)44 b(wird)e(v)m(ersuc)m(h)m(t,)j(je)f(nac)
m(h)g(der)f(zugrundeliegenden)390 5609 y(mathematisc)m(hen)c
(Problemstellung,)d(die)i(zu)g(v)m(erw)m(endenden)g(P)m(arameter)i(zu)e
(v)m(erein-)p eop
%%Page: 21 22
21 21 bop 270 82 a Fw(4.2.)62 b(Der)31 b(Standard)e(IEEE)h(P1363)i(/)e
(D8)1545 b(21)270 415 y(heitlic)m(hen.)33 b(Dab)s(ei)i(sollen)e(w)m
(eder)i(b)s(estimm)m(te)f(V)-8 b(erfahren)35 b(no)s(c)m(h)f(v)m(er)2715
419 y(\177)2715 415 y(anderlic)m(he)g(Merk-)270 528 y(male)44
b(wie)g(Sc)m(hl)845 532 y(\177)842 528 y(ussell)1057
532 y(\177)1057 528 y(angen)e(empfohlen)g(w)m(erden.)i(Der)h(Sinn)d
(liegt)i(vielmehr)e(in)h(der)270 641 y(standardisierten)38
b(Au\015istung)h(v)m(on)i(tauglic)m(hen)f(V)-8 b(erfahren)41
b(im)e(Rahmen)h(der)g(Public)270 754 y(Key)30 b(Kryptographie.)411
873 y(Die)i(erste)g(V)-8 b(ersion)31 b(des)g(Standards)f(stamm)m(t)j
(aus)e(dem)g(Jahre)g(1994,)i(damals)e(no)s(c)m(h)270
986 y(un)m(ter)45 b(dem)g(Titel)f("Standard)g(for)h(Riv)m
(est-Shamir-Adleman,)e(Di\016e-Hellman,)i(and)270 1098
y(Related)i(Public-Key)e(Cryptograph)m(y\\.)i(Mit)g(zunehmender)e(Zeit)
i(v)m(ergr)2951 1102 y(\177)2951 1098 y(o\031erte)i(sic)m(h)270
1211 y(der)41 b(Umfang.)h(So)g(k)-5 b(amen)42 b(b)s(ei)e(den)h
(mathematisc)m(hen)h(Grundlagen)f(die)g(elliptisc)m(hen)270
1324 y(Kurv)m(en)28 b(mit)f(den)h(dazugeh)1249 1328 y(\177)1249
1324 y(origen)g(V)-8 b(erfahren)29 b(hinzu.)d(Leider)i(sind)e(zum)i
(jetzigen)h(Zeit-)270 1437 y(punkt)21 b(lediglic)m(h)f(Sc)m(hl)1043
1441 y(\177)1040 1437 y(usselgenerierungen)f(und)i(Signaturen)f
(mithilfe)g(elliptisc)m(her)g(Kur-)270 1550 y(v)m(en)32
b(v)m(orhanden.)e(Die)i(Aufnahme)e(v)m(on)h(V)-8 b(ersc)m(hl)1976
1554 y(\177)1973 1550 y(usselungsv)m(erfahren)30 b(in)g(den)g(Standard)
270 1663 y(ist)g(jedo)s(c)m(h)g(geplan)m(t.)411 1782
y(Inzwisc)m(hen)23 b(umfa\031t)h(der)f(P)m(ersonenkreis)g(dieses)g(Pro)
5 b(jekts)24 b(b)s(ek)-5 b(ann)m(te)24 b(Exp)s(erten)f(auf)270
1895 y(dem)29 b(Gebiet)h(der)g(Kryptographie,)f(wie)g(z.)16
b(B.)30 b Ft(Alfred)i(Menezes)c Fw(und)g Ft(Scott)33
b(V)-11 b(an-)270 2008 y(stone)34 b Fw(\(siehe)g(De\014nition)g(9\).)i
(Au\031erdem)f(ist)g(die)f(Institution)f Ft(IEEE)j Fw(eine)f(f)3075
2012 y(\177)3072 2008 y(uhrende)270 2121 y(Autorit)566
2125 y(\177)566 2121 y(at)i(auf)e(dem)h(T)-8 b(ec)m(hnologiesektor.)37
b(Man)g(k)-5 b(ann)35 b(also)h(erw)m(arten,)h(da\031)f(sic)m(h)g(viele)
270 2233 y(An)m(w)m(ender)30 b(der)g(Public)e(Key)i(Kryptographie)f(in)
g(Zukunft)g(danac)m(h)i(ric)m(h)m(ten.)411 2352 y(Wir)42
b(k)m(ommen)g(n)m(un)f(zu)h(den)g(f)1522 2356 y(\177)1519
2352 y(ur)f(uns)g(relev)-5 b(an)m(ten)43 b(Absc)m(hnitten)e(aus)h(dem)g
(Stan-)270 2465 y(dard.)53 b(Da)i(wir)e(uns)f(mit)i(Kryptographie)e
(auf)i(der)f(Grundlage)g(elliptisc)m(her)f(Kur-)270 2578
y(v)m(en)60 b(b)s(efassen,)f(w)m(ar)h(das)f(Kapitel)f(7.1)j(in)d
([ISD98)q(])i(v)m(on)g(b)s(esonderem)e(In)m(teresse.)270
2691 y(Dort)40 b(w)m(erden)f(die)g(P)m(arameter)h(f)1454
2695 y(\177)1451 2691 y(ur)f(V)-8 b(erfahren)39 b(mit)g(elliptisc)m
(hen)d(Kurv)m(en)j(festgelegt.)270 2804 y(Dar)423 2808
y(\177)420 2804 y(ub)s(erhinaus)52 b(wird)h(gesagt,)k(wie)e(der)1841
2808 y(\177)1841 2804 y(o\013en)m(tlic)m(he)h(und)d(der)i(geheime)g(T)
-8 b(eil)54 b(des)270 2917 y(Sc)m(hl)438 2921 y(\177)435
2917 y(ussels)36 b(aussehen)h(sollen.)g(Eingang)h(fanden)f(diese)h(Sp)s
(ezi\014k)-5 b(ationen)36 b(v)m(or)i(allem)f(in)270 3030
y(der)28 b(Klasse)g Ff(LECDomainParametersP1363)p Fw(.)22
b(W)-8 b(as)30 b(die)e(Sc)m(hl)2415 3034 y(\177)2412
3030 y(ussel)e(b)s(etri\013t,)i(so)h(b)s(eruhen)270 3143
y Ff(ECElGamalGFPPrivateKey)c Fw(und)30 b Ff(ECElGamalGFPPublicKey)25
b Fw(auf)31 b(den)f(en)m(tsprec)m(hen-)270 3256 y(den)g(F)-8
b(estlegungen)31 b(im)e(Standard.)270 3533 y Fe(4.2.1)112
b(Die)37 b(Domain)g(P)m(arameter)270 3716 y Fw(Die)30
b(Domain)h(P)m(arameter)g(f)1252 3720 y(\177)1249 3716
y(ur)e(elliptisc)m(he)g(Kurv)m(en)g(sind:)406 3958 y
Fo(\017)46 b Fw(Eine)30 b(ungerade)g(Primzahl)e Fq(p)406
4169 y Fo(\017)46 b Fw(Zw)m(ei)30 b(Ko)s(e\016zien)m(ten)g
Fq(a;)15 b(b)26 b Fo(2)f Fn(F)1546 4183 y Fl(p)1621 4169
y Fw(und)k(damit)h(die)f(Kurv)m(e)h Fq(E)h Fw(:)25 b
Fq(y)2681 4136 y Fp(2)2746 4169 y Fw(=)g Fq(x)2894 4136
y Fp(3)2954 4169 y Fw(+)19 b Fq(ax)i Fw(+)f Fq(b)406
4380 y Fo(\017)46 b Fw(Ein)34 b(Kurv)m(enpunkt)g Fq(G)i
Fw(\(der)f(Erzeuger)h(einer)f(Un)m(tergrupp)s(e)f(v)m(on)i(Kurv)m
(enpunk-)497 4493 y(ten\))406 4704 y Fo(\017)46 b Fw(Die)31
b(Ordn)m(ung)d Fq(r)33 b Fw(v)m(on)e Fq(G)406 4915 y
Fo(\017)46 b Fw(Der)36 b(Kofaktor)f Fq(k)s Fw(,)h(w)m(ob)s(ei)e
Fo(j)p Fq(E)5 b Fo(j)34 b Fw(=)f Fq(k)s(r)k Fw(\(Mit)f
Fo(j)p Fq(E)5 b Fo(j)35 b Fw(wird)e(nac)m(h)j(wie)e(v)m(or)i(die)e
(Anzahl)497 5028 y(der)c(Punkte)g(auf)g(der)g(Kurv)m(e)g(b)s(ezeic)m
(hnet.\))411 5270 y(Sie)24 b(b)s(einhalten)e(die)i(W)-8
b(erte,)26 b(die)e(allgemein)f(f)1991 5274 y(\177)1988
5270 y(ur)g(die)h(V)-8 b(erw)m(endung)24 b(elliptisc)m(her)e(Kur-)270
5383 y(v)m(en)41 b(in)e(der)h(Kryptographie)g(b)s(en)1495
5387 y(\177)1495 5383 y(otigt)g(w)m(erden.)g(Das)i(Sc)m(hl)2427
5387 y(\177)2424 5383 y(usselpaar,)d(das)h(f)3057 5387
y(\177)3054 5383 y(ur)f(jeden)270 5496 y(T)-8 b(eilnehmer)28
b(eigens)i(generiert)f(wird,)f(geh)1756 5500 y(\177)1756
5496 y(ort)j(nic)m(h)m(t)e(dazu)h(und)e(wird)g(im)h(n)2916
5500 y(\177)2916 5496 y(ac)m(hsten)h(Un-)270 5609 y(terabsc)m(hnitt)g
(angespro)s(c)m(hen.)p eop
%%Page: 22 23
22 22 bop 390 82 a Fw(22)1313 b(KAPITEL)30 b(4.)61 b(DIE)30
b(IMPLEMENTIER)m(UNG)390 415 y Fe(4.2.2)112 b(Das)38
b(Sc)m(hl)1160 420 y(\177)1157 415 y(usselpaar)390 587
y Fw(Der)31 b Fu(Private)h(Key)39 b Fw(und)28 b(der)i
Fu(Public)i(Key)38 b Fw(bilden)28 b(zusammen)i(das)g
Fu(Schl)2933 591 y(\177)2933 587 y(usselp)-5 b(aar)10
b Fw(:)526 787 y Fo(\017)46 b Fw(Eine)30 b(ganze)h(Zahl)e
Fq(s)c Fo(2)g Fw([1)p Fq(;)15 b(r)24 b Fo(\000)c Fw(1])31
b(wird)d(als)i(Priv)-5 b(ate)30 b(Key)g(b)s(ezeic)m(hnet.)526
970 y Fo(\017)46 b Fw(Der)38 b(Punkt)f Fq(W)49 b Fw(auf)37
b(der)g(Kurv)m(e)g Fq(E)43 b Fw(mit)36 b Fq(W)50 b Fw(=)36
b Fq(sG)i Fw(ist)f(der)f(zugeh)3026 974 y(\177)3026 970
y(orige)i(Public)617 1083 y(Key)-8 b(.)531 1283 y(Un)m(ter)34
b(der)e(Annahme,)h(da\031)g(das)g(DLP)g(f)1982 1287 y(\177)1979
1283 y(ur)f(elliptisc)m(he)f(Kurv)m(en)i(w)m(eiterhin)e(sc)m(h)m(w)m
(er)390 1396 y(zu)c(l)534 1400 y(\177)534 1396 y(osen)f(ist,)h(b)s
(eruh)m(t)f(die)g(Sic)m(herheit)g(des)h(V)-8 b(erfahrens,)27
b(wie)f(b)s(ei)g(jedem)h(Kryptosystem,)390 1508 y(auf)e(der)g
(Geheimhaltung)f(des)i(Priv)-5 b(ate)25 b(Key)-8 b(.)26
b(Es)f(ist)g(klar,)g(da\031)h(dieser)f(auf)g(eine)g(nic)m(h)m(tde-)390
1621 y(terministisc)m(he)32 b(W)-8 b(eise)34 b(erzeugt)h(w)m(erden)e(m)
m(u\031.)h(Des)h(w)m(eiteren)e(m)m(u\031)h(er)g(so)g(gesp)s(eic)m(hert)
390 1734 y(w)m(erden,)c(da\031)h(n)m(ur)e(die)h(b)s(erec)m(h)m(tigten)h
(P)m(ersonen)f(darauf)g(zugreifen)f(k)2835 1738 y(\177)2835
1734 y(onnen.)390 2019 y Fz(4.3)135 b(Die)46 b(C++-Klassen)l
(bibliothek)h(LiDIA)390 2222 y Fw(In)23 b(diesem)g(Absc)m(hnitt)h(soll)
e(no)s(c)m(h)i(ein)f(Asp)s(ekt)h(angespro)s(c)m(hen)g(w)m(erden,)g(der)
f(die)g(V)-8 b(erw)m(en-)390 2335 y(dung)32 b(v)m(on)g(JA)-10
b(V)g(A)33 b(als)f(Programmiersprac)m(he)g(b)s(eein)m(tr)2326
2339 y(\177)2326 2335 y(ac)m(h)m(tigt.)i(Bei)e(An)m(w)m(endungen,)g(in)
390 2447 y(denen)f(die)f(E\016zienz)h(eine)g(gro\031e)i(Rolle)e
(spielt,)f(sind)g(andere)h(masc)m(hinenn)3055 2451 y(\177)3055
2447 y(ahere)g(Spra-)390 2560 y(c)m(hen)i(b)s(esser)f(geeignet.)j(Das)f
(ist)e(auc)m(h)h(der)g(Grund,)f(w)m(arum)g(in)g(dem)g(Pro)m(vider,)h
(der)f(in)390 2673 y(dieser)d(Arb)s(eit)h(en)m(t)m(wic)m(k)m(elt)h
(wird,)e(f)1627 2677 y(\177)1624 2673 y(ur)g(rec)m(henin)m(tensiv)m(e)h
(Metho)s(den)h(auf)f(die)f(Klassen)m(bi-)390 2786 y(bliothek)23
b Fu(LiDIA)h Fw(zur)1146 2790 y(\177)1143 2786 y(uc)m(kgreift.)g(Man)h
(gibt)f(damit)f(ab)s(er)h(die)g(Plattform)m(unabh)3153
2790 y(\177)3153 2786 y(angigk)m(eit)390 2899 y(v)m(on)31
b(JA)-10 b(V)g(A)30 b(auf.)531 3012 y(LiDIA)c(ist)g(eine)f(in)g(der)h
(Programmiersprac)m(he)g(C++)f(am)i(F)-8 b(ac)m(h)m(b)s(ereic)m(h)27
b(Informatik)390 3125 y(der)38 b Ft(Technischen)k(Universit)1616
3121 y(\177)1613 3125 y(at)g(D)m(armst)-6 b(adt)38 b
Fw(en)m(t)m(wic)m(k)m(elte)j(Klassen)m(bibliothek.)390
3238 y(Sie)31 b(stellt)h(v)m(ersc)m(hiedene)g(Algorithmen)f(aus)h(dem)g
(Bereic)m(h)h(der)e(algorithmisc)m(hen)g(Zah-)390 3351
y(len)m(theorie)f(zur)g(V)-8 b(erf)1131 3355 y(\177)1128
3351 y(ugung.)531 3464 y(Es)35 b(gibt)g(b)s(ei)e(dem)i(Pro)m(vider)f
(zw)m(ei)i(Stellen,)e(an)h(denen)f(LiDIA)h(b)s(en)m(utzt)g(wird.)e(Bei)
390 3577 y(der)40 b(Multiplik)-5 b(ation)37 b(eines)j(Kurv)m(enpunktes)
f(mit)h(gro\031en)i(Zahlen)d(ist)h(eine)g(e\016zien)m(te)390
3689 y(Implemen)m(tierung)f(sehr)h(wic)m(h)m(tig.)h(Die)g(Generierung)f
(v)m(on)i(kryptographisc)m(h)d(stark)m(en)390 3802 y(Kurv)m(en)26
b(\(siehe)g(hierzu)f(Kapitel)g(2.3\))j(ist)e(in)g(LiDIA)g(sc)m(hon)g
(implemen)m(tiert)f(w)m(orden,)h(so-)390 3915 y(da\031)i(sic)m(h)f
(hier)f(ein)h(Zugri\013)f(darauf)h(angeb)s(oten)h(hat.)g(Realisiert)f
(wurde)f(die)h(An)m(bindung)390 4028 y(an)j(b)s(estimm)m(te)g
(LiDIA-Klassen)f(mittels)h(des)g Fu(JA)-9 b(V)g(A)30
b(Native)j(Interfac)-5 b(e)33 b(\(JNI\))p Fw(.)531 4141
y(In)41 b(LiDIA)h(gibt)f(es)h(die)f(Klasse)h Ff(EC)p
1823 4141 29 4 v 33 w(domain)p 2144 4141 V 33 w(parameter)p
2609 4141 V 32 w(P1363)f Fw(\(eine)g(Besc)m(hrei-)390
4254 y(bung)63 b(\014ndet)h(man)g(in)f(dem)h(Man)m(ual)g([LiD99)q(]\).)
h(Ihre)f(Hauptfunktionalit)3271 4258 y(\177)3271 4254
y(at)f(ist)390 4367 y(die)g(Generierung)g(v)m(on)h(elliptisc)m(hen)d
(Kurv)m(en,)j(die)f(f)2425 4371 y(\177)2422 4367 y(ur)f(die)h
(Kryptographie)g(ge-)390 4480 y(eignet)d(sind,)d(w)m(as)j(in)e(der)h
(Metho)s(de)g Ff(generate)p 2255 4480 V 33 w(parameters)d
Fw(erfolgt.)k(Um)f(n)m(un)390 4593 y(auf)69 b(diese)g(F)-8
b(unktionalit)1335 4597 y(\177)1335 4593 y(at)68 b(v)m(on)i(der)f(JA)
-10 b(V)g(A-Seite)70 b(aus)f(zugreifen)f(zu)h(k)3247
4597 y(\177)3247 4593 y(onnen,)390 4706 y(wurde)56 b(mit)h
Ff(LECDomainParametersP136)o(3)51 b Fw(eine)57 b(sogenann)m(te)i
Fu(Spie)-5 b(gelklasse)65 b Fw(en)m(t-)390 4819 y(wic)m(k)m(elt.)60
b(Die)g(b)s(eiden)d(T)-8 b(eile)59 b(dieser)g(Klasse)g(sind)e(auf)j(zw)
m(ei)f(Dateien)i(aufgeteilt.)390 4932 y Ff(LECDomainParametersP1363)o
(.jav)o(a)50 b Fw(en)m(th)2004 4936 y(\177)2004 4932
y(alt)56 b(die)g(De\014nition)e(der)i(JA)-10 b(V)g(A-Klasse.)390
5044 y(Dort)36 b(wird)d(u.a.)j(eine)e(Metho)s(de)i Ff
(generateParameters)30 b Fw(als)35 b Fu(native)42 b Fw(deklariert.)34
b(Dies)390 5157 y(b)s(edeutet,)54 b(da\031)h(die)f(Implemen)m(tierung)e
(dieser)h(Metho)s(de)i(in)e(einer)g(anderen)h(Pro-)390
5270 y(grammiersprac)m(he)28 b(erfolgt.)i(In)e Ff
(LECDomainParametersP1363)o(.cpp)22 b Fw(sind)27 b(alle)h(nativ)m(e-)
390 5383 y(Metho)s(den)42 b(dieser)f(Klasse)g(in)g(C++)g(implemen)m
(tiert.)f(Dort)j(wird)d(dann)h(ein)g(LiDIA-)390 5496
y(Ob)5 b(jekt)23 b(erzeugt,)i(das)f(die)f(P)m(arameter)i(generiert.)f
(Auf)f(die)g(gleic)m(he)g(W)-8 b(eise)25 b(wurden)d(auc)m(h)390
5609 y(die)28 b(Zugri\013smetho)s(den)f(realisiert,)i(so)s(da\031)g
(das)g(LiDIA-Ob)5 b(jekt)29 b(sc)m(hlie\031lic)m(h)f(wieder)g(aus)p
eop
%%Page: 23 24
23 23 bop 270 82 a Fw(4.4.)62 b(Der)31 b(ElGamal-Pro)m(vider)e(mit)h
(elliptisc)m(hen)d(Kurv)m(en)967 b(23)270 415 y(dem)28
b(Sp)s(eic)m(her)f(en)m(tfern)m(t)j(w)m(erden)e(k)-5
b(ann)28 b(und)f(die)h(P)m(arameter)i(in)d(der)h(JA)-10
b(V)g(A-Klasse)28 b(zur)270 528 y(V)-8 b(erf)437 532
y(\177)434 528 y(ugung)30 b(stehen.)h(Zu)e(w)m(eiteren)h(Details)h
(wird)d(auf)i(Anhang)g(C)g(v)m(erwiesen.)411 645 y(Das)e(P)m(ak)m(et)g
Ff(cdc.lidia)d Fw(en)m(th)1472 649 y(\177)1472 645 y(alt)i(die)f(ob)s
(engenann)m(te)h(Klasse.)f(Zus)2768 649 y(\177)2768 645
y(atzlic)m(h)g(b)s(e\014nden)270 758 y(sic)m(h)f(darin)e(auc)m(h)j(no)s
(c)m(h)f(andere)h(Hilfsklassen,)d(die)h(f)2074 762 y(\177)2071
758 y(ur)h(die)f(An)m(bindung)f(an)i(LiDIA)g(n)3239 762
y(\177)3239 758 y(otig)270 871 y(w)m(aren.)40 b(F)632
875 y(\177)629 871 y(ur)f(deren)g(F)-8 b(unktionalit)1502
875 y(\177)1502 871 y(at)39 b(wird)f(auf)h(den)g(Quellco)s(de)f(direkt)
h(bzw.)g(die)g(Di-)270 984 y(plomarb)s(eiten)24 b(der)i(en)m(tsprec)m
(henden)g(Autoren)g(v)m(erwiesen.)g(Als)g(Beispiel)e(seien)i(hier)f
(die)270 1097 y(Klassen)30 b Ff(LECCurveGFP)e Fw(und)h
Ff(LECPointGFP)f Fw(v)m(on)j Ft(Bir)n(git)36 b(Henhapl)29
b Fw(genann)m(t)i(\(siehe)270 1210 y(auc)m(h)g([Hen99)q(]\).)270
1521 y Fz(4.4)135 b(Der)46 b(ElGamal-Pro)l(vider)g(mit)g(elliptisc)l
(hen)h(Kurv)l(en)270 1733 y Fw(Es)25 b(sei)g(no)s(c)m(h)g(einmal)f
(kurz)h(erw)1362 1737 y(\177)1362 1733 y(ahn)m(t,)g(da\031)h(sic)m(h)f
(die)g(genaue)h(Erl)2497 1737 y(\177)2497 1733 y(auterung)e(der)g
(einzelnen)270 1845 y(Klassen)30 b(und)g(ihrer)f(Metho)s(den)i(im)f
(Quellco)s(de)g(selbst)g(b)s(e\014ndet.)g(Dort)i(wurde)e(f)3144
1849 y(\177)3141 1845 y(ur)g(das)270 1958 y(T)-8 b(o)s(ol)36
b Fu(javado)-5 b(c)43 b Fw(eine)35 b(umfassende)g(Dokumen)m(tation)i
(erstellt.)f(Diese)h(sollte)e(zusammen)270 2071 y(mit)43
b(den)h(Kommen)m(tarzeilen)f(innerhalb)e(der)j(einzelnen)f(Metho)s(den)
h(als)f(Erg)3107 2075 y(\177)3107 2071 y(anzung)270 2184
y(dieses)30 b(Absc)m(hnitts)f(f)1015 2188 y(\177)1012
2184 y(ur)g(eine)h(v)m(ollst)1528 2188 y(\177)1528 2184
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(viders)g(gen)2947 2188 y(\177)2944 2184 y(ugen.)270
2453 y Fe(4.4.1)112 b(Klassendiagramme)270 2633 y Fw(Der)36
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(mehreren)f(Klassen.Deren)h(Bezieh)m(un-)270 2746 y(gen)e(sollen)e(im)h
(folgenden)f(in)h(Diagrammen)g(dargestellt)h(w)m(erden.)f(Dab)s(ei)g
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b(eildiagramme)29 b(aufgeteilt,)h(w)m(as)h(nic)m(h)m(t)f(hei\031en)g
(soll,)g(da\031)h(diese)270 2971 y(unabh)519 2975 y(\177)519
2971 y(angig)e(v)m(oneinander)g(sind.)327 3809 y Fd(\033)p
327 3926 7 4 v 327 4046 a(\032)1744 3809 y(\030)p 1744
3926 V 1744 4046 a(\031)p 446 4046 1185 7 v 446 3809
V 421 3946 a Fw(ja)m(v)-5 b(a.securit)m(y)d(.KeyP)m(airGenerator)p
1747 3927 237 7 v 1977 3806 250 7 v 1977 4042 7 237 v
2011 3955 a(JCA)p 2220 4042 V 1977 4049 250 7 v 1972
3801 259 7 v 1972 4047 7 246 v 2224 4047 V 1972 4054
259 7 v 1967 3797 269 7 v 1967 4052 7 256 v 2229 4052
V 1967 4059 269 7 v 2098 3806 7 473 v 1815 3101 a Fd(\033)p
1815 3217 7 4 v 1815 3337 a(\032)3327 3101 y(\030)p 3327
3217 V 3327 3337 a(\031)p 1934 3337 1280 7 v 1934 3101
V 1893 3238 a Fw(ja)m(v)j(a.securit)m(y)d(.KeyP)m(airGeneratorSpi)p
2571 4515 7 1182 v 2574 3417 a Fc(6)2598 3942 y Ff(implements)1817
4516 y Fb(\033)p 1817 4635 4 4 v 1817 4753 a(\032)3328
4516 y(\030)p 3328 4635 V 3328 4753 a(\031)p 1934 4753
1280 4 v 1934 4516 V 1876 4656 a Fw(ECElGamalGFPKeyP)m(airGenerator)852
5046 y(Abbildung)27 b(4.1:)32 b Ff(ECElGamalGFPKeyPairGenera)o(tor)411
5270 y Fw(An)47 b(dieser)g(Stelle)f(soll)h(auf)g(die)g(einzelnen)f
(Gestaltungselemen)m(te)i(in)f(den)g(Dia-)270 5383 y(grammen)d
(eingegangen)h(w)m(erden.)e(Die)h(abgerundeten)g(K)2386
5387 y(\177)2386 5383 y(astc)m(hen)g(stellen)f(die)g(Klas-)270
5496 y(sen)f(der)h(JA)-10 b(V)g(A)42 b(Cryptograph)m(y)g(Arc)m
(hitecture)i(wie)e(auc)m(h)h(die)e(des)i(implemen)m(tierten)270
5609 y(Pro)m(viders)h(dar.)g(V)-8 b(ererbungsb)s(ezieh)m(ungen)43
b(w)m(erden)h(durc)m(h)f(Pfeile)h(sym)m(b)s(olisiert.)e(Da)p
eop
%%Page: 24 25
24 24 bop 390 82 a Fw(24)1313 b(KAPITEL)30 b(4.)61 b(DIE)30
b(IMPLEMENTIER)m(UNG)390 415 y(es)43 b(in)e(JA)-10 b(V)g(A)43
b(zw)m(ei)g(un)m(tersc)m(hiedlic)m(he)e(Arten)i(da)m(v)m(on)g(gibt,)g
(stehen)f(die)g(en)m(tsprec)m(hen-)390 528 y(den)29 b(Sc)m(hl)729
532 y(\177)726 528 y(usselw)981 532 y(\177)981 528 y(orter)f
Ff(extends)g Fw(bzw.)h Ff(implements)e Fw(mit)i(dab)s(ei.)f(In)h(den)g
(Abbildungen)390 641 y(4.1)37 b(und)e(4.2)i(\014ndet)f(sic)m(h)g(jew)m
(eils)f(ein)g(K)1849 645 y(\177)1849 641 y(astc)m(hen,)i(das)f(div)m
(erse)g(Klassen)f(der)h(JCA)g(re-)390 754 y(pr)477 758
y(\177)477 754 y(asen)m(tieren)28 b(soll.)g(Die)h(genaue)h(Aufrufk)-5
b(ask)g(ade)28 b(b)s(ei)f(Anforderung)h(einer)g(b)s(estimm)m(ten)390
867 y(Implemen)m(tierung)f(darzustellen,)h(w)1690 871
y(\177)1687 867 y(urde)g(die)2050 848 y(\177)2038 867
y(Ub)s(ersic)m(h)m(tlic)m(hk)m(eit)g(der)h(Diagramme)h(n)m(ur)390
980 y(unn)543 984 y(\177)543 980 y(otig)i(b)s(ein)m(tr)965
984 y(\177)965 980 y(ac)m(h)m(tigen.)i(In)f(Kapitel)f(4.4.2)k(wird)31
b(sie)i(ab)s(er)g(kurz)g(angespro)s(c)m(hen)h(w)m(er-)390
1093 y(den.)707 1957 y Fd(\033)p 707 2074 7 4 v 707 2194
a(\032)1651 1957 y(\030)p 1651 2074 V 1651 2194 a(\031)p
826 2194 713 7 v 826 1957 V 792 2095 a Fw(ja)m(v)-5 b(ax.crypto.Cipher)
p 1655 2076 237 7 v 1884 1954 250 7 v 1884 2190 7 237
v 1919 2103 a(JCA)p 2127 2190 V 1884 2197 250 7 v 1880
1949 259 7 v 1880 2195 7 246 v 2132 2195 V 1880 2202
259 7 v 1875 1945 269 7 v 1875 2200 7 256 v 2137 2200
V 1875 2207 269 7 v 2006 1954 7 473 v 1770 1249 a Fd(\033)p
1770 1365 7 4 v 1770 1485 a(\032)2951 1249 y(\030)p 2951
1365 V 2951 1485 a(\031)p 1889 1485 949 7 v 1889 1249
V 1910 1386 a Fw(ja)m(v)g(ax.crypto.CipherSpi)p 2360
2663 7 1182 v 2363 1565 a Fc(6)2387 2090 y Ff(implements)1770
2666 y Fd(\033)p 1770 2783 7 4 v 1770 2902 a(\032)2951
2666 y(\030)p 2951 2783 V 2951 2902 a(\031)p 1889 2902
949 7 v 1889 2666 V 1894 2804 a Fw(ECElGamalGFPCipher)1211
3195 y(Abbildung)27 b(4.2:)32 b Ff(ECElGamalGFPCipher)531
3481 y Fw(Abbildung)19 b(4.3)k(zeigt)f(die)f(Klasse)h
Ff(LECDomainParametersP136)o(3)p Fw(,)16 b(die)21 b(die)g(Domain)390
3593 y(P)m(arameter)i(f)856 3597 y(\177)853 3593 y(ur)e(elliptisc)m(he)
f(Kurv)m(en)h(b)s(einhaltet)f(und)h(f)2318 3597 y(\177)2315
3593 y(ur)f(die)i(Generierung)e(der)i(Kurv)m(en)390 3706
y(zust)552 3710 y(\177)552 3706 y(andig)30 b(ist.)1002
3880 y Fd(\033)p 1002 3997 7 4 v 1002 4116 a(\032)2892
3880 y(\030)p 2892 3997 V 2892 4116 a(\031)p 1121 4116
1658 7 v 1121 3880 V 1095 4018 a Fw(ja)m(v)-5 b(a.securit)m(y)d(.sp)s
(ec.AlgorithmP)m(arameterSp)s(ec)p 1947 4586 7 473 v
1950 4196 a Fc(6)1974 4367 y Ff(implements)1238 4589
y Fd(\033)p 1238 4706 7 4 v 1238 4825 a(\032)2655 4589
y(\030)p 2655 4706 V 2655 4825 a(\031)p 1358 4825 1185
7 v 1358 4589 V 1355 4735 a Fw(LECDomainP)m(arametersP1363)1068
5117 y(Abbildung)27 b(4.3:)32 b Ff(LECDomainParametersP136)o(3)531
5383 y Fw(Ein)21 b(gutes)i(Beispiel)e(f)1282 5387 y(\177)1279
5383 y(ur)h(die)g(durc)m(hdac)m(h)m(te)h(Konzeption)f(der)g(JCA)g(sieh)
m(t)g(man)h(in)e(Ab-)390 5496 y(bildung)h(4.4.)27 b(Die)e(Klassen)f
(des)h(Pro)m(viders)g(implemen)m(tieren,)e(als)i(eine)g(Sp)s
(ezialisierung,)390 5609 y(die)k(durc)m(h)h(die)f(JCA)h(v)m(orgegeb)s
(ene)i(Sc)m(hnittstelle.)p eop
%%Page: 25 26
25 25 bop 270 82 a Fw(4.4.)62 b(Der)31 b(ElGamal-Pro)m(vider)e(mit)h
(elliptisc)m(hen)d(Kurv)m(en)967 b(25)1354 327 y Fd(\033)p
1354 444 7 4 v 1354 563 a(\032)2299 327 y(\030)p 2299
444 V 2299 563 a(\031)p 1474 563 713 7 v 1474 327 V 1495
464 a Fw(ja)m(v)-5 b(a.securit)m(y)d(.Key)885 1032 y
Fc(\021)968 977 y(\021)1051 922 y(\021)1134 866 y(\021)1217
811 y(\021)1300 756 y(\021)1383 700 y(\021)1466 645 y(\021)1511
615 y(\021)-83 b(3)1287 824 y Ff(extends)p 1827 1505
7 945 v 1830 643 a Fc(6)1854 932 y Ff(implements)2692
1032 y Fc(Q)2609 977 y(Q)2526 922 y(Q)2443 866 y(Q)2360
811 y(Q)2277 756 y(Q)2194 700 y(Q)2111 645 y(Q)2066 615
y(Q)g(k)2491 824 y Ff(extends)409 1036 y Fd(\033)p 409
1152 7 4 v 409 1272 a(\032)1590 1036 y(\030)p 1590 1152
V 1590 1272 a(\031)p 529 1272 949 7 v 529 1036 V 525
1173 a Fw(ja)m(v)-5 b(a.securit)m(y)d(.Priv)j(ateKey)2063
1036 y Fd(\033)p 2063 1152 7 4 v 2063 1272 a(\032)3244
1036 y(\030)p 3244 1152 V 3244 1272 a(\031)p 2182 1272
949 7 v 2182 1036 V 2195 1173 a Fw(ja)m(v)g(a.securit)m(y)d(.PublicKey)
1354 1508 y Fd(\033)p 1354 1625 7 4 v 1354 1744 a(\032)2299
1508 y(\030)p 2299 1625 V 2299 1744 a(\031)p 1474 1744
713 7 v 1474 1508 V 1415 1646 a Fw(ECElGamalGFPKey)p
882 2214 7 945 v 885 1352 a Fc(6)909 1877 y Ff(implements)p
1531 2214 7 473 v 1535 1824 a Fc(6)1558 2005 y Ff(extends)p
2122 2214 V 2125 1824 a Fc(6)2149 2005 y Ff(extends)p
2772 2214 7 945 v 2775 1352 a Fc(6)2798 1877 y Ff(implements)527
2217 y Fd(\033)p 527 2334 7 4 v 527 2453 a(\032)1709
2217 y(\030)p 1709 2334 V 1709 2453 a(\031)p 647 2453
949 7 v 647 2217 V 563 2354 a Fw(ECElGamalGFPPriv)j(ateKey)1945
2217 y Fd(\033)p 1945 2334 7 4 v 1945 2453 a(\032)3126
2217 y(\030)p 3126 2334 V 3126 2453 a(\031)p 2064 2453
949 7 v 2064 2217 V 1997 2354 a Fw(ECElGamalGFPPublicKey)1163
2745 y(Abbildung)27 b(4.4:)k Ff(ECElGamalGFPKey)270 3037
y Fe(4.4.2)112 b(Besc)m(hreibung)38 b(der)f(Programmsc)m(hnittstelle)
270 3223 y Fw(In)i(diesem)g(Absc)m(hnitt)g(soll)g(der)g(Pro)m(vider)g
(v)m(orgestellt)h(w)m(erden.)g(Den)g(Hauptteil)f(bil-)270
3336 y(det)c(dab)s(ei)d(die)i(Sc)m(hnittstelle)f(zum)h(Ben)m(utzer,)i
(w)m(oran)e(sic)m(h)g(einige)f(Bemerkungen)h(zur)270
3449 y(in)m(ternen)g(F)-8 b(unktionsw)m(eise)33 b(ansc)m(hlie\031en.)i
(Die)f(folgenden)g(Erl)2475 3453 y(\177)2475 3449 y(auterungen)f(sind)g
(dab)s(ei,)270 3562 y(so)m(w)m(eit)e(sie)f(die)g(JCA)f(allgemein)g(b)s
(etre\013en,)i(an)f([Kn)m(u98])h(angelehn)m(t.)411 3682
y(Die)26 b(Klasse)e Ff(cdc.ec.Provider)d Fw(v)m(erw)m(altet)27
b(die)e(b)s(eiden)e(Implemen)m(tierungen,)g(d.)15 b(h.)270
3795 y Ff(ECElGamalGFPKeyPairGener)o(ator)25 b Fw(und)30
b Ff(ECElGamalGFPCipher)p Fw(.)d(Diese)k(w)m(erden)h(je-)270
3908 y(w)m(eils)c(mit)h(dem)g(K)925 3912 y(\177)922 3908
y(urzel)f("ECElGamalGFP\\)i(iden)m(ti\014ziert.)e(Der)i(eigen)m(tlic)m
(hen)f(Haupt-)270 4021 y(klasse)38 b(der)f(JCA,)g Ff
(java.security.Security)p Fw(,)32 b(die)37 b(die)g(V)-8
b(erbindung)35 b(zwisc)m(hen)j(den)270 4134 y(Konzeptklassen)49
b(und)f(den)h(Implemen)m(tierungen)e(herstellt,)i(m)m(u\031)h(n)m(un)e
(unser)h(Pro-)270 4247 y(vider)44 b(mit)g(der)g(Metho)s(de)h
Ff(addProvider)1833 4251 y Fw(\177)1830 4247 y(ub)s(ergeb)s(en)e(w)m
(erden.)i(Ansc)m(hlie\031end)f(k)-5 b(ann)270 4360 y(mit)26
b(den)g Ff(getInstance)p Fw(-Metho)s(den)e(v)m(on)j Ff
(java.security.KeyPairGen)o(era)o(tor)20 b Fw(bzw.)270
4473 y Ff(javax.crypto.Cipher)30 b Fw(un)m(ter)35 b(Angab)s(e)g(des)f
(Kurznamens)g(ein)g(Ob)5 b(jekt)35 b(der)f(jew)m(eili-)270
4585 y(gen)e(Klasse)e(erzeugt)j(w)m(erden,)e(mit)f(denen)h(man)f
(direkt)h(auf)g(unsere)f(Implemen)m(tierung)270 4698
y(zugreifen)e(k)-5 b(ann.)28 b(An)g(dieser)g(Stelle)g(ist)g(es)h(auc)m
(h)f(m)2066 4702 y(\177)2066 4698 y(oglic)m(h,)h(un)m(ter)f(anderen)g
(Kurznamen)270 4811 y(die)36 b(Implemen)m(tierungen)e(w)m(eiterer)j
(Pro)m(vider)e(anzusprec)m(hen,)i(sofern)f(diese)g(auc)m(h)g(der)270
4924 y(Klasse)30 b Ff(java.security.Security)24 b Fw(b)s(ek)-5
b(ann)m(t)30 b(sind.)411 5044 y(Nac)m(h)46 b(au\031en)f(hin)f(deutet)h
(nic)m(h)m(ts)g(auf)g(die)f(Aufrufk)-5 b(ask)g(ade)44
b(hin,)f(die)i(durc)m(h)f(den)270 5157 y(Aufruf)38 b(einer)g
Ff(getInstance)p Fw(-Metho)s(de)f(ausgel)1987 5161 y(\177)1987
5157 y(ost)j(wird.)e(In)m(tern)g(wird)g(durc)m(h)g(diesen)270
5270 y(Befehl)h(die)g(Klasse)g Ff(java.security.Security)34
b Fw(angespro)s(c)m(hen,)40 b(die)e(un)m(ter)i(den)f(ihr)270
5383 y(b)s(ek)-5 b(ann)m(ten)33 b(Pro)m(vidern)f(suc)m(h)m(t,)j(ob)e
(die)g(en)m(tsprec)m(hende)h(Implemen)m(tierung)e(v)m(orhanden)270
5496 y(ist.)27 b(Bei)f(Erfolg)h(erh)971 5500 y(\177)971
5496 y(alt)f(man)h(eine)f(Instanz)h(v)m(on)g(z.)16 b(B.)27
b Ff(javax.crypto.Cipher)22 b Fw(zur)3232 5500 y(\177)3229
5496 y(uc)m(k,)270 5609 y(w)m(elc)m(he)31 b(die)e(gew)858
5613 y(\177)855 5609 y(unsc)m(h)m(te)j(Implemen)m(tierung)c(b)s(en)m
(utzt.)p eop
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26 26 bop 390 82 a Fw(26)1313 b(KAPITEL)30 b(4.)61 b(DIE)30
b(IMPLEMENTIER)m(UNG)542 396 y(\177)531 415 y(Ub)s(er)21
b(die)h(Sc)m(hnittstelle)f(v)m(on)h Ff(java.security.KeyPairGene)o
(rato)o(r)16 b Fw(m)m(u\031)23 b(n)m(un)e(un-)390 528
y(ser)27 b(Ob)5 b(jekt)26 b(initialisiert)d(w)m(erden.)k(Da)g(die)f
(Metho)s(de)h Ff(initialize)2789 532 y Fw(\177)2786 528
y(ub)s(erladen)e(ist,)h(gibt)390 641 y(es)g(im)e(w)m(esen)m(tlic)m(hen)
h(zw)m(ei)g(M)1413 645 y(\177)1413 641 y(oglic)m(hk)m(eiten.)h(Man)g(k)
-5 b(ann)25 b(als)f(Argumen)m(t)i(einen)e Ff(int)p Fw(-W)-8
b(ert)393 758 y(\177)390 754 y(ub)s(ergeb)s(en,)34 b(der)h(in)g(den)g
(meisten)g(F)1709 758 y(\177)1709 754 y(allen)f(als)i(Bitl)2225
758 y(\177)2225 754 y(ange)f(des)g(Sc)m(hl)2771 758 y(\177)2768
754 y(ussels)f(in)m(terpretiert)390 867 y(wird.)e(Denkbar)i(ist)f(ab)s
(er)g(auc)m(h,)i(sic)m(h)e(die)g(grundlegenden)f(P)m(arameter)j(f)3003
871 y(\177)3000 867 y(ur)e(den)g(Algo-)390 980 y(rithm)m(us)d(\(s.)i
(Kapitel)f(4.2.1\))j(sc)m(hon)e(v)m(orher)g(zu)g(erzeugen,)g(die)f
(dann)g(in)g(die)g(Sc)m(hl)3296 984 y(\177)3293 980 y(ussel-)390
1093 y(generierung)h(eingehen.)h(Das)g(funktioniert)e(jedo)s(c)m(h)i(n)
m(ur,)f(w)m(enn)h(die)f(Implemen)m(tierung)390 1205 y(dies)f(un)m
(terst)858 1209 y(\177)855 1205 y(utzt,)i(w)m(as)g(b)s(ei)e(unserem)g
(Pro)m(vider)h(der)f(F)-8 b(all)32 b(ist.)g(Durc)m(h)h(Aufruf)d(der)i
(Me-)390 1318 y(tho)s(de)39 b Ff(generateKeyPair)c Fw(erh)1536
1322 y(\177)1536 1318 y(alt)k(man)f(sc)m(hlie\031lic)m(h)g(als)h(R)2546
1322 y(\177)2543 1318 y(uc)m(kgab)s(ew)m(ert)h(ein)e(Ob)5
b(jekt)390 1431 y(der)30 b(Klasse)f Ff(java.security.KeyPair)p
Fw(,)c(w)m(elc)m(hes)30 b(lediglic)m(h)e(einen)h(Priv)-5
b(ate)30 b(und)f(einen)390 1544 y(Public)d(Key)j(en)m(th)1034
1548 y(\177)1034 1544 y(alt,)g(auf)f(w)m(elc)m(he)h(mit)f(den)g(Metho)s
(den)h Ff(getPrivate)c Fw(und)i Ff(getPublic)390 1657
y Fw(zugegri\013en)j(w)m(erden)g(k)-5 b(ann.)531 1770
y(Beim)48 b Ff(Cipher)p Fw(-Ob)5 b(jekt)47 b(sieh)m(t)h(die)f
(Initialisierung)2422 1774 y(\177)2422 1770 y(ahnlic)m(h)f(aus.)j(Eine)
e(wic)m(h)m(tige)390 1883 y(Angab)s(e)39 b(ist)g(der)g(Mo)s(dus,)g(d.)
15 b(h.)39 b(ob)g(v)m(er-)i(o)s(der)e(en)m(tsc)m(hl)2390
1887 y(\177)2387 1883 y(usselt)f(w)m(erden)h(soll.)g(Des)h(w)m(ei-)390
1996 y(teren)46 b(b)s(en)783 2000 y(\177)783 1996 y(otigt)g(man)g
(einen)f(Sc)m(hl)1654 2000 y(\177)1651 1996 y(ussel,)f(der)i(nat)2216
2000 y(\177)2213 1996 y(urlic)m(h)e(zum)i(Mo)s(dus)f(passen)h(m)m
(u\031.)390 2109 y(Soll)g(v)m(ersc)m(hl)858 2113 y(\177)855
2109 y(usselt)h(w)m(erden,)g(so)h(ist)f(ein)f(Public)g(Key)h(dieses)g
(V)-8 b(erfahrens,)47 b(also)h(ein)390 2222 y Ff(ECElGamalGFPPublicKey)
p Fw(-Ob)5 b(jekt)22 b(erforderlic)m(h.)j(Bei)i(der)g(En)m(tsc)m(hl)
2839 2226 y(\177)2836 2222 y(usselung)e(m)m(u\031)j(ein)390
2335 y(en)m(tsprec)m(hender)22 b(Priv)-5 b(ate)21 b(Key)1495
2339 y(\177)1493 2335 y(ub)s(ergeb)s(en)f(w)m(erden.)h(Aufgrund)f(der)h
(langen)h(Klassenna-)390 2447 y(men,)34 b(soll)f(b)s(ei)g(unseren)g
(Pro)m(viderklassen)f(der)i(P)m(ak)m(etpfad)h(w)m(eggelassen)h(w)m
(erden,)e(w)m(as)390 2560 y(jedo)s(c)m(h)42 b(durc)m(h)f(das)h(Pr)1226
2564 y(\177)1226 2560 y(a\014x)f("ECElGamalGFP\\)h(nic)m(h)m(t)g(zu)g
(V)-8 b(erw)m(ec)m(hslungen)42 b(f)3286 2564 y(\177)3283
2560 y(uhren)390 2673 y(wird.)25 b(Auc)m(h)i(hier)e(k)-5
b(ann)26 b(man)h(wieder)e(die)h(sp)s(ezi\014sc)m(hen)g(P)m(arameter)i
(zus)2928 2677 y(\177)2928 2673 y(atzlic)m(h)3217 2677
y(\177)3214 2673 y(ub)s(erge-)390 2786 y(b)s(en.)34 b(Sie)f(m)823
2790 y(\177)820 2786 y(ussen)h(ab)s(er)f(iden)m(tisc)m(h)h(mit)g(denen)
f(sein,)h(die)g(f)2491 2790 y(\177)2488 2786 y(ur)f(die)h(Sc)m(hl)2927
2790 y(\177)2924 2786 y(usselerzeugung)390 2899 y(v)m(erw)m(endet)d
(wurden.)531 3012 y(Es)24 b(wurde)g(sc)m(hon)g(gesagt,)j(da\031)e(die)f
(Erzeugung)g(des)g(Sc)m(hl)2534 3016 y(\177)2531 3012
y(ussels)e(nic)m(h)m(t)j(nac)m(h)m(v)m(ollzieh-)390 3125
y(bar)36 b(sein)f(darf,)h(um)g(die)g(Sic)m(herheit)f(nic)m(h)m(t)h(in)f
(F)-8 b(rage)39 b(zu)d(stellen.)g(Das)h(ist)f(ein)f(Grund,)390
3238 y(w)m(arum)24 b(es)i(m)863 3242 y(\177)863 3238
y(oglic)m(h)e(sein)g(sollte,)h(die)f(Domain)h(P)m(arameter)h(selbst)e
(zu)h(generieren.)g(Gera-)390 3351 y(de)33 b(b)s(ei)f(den)h
(Pseudozufallszahlengeneratoren)e(stellt)i(sic)m(h)g(immer)f(wieder)g
(die)g(F)-8 b(rage,)390 3464 y(wie)30 b(zuf)671 3468
y(\177)671 3464 y(allig)e(die)i(Ergebnisse)f(wirklic)m(h)e(sind.)531
3577 y(Jetzt)35 b(sind)c(wir)h(in)g(der)h(Lage,)i(das)e
Ff(Cipher)p Fw(-Ob)5 b(jekt)32 b(zu)i(b)s(en)m(utzen,)f(um)f(Nac)m
(hric)m(h-)390 3689 y(ten)g(zu)f(v)m(er-)h(bzw.)f(zu)g(en)m(tsc)m(hl)
1464 3693 y(\177)1461 3689 y(usseln.)f(Dazu)j(gibt)d(es)i(mehrere)f
(Alternativ)m(en.)g(Mit)h(den)390 3802 y Ff(update)p
Fw(-Metho)s(den)24 b(k)1179 3806 y(\177)1179 3802 y(onnen)g(b)s
(eliebig)f(viele)i(Daten)h(zur)f(Bearb)s(eitung)2909
3806 y(\177)2906 3802 y(ub)s(ergeb)s(en)f(w)m(er-)390
3915 y(den.)f(Alle)f(ganzen)h(Bl)1147 3919 y(\177)1147
3915 y(oc)m(k)m(e)i(w)m(erden)e(hierb)s(ei)d(auc)m(h)j(v)m(er-)h(bzw.)f
(en)m(tsc)m(hl)2799 3919 y(\177)2796 3915 y(usselt)f(und)g(zur)3347
3919 y(\177)3344 3915 y(uc)m(k-)390 4028 y(gegeb)s(en.)44
b(Durc)m(h)g(die)1233 4009 y(\177)1221 4028 y(Ub)s(erladung)d(dieser)i
(Metho)s(de)h(hat)f(der)g(Ben)m(utzer)i(dab)s(ei)d(die)390
4141 y(M)473 4145 y(\177)473 4141 y(oglic)m(hk)m(eit,)27
b(zwisc)m(hen)f(der)g(R)1497 4145 y(\177)1494 4141 y(uc)m(kgab)s(e)h
(als)f(Metho)s(denargumen)m(t)h(und)e(als)h(F)-8 b(unktions-)390
4254 y(w)m(ert)30 b(auszu)m(w)880 4258 y(\177)880 4254
y(ahlen.)e(Die)h Ff(doFinal)p Fw(-Metho)s(den)f(un)m(tersc)m(heiden)g
(sic)m(h)h(v)m(on)h(den)e Ff(update)p Fw(-)390 4367 y(Metho)s(den)41
b(insofern,)e(als)h(sie)g(die)g(V)-8 b(ersc)m(hl)1942
4371 y(\177)1939 4367 y(usselung)39 b(absc)m(hlie\031en,)i(d.)15
b(h.)40 b(den)g(letzten,)390 4480 y(ev)m(en)m(tuell)31
b(un)m(v)m(ollst)1084 4484 y(\177)1084 4480 y(andigen,)e(Blo)s(c)m(k)j
(auc)m(h)f(no)s(c)m(h)g(v)m(ersc)m(hl)2409 4484 y(\177)2406
4480 y(usseln.)f(Das)h(Ob)5 b(jekt)31 b(k)-5 b(ann)31
b(je-)390 4593 y(do)s(c)m(h)44 b(w)m(eiter)g(v)m(erw)m(endet)h(w)m
(erden.)e(Sogar)i(eine)e(erneute)i(Initialisierung)39
b(mit)44 b(neuen)390 4706 y(P)m(arametern)31 b(und)e(neuem)h(Mo)s(dus)f
(ist)h(m)1855 4710 y(\177)1855 4706 y(oglic)m(h.)531
4819 y(Absc)m(hlie\031end)77 b(w)m(ollen)h(wir)f(uns)h(no)s(c)m(h)g
(der)g(Implemen)m(tierung)f(zu)m(w)m(enden.)390 4932
y(Um)71 b(einen)f(Pro)m(vider)h(zu)g(en)m(t)m(wic)m(k)m(eln,)h(ist)f
(es)g(erforderlic)m(h,)f(die)g(en)m(tsprec)m(hen-)390
5044 y(de)86 b(Klasse)g(zu)g(b)s(eerb)s(en)f(und)g(deren)g(abstrakte)j
(Metho)s(den)e(zu)g(rede\014nie-)390 5157 y(ren)73 b(bzw.)h(zu)g
(implemen)m(tieren.)e(In)h(unserem)g(F)-8 b(all)74 b(w)m(aren)g(das)g
(die)f(Klassen)390 5270 y Ff(java.security.KeyPairGen)o(erat)o(orSp)o
(i)41 b Fw(und)46 b Ff(javax.crypto.CipherSpi)p Fw(.)41
b(Zwi-)390 5383 y(sc)m(hen)65 b Ff(javax.crypto.CipherSpi)58
b Fw(und)64 b Ff(ECElGamalGFPCipher)59 b Fw(wurde)64
b(zus)3359 5387 y(\177)3359 5383 y(atz-)390 5496 y(lic)m(h)50
b(no)s(c)m(h)i(eine)f(Klasse)g Ff(ECElGamalGFPBlockCipher)45
b Fw(gesetzt,)55 b(die)50 b(sic)m(h)h(um)g(das)390 5609
y(Blo)s(c)m(khandling)h(k)1062 5613 y(\177)1059 5609
y(ummert.)i(Alle)f(anderen)g(Metho)s(den,)i(die)e(direkt)g(v)m(on)i
(dem)e(Al-)p eop
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27 27 bop 270 82 a Fw(4.4.)62 b(Der)31 b(ElGamal-Pro)m(vider)e(mit)h
(elliptisc)m(hen)d(Kurv)m(en)967 b(27)270 415 y(gorithm)m(us)67
b(abhingen,)f(wurden)g(nic)m(h)m(t)h(implemen)m(tiert.)f(Der)i(Co)s(de)
f(der)g(Klasse)270 528 y Ff(ECElGamalGFPBlockCipher)24
b Fw(en)m(tstamm)m(t)32 b(haupts)2122 532 y(\177)2122
528 y(ac)m(hlic)m(h)d([Kn)m(u98)q(].)411 641 y(In)23
b Ff(ECElGamalGFPCipher)c Fw(steh)m(t)25 b(sc)m(hlie\031lic)m(h)d(die)h
(Implemen)m(tierung)e(des)j(Algorith-)270 754 y(m)m(us)29
b(aus)g(Kapitel)g(3.2.)h(Die)g(eigen)m(tlic)m(he)f(Berec)m(hn)m(ung)h
(\014ndet)f(dab)s(ei)f(in)g(den)h(Metho)s(den)270 867
y Ff(encryptBlock)24 b Fw(und)h Ff(decryptBlock)f Fw(statt.)29
b(Ein)c(V)-8 b(orteil)27 b(dieses)g(Aufbaus)f(liegt)g(darin,)270
980 y(da\031)36 b(b)s(ei)e(einer)h(Mo)s(di\014zierung)f(des)h
(Algorithm)m(us,)g(als)g(Kernst)2539 984 y(\177)2536
980 y(uc)m(k)h(der)f(Implemen)m(tie-)270 1093 y(rung,)30
b(haupts)777 1097 y(\177)777 1093 y(ac)m(hlic)m(h)f(diese)h(b)s(eiden)e
(Metho)s(den)i(angepa\031t)i(w)m(erden)e(m)2800 1097
y(\177)2797 1093 y(ussen.)411 1205 y(Das)87 b(Konzept)f(der)g(JCA)f(f)
1577 1209 y(\177)1574 1205 y(uhrt)g(zu)h(relativ)g(vielen)e(Klassen,)i
(die)f(di-)270 1318 y(rekt)80 b(den)f(Sc)m(hl)898 1322
y(\177)895 1318 y(ussel)f(b)s(etre\013en.)i(Abbildung)d(4.4)k(mac)m(h)m
(te)g(das)f(sc)m(hon)g(deut-)270 1431 y(lic)m(h,)56 b(an)h(dieser)f
(Stelle)h(soll)e(no)s(c)m(h)i(eine)g(kurze)g(Erl)2246
1435 y(\177)2246 1431 y(auterung)f(folgen.)h(Durc)m(h)g(die)270
1544 y(JCA)63 b(sind)f(die)h(drei)g(Klassen)g(\(eigen)m(tlic)m(h)h
Ff(interfaces)p Fw(\))d Ff(java.security.Key)p Fw(,)270
1657 y Ff(java.security.PublicKey)24 b Fw(und)30 b Ff
(java.security.PrivateKe)o(y)25 b Fw(v)m(orgegeb)s(en.)32
b(Die)270 1770 y(erste)h(wird)e(durc)m(h)h Ff(ECElGamalGFPKey)d
Fw(implemen)m(tiert.)i(Da)m(v)m(on)j(wiederum)d(erb)s(en)h(die)270
1883 y(Klassen)38 b Ff(ECElGamalGFPPublicKey)32 b Fw(und)37
b Ff(ECElGamalGFPPrivateKey)p Fw(,)c(die)38 b(zus)3239
1887 y(\177)3239 1883 y(atz-)270 1996 y(lic)m(h)e(das)g(jew)m(eilige)g
Ff(interface)e Fw(der)i(JCA)h(implemen)m(tieren.)d(Durc)m(h)j(diese)f
(T)-8 b(renn)m(ung)270 2109 y(des)33 b(Aufgab)s(en)m(b)s(ereic)m(hs)e
(erh)1287 2113 y(\177)1287 2109 y(alt)i(man)f(somit)h(eine)f(Struktur,)
g(b)s(ei)f(der)h(sc)m(hon)h(die)f(Klas-)270 2222 y(sendiagramme)e
(deutlic)m(h)f(mac)m(hen,)i(w)m(elc)m(hen)f(Sinn)e(die)i(einzelnen)f
(Klassen)g(hab)s(en.)p eop
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28 28 bop eop
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29 29 bop 270 1039 a Fr(Kapitel)66 b(5)270 1455 y Fv(E\016zienz)270
1901 y Fw(Die)22 b(E\016zienz)f(ist)g(b)s(ei)f(kryptographisc)m(hen)h
(An)m(w)m(endungen)f(in)h(der)g(Regel)h(ein)e(sehr)h(wic)m(h-)270
2014 y(tiger)27 b(Gesic)m(h)m(tspunkt.)h(V)-8 b(on)28
b(JA)-10 b(V)g(A)27 b(darf)g(man)g(in)f(dieser)h(Hinsic)m(h)m(t)f
(allerdings)f(nic)m(h)m(t)j(zu-)270 2127 y(viel)21 b(erw)m(arten.)h(Es)
g(wurde)e(sc)m(hon)i(einmal)f(erw)1877 2131 y(\177)1877
2127 y(ahn)m(t,)h(da\031)g(dies)f(w)m(ohl)g(der)h(gr)2876
2131 y(\177)2876 2127 y(o\031te)h(Nac)m(h)m(teil)270
2240 y(v)m(on)38 b(JA)-10 b(V)g(A)37 b(sei.)h(Im)f(gro\031en)h(und)e
(ganzen)j(hat)e(sic)m(h)g(das)h(auc)m(h)g(b)s(ei)e(der)h(v)m
(orliegenden)270 2353 y(Implemen)m(tierung)31 b(gezeigt.)k(Neb)s(en)d
(einer)g(Au\015istung)f(der)i(Timings)e(soll)g(ferner)i(auc)m(h)270
2466 y(v)m(ersuc)m(h)m(t)e(w)m(erden,)f(die)g(Gr)1222
2470 y(\177)1219 2466 y(unde)f(hierf)1625 2470 y(\177)1622
2466 y(ur)f(herauszu\014nden.)411 2580 y(Der)43 b(Pro)m(vider)f(wurde)f
(f)1298 2584 y(\177)1295 2580 y(ur)g(die)h(Timings)e(auf)j(einer)e
(UltraSP)-8 b(AR)m(C-I)s(Ii)42 b(v)m(on)h Fg(SUN)270
2693 y Fw(mit)33 b(333)i(MHz)f(und)e(512)j(MByte)g(Hauptsp)s(eic)m(her)
d(ausgef)2345 2697 y(\177)2342 2693 y(uhrt.)i(Gemessen)g(wurde)e(die)
270 2806 y(v)m(erbrauc)m(h)m(te)38 b(CPU-Zeit,)e(w)m(ob)s(ei)g(f)1511
2810 y(\177)1508 2806 y(ur)g(die)g(V)-8 b(er-)37 b(und)e(En)m(tsc)m(hl)
2473 2810 y(\177)2470 2806 y(usselungen)g(der)h(Pro\014ler)270
2919 y(v)m(on)d Fg(JProb)s(e)g Fw(v)m(erw)m(endet)g(wurde.)e(Mit)i
(diesem)e(Programm)i(w)m(ar)f(es)h(auc)m(h)g(m)2987 2923
y(\177)2987 2919 y(oglic)m(h,)f(die)270 3031 y(Stellen)25
b(innerhalb)e(des)j(Pro)m(viders)f(herauszu\014nden,)f(die)h(die)h
(meiste)g(Zeit)g(in)e(Anspruc)m(h)270 3144 y(nahmen.)270
3437 y Fz(5.1)135 b(Kurv)l(engenerierung)953 3800 y Fw(T)-8
b(ab)s(elle)29 b(5.1:)j(Timings)c(der)i(Kurv)m(engenerierung)p
833 3929 1995 4 v 831 4041 4 113 v 883 4008 a(Sc)m(hl)1051
4012 y(\177)1048 4008 y(ussell)1263 4012 y(\177)1263
4008 y(ange)p 1490 4041 V 97 w(CPU-Zeit)p 1972 4041 V
99 w(CPU-Zeit)g(je)h(Kurv)m(e)p 2826 4041 V 833 4045
1995 4 v 833 4061 V 831 4174 4 113 v 1150 4140 a(120)h(Bit)p
1490 4174 V 254 w(85.5)g(s)p 1972 4174 V 626 w(22.5)g(s)p
2826 4174 V 833 4178 1995 4 v 831 4291 4 113 v 1150 4257
a(160)g(Bit)p 1490 4291 V 209 w(537.9)g(s)p 1972 4291
V 580 w(104.9)h(s)p 2826 4291 V 833 4294 1995 4 v 831
4407 4 113 v 1150 4373 a(200)f(Bit)p 1490 4407 V 163
w(1840.7)h(s)p 1972 4407 V 580 w(480.0)g(s)p 2826 4407
V 833 4410 1995 4 v 831 4523 4 113 v 1150 4489 a(240)f(Bit)p
1490 4523 V 163 w(3079.9)h(s)p 1972 4523 V 580 w(968.3)g(s)p
2826 4523 V 833 4526 1995 4 v 411 4817 a(Bei)25 b(diesen)f(Timings)f
(wurden)g(kryptographisc)m(h)h(stark)m(e)i(Kurv)m(en)e(erzeugt.)i(Es)f
(wur-)270 4930 y(den)35 b(dab)s(ei)g(solange)h(Kurv)m(en)f(generiert,)i
(bis)d(eine)i(die)2256 4934 y(\177)2254 4930 y(ub)s(erpr)2525
4934 y(\177)2522 4930 y(uften)d(Kriterien)h(erf)3262
4934 y(\177)3259 4930 y(ull-)270 5043 y(te.)42 b(Dadurc)m(h)f(k)-5
b(amen)40 b(sehr)h(un)m(tersc)m(hiedlic)m(he)e(W)-8 b(erte)42
b(f)2271 5047 y(\177)2268 5043 y(ur)e(die)g(einzelnen)g(Durc)m(hl)3227
5047 y(\177)3227 5043 y(aufe)270 5156 y(zustande,)g(w)m(as)f(nat)1001
5160 y(\177)998 5156 y(urlic)m(h)f(da)m(v)m(on)i(abhing,)e(wie)h(sc)m
(hnell)f(eine)h(gute)h(Kurv)m(e)f(gefunden)270 5269 y(wurde.)411
5383 y(Man)22 b(erk)m(enn)m(t)h(jedo)s(c)m(h)f(auc)m(h)g(den)g(Anstieg)
g(der)f(v)m(erbrauc)m(h)m(ten)i(Zeit)e(in)g(Abh)3032
5387 y(\177)3032 5383 y(angigk)m(eit)270 5496 y(der)41
b(Bitl)589 5500 y(\177)589 5496 y(ange)h(des)f(Sc)m(hl)1148
5500 y(\177)1145 5496 y(ussels.)f(Die)h(LiDIA-Klasse)g
Ff(EC)p 2291 5496 29 4 v 34 w(domain)p 2613 5496 V 33
w(parameters)p 3126 5496 V 31 w(P1363)270 5609 y Fw(f)301
5613 y(\177)298 5609 y(uhrt)24 b(extrem)h(viele)f(Berec)m(hn)m(ungen)i
(aus,)f(um)f(eine)h(kryptographisc)m(h)f(geeignete)i(Kurv)m(e)1785
5858 y(29)p eop
%%Page: 30 31
30 30 bop 390 82 a Fw(30)1963 b(KAPITEL)29 b(5.)62 b(EFFIZIENZ)390
415 y(zu)21 b(\014nden.)f(Die)i(Kurv)m(e,)f(die)g(dann)f(sp)1680
419 y(\177)1680 415 y(ater)i(f)1889 419 y(\177)1886 415
y(ur)e(die)h(V)-8 b(ersc)m(hl)2420 419 y(\177)2417 415
y(usselung)19 b(b)s(en)m(utzt)j(wird,)e(k)-5 b(ann)390
528 y(jedo)s(c)m(h)34 b(v)m(orb)s(erec)m(hnet)h(w)m(erden,)f(so)h
(da\031)g(sic)m(h)f(der)f(hohe)i(Zeitv)m(erbrauc)m(h)e(nic)m(h)m(t)i
(auf)f(die)390 641 y(E\016zienz)c(der)g(V)-8 b(ersc)m(hl)1201
645 y(\177)1198 641 y(usselung)28 b(auswirkt.)531 754
y(Zur)k(Zeit)h(wird)f(eine)h(Bitl)1448 758 y(\177)1448
754 y(ange)g(des)h(Sc)m(hl)1991 758 y(\177)1988 754 y(ussels)d(v)m(on)j
(200)h(empfohlen,)d(um)h(eine)g(re-)390 867 y(lativ)k(hohe)g(Sic)m
(herheit)f(zu)i(erhalten.)f(Diese)h(Zahl)2238 871 y(\177)2238
867 y(andert)f(sic)m(h)g(jedo)s(c)m(h)h(parallel)d(zum)390
980 y(tec)m(hnisc)m(hen)g(F)-8 b(ortsc)m(hritt)36 b(in)e(der)h
(Computerindustrie.)d(Auc)m(h)j(die)f(En)m(tdec)m(kung)i(eines)390
1093 y(e\016zien)m(teren)28 b(Algorithm)m(us)f(zur)g(Berec)m(hn)m(ung)h
(v)m(on)h(Diskreten)e(Logarithmen)h(b)s(ei)e(ellip-)390
1205 y(tisc)m(hen)31 b(Kurv)m(en)g(k)1070 1209 y(\177)1070
1205 y(onn)m(te)h(dazu)g(f)1571 1209 y(\177)1568 1205
y(uhren,)e(da\031)i(obige)f(Empfehlung)e(k)m(eine)j(ausreic)m(hende)390
1318 y(Sic)m(herheit)d(mehr)h(bietet.)390 1604 y Fz(5.2)135
b(V)-11 b(ersc)l(hl)1128 1610 y(\177)1124 1604 y(usselung)390
1807 y Fw(In)25 b(den)g(folgenden)g(T)-8 b(ab)s(ellen)24
b(sind)f(der)j(Zeitv)m(erbrauc)m(h)f(und)f(die)g(daraus)h
(resultierenden)390 1920 y(V)-8 b(er-)26 b(bzw.)f(En)m(tsc)m(hl)1087
1924 y(\177)1084 1920 y(usselungsraten)e(dargestellt.)i(Es)g(wurde)f
(dab)s(ei)g(eine)g(Datei)i(v)m(erw)m(en-)390 2033 y(det,)32
b(die)e(eine)h(Gr)1015 2037 y(\177)1015 2033 y(o\031e)i(v)m(on)f(et)m
(w)m(a)h(4)f(kByte)g(hatte.)h(Die)e(W)-8 b(erte)33 b(wurden)d(auc)m(h)h
(hier)3333 2037 y(\177)3330 2033 y(ub)s(er)390 2145 y(die)g(Durc)m(hl)
808 2149 y(\177)808 2145 y(aufe)g(gemittelt,)i(um)e(aussagekr)1959
2149 y(\177)1959 2145 y(aftig)h(zu)g(sein.)f(T)-8 b(rotzdem)32
b(ist)f(der)h(Zeitv)m(er-)390 2258 y(brauc)m(h)g(v)m(erh)865
2262 y(\177)865 2258 y(altnism)1160 2262 y(\177)1160
2258 y(a\031ig)f(ho)s(c)m(h.)i(Es)f(soll)f(daher)h(auc)m(h)h(v)m(ersuc)
m(h)m(t)g(w)m(erden,)g(die)e(Gr)3321 2262 y(\177)3318
2258 y(unde)390 2371 y(hierf)574 2375 y(\177)571 2371
y(ur)d(anzugeb)s(en.)1139 2632 y(T)-8 b(ab)s(elle)30
b(5.2:)h(Timings)d(der)i(V)-8 b(ersc)m(hl)2429 2636 y(\177)2426
2632 y(usselung)p 923 2761 2054 4 v 921 2874 4 113 v
973 2840 a(Sc)m(hl)1141 2844 y(\177)1138 2840 y(ussell)1353
2844 y(\177)1353 2840 y(ange)p 1580 2874 V 98 w(CPU-Zeit)p
2062 2874 V 99 w(V)g(ersc)m(hl)2403 2844 y(\177)2400
2840 y(usselungsrate)p 2975 2874 V 923 2878 2054 4 v
923 2894 V 921 3007 4 113 v 1240 2973 a(120)32 b(Bit)p
1580 3007 V 254 w(18.1)g(s)p 2062 3007 V 424 w(1812.2)h(Bit/s)p
2975 3007 V 923 3010 2054 4 v 921 3123 4 113 v 1240 3089
a(160)f(Bit)p 1580 3123 V 254 w(23.6)g(s)p 2062 3123
V 424 w(1389.8)h(Bit/s)p 2975 3123 V 923 3127 2054 4
v 921 3240 4 113 v 1240 3206 a(200)f(Bit)p 1580 3240
V 254 w(29.0)g(s)p 2062 3240 V 424 w(1131.0)h(Bit/s)p
2975 3240 V 923 3243 2054 4 v 921 3356 4 113 v 1240 3322
a(240)f(Bit)p 1580 3356 V 254 w(36.2)g(s)p 2062 3356
V 469 w(906.1)h(Bit/s)p 2975 3356 V 923 3359 2054 4 v
1135 3800 a(T)-8 b(ab)s(elle)29 b(5.3:)j(Timings)c(der)i(En)m(tsc)m(hl)
2434 3804 y(\177)2431 3800 y(usselung)p 919 3929 2063
4 v 917 4042 4 113 v 969 4008 a(Sc)m(hl)1137 4012 y(\177)1134
4008 y(ussell)1349 4012 y(\177)1349 4008 y(ange)p 1576
4042 V 97 w(CPU-Zeit)p 2058 4042 V 100 w(En)m(tsc)m(hl)2408
4012 y(\177)2405 4008 y(usselungsrate)p 2979 4042 V 919
4045 2063 4 v 919 4061 V 917 4174 4 113 v 1236 4140 a(120)i(Bit)p
1576 4174 V 300 w(8.8)f(s)p 2058 4174 V 432 w(3727.3)i(Bit/s)p
2979 4174 V 919 4178 2063 4 v 917 4291 4 113 v 1236 4257
a(160)f(Bit)p 1576 4291 V 254 w(11.8)g(s)p 2058 4291
V 432 w(2779.7)h(Bit/s)p 2979 4291 V 919 4294 2063 4
v 917 4407 4 113 v 1236 4373 a(200)f(Bit)p 1576 4407
V 254 w(14.4)g(s)p 2058 4407 V 432 w(2277.8)h(Bit/s)p
2979 4407 V 919 4410 2063 4 v 917 4523 4 113 v 1236 4489
a(240)f(Bit)p 1576 4523 V 254 w(17.8)g(s)p 2058 4523
V 432 w(1842.7)h(Bit/s)p 2979 4523 V 919 4526 2063 4
v 531 4819 a(Ein)i(Asp)s(ekt)g(ist)h(dab)s(ei)e(die)h(V)-8
b(erw)m(endung)36 b(v)m(on)g(JA)-10 b(V)g(A)36 b(als)f
(Programmiersprac)m(he.)390 4932 y(Einige)22 b(ihrer)g(V)-8
b(orz)1057 4936 y(\177)1054 4932 y(uge,)25 b(wie)e(z.)16
b(B.)24 b(die)e(Sp)s(eic)m(herv)m(erw)m(altung,)h(ziehen)f(eine)h
(Beein)m(tr)3287 4936 y(\177)3287 4932 y(ac)m(h)m(ti-)390
5044 y(gung)34 b(der)g(E\016zienz)h(unmittelbar)d(nac)m(h)j(sic)m(h.)f
(In)g(Kapitel)g(6)h(folgt)f(eine)g(detailliertere)390
5157 y(Beurteilung)29 b(der)h(JCA.)531 5270 y(Innerhalb)41
b(des)h(Pro)m(viders)g(sind)g(haupts)2019 5274 y(\177)2019
5270 y(ac)m(hlic)m(h)f(drei)h(Befehle)h(f)2891 5274 y(\177)2888
5270 y(ur)f(den)h(Zeitv)m(er-)390 5383 y(brauc)m(h)22
b(v)m(eran)m(t)m(w)m(ortlic)m(h.)j(Zum)c(einen)h(ist)h(dies)f(die)g
(Erzeugung)g(der)h(Zufallszahl)d Fq(k)26 b Fw(\(siehe)390
5496 y(hierzu)k(De\014nition)h(9\))i(in)d(der)i(Metho)s(de)g
Ff(encryptBlock)c Fw(v)m(on)33 b Ff(ECElGamalGFPCipher)p
Fw(.)390 5609 y(Auf)k(der)f(anderen)h(Seite)g(b)s(en)1462
5613 y(\177)1462 5609 y(otigen)g(die)f(Punktm)m(ultiplik)-5
b(ationen)33 b(in)j Ff(encryptBlock)p eop
%%Page: 31 32
31 31 bop 270 82 a Fw(5.2.)62 b(V)-8 b(ersc)m(hl)761
86 y(\177)758 82 y(usselung)2205 b(31)270 415 y(bzw.)24
b Ff(decryptBlock)e Fw(erw)m(artungsgem)1673 419 y(\177)1673
415 y(a\031)k(viel)d(Zeit.)i(Den)g(prozen)m(tualen)f(An)m(teil)g(an)g
(der)270 528 y(Gesam)m(tzeit)h(zeigt)f(T)-8 b(ab)s(elle)22
b(5.4.)j(Man)e(sieh)m(t,)g(da\031)h(an)f(diesen)f(Stellen)g(in)g(der)g
(Regel)i(mehr)270 641 y(als)30 b(90\045)h(der)f(Zeit)g(v)m(erbrauc)m(h)
m(t)h(wird.)1072 892 y(T)-8 b(ab)s(elle)29 b(5.4:)j(Zeitin)m(tensiv)m
(e)d(Op)s(erationen)p 270 1021 3120 4 v 268 1249 4 228
v 320 1166 a(Sc)m(hl)488 1170 y(\177)485 1166 y(ussell)700
1170 y(\177)700 1166 y(ange)p 928 1249 V 979 1101 a(Generierung)979
1214 y(der)h(Zufallszahl)p 1618 1249 V 1669 1101 a(Multiplik)-5
b(ation)27 b(b)s(ei)1669 1214 y(der)j(V)-8 b(ersc)m(hl)2115
1218 y(\177)2112 1214 y(usselung)p 2497 1249 V 2548 1101
a(Multiplik)j(ation)27 b(b)s(ei)2548 1214 y(der)j(En)m(tsc)m(hl)3003
1218 y(\177)3000 1214 y(usselung)p 3388 1249 V 270 1252
3120 4 v 270 1269 V 268 1381 4 113 v 588 1348 a(120)h(Bit)p
928 1381 V 423 w(29.4)h(\045)p 1618 1381 V 611 w(29.4)g(\045)p
2497 1381 V 623 w(27.5)g(\045)p 3388 1381 V 270 1385
3120 4 v 268 1498 4 113 v 588 1464 a(160)f(Bit)p 928
1498 V 423 w(30.3)h(\045)p 1618 1498 V 611 w(30.4)g(\045)p
2497 1498 V 623 w(30.3)g(\045)p 3388 1498 V 270 1501
3120 4 v 268 1614 4 113 v 588 1580 a(200)f(Bit)p 928
1614 V 423 w(31.2)h(\045)p 1618 1614 V 611 w(31.3)g(\045)p
2497 1614 V 623 w(30.5)g(\045)p 3388 1614 V 270 1617
3120 4 v 268 1730 4 113 v 588 1696 a(240)f(Bit)p 928
1730 V 423 w(32.1)h(\045)p 1618 1730 V 611 w(32.2)g(\045)p
2497 1730 V 623 w(30.8)g(\045)p 3388 1730 V 270 1733
3120 4 v 411 2041 a(Den)22 b(Zeitaufw)m(and)f(f)1136
2045 y(\177)1133 2041 y(ur)f(die)h(Erzeugung)g(der)g(Zufallszahlen)e(k)
2556 2045 y(\177)2556 2041 y(onn)m(te)j(man)f(einsparen,)270
2153 y(indem)32 b(man)h(eine)h(b)s(estimm)m(te)f(Anzahl)g(v)m
(orgeneriert.)h(Ist)g(das)f(der)g(F)-8 b(all,)34 b(und)e(gilt)h(das)270
2266 y(auc)m(h)38 b(f)523 2270 y(\177)520 2266 y(ur)f(die)h(elliptisc)m
(he)e(Kurv)m(e)h(inkl.)g(des)g(Sc)m(hl)2054 2270 y(\177)2051
2266 y(usselpaares,)g(so)h(l)2700 2270 y(\177)2700 2266
y(a\031t)h(sic)m(h)e(auc)m(h)i(die)270 2379 y(Punktm)m(ultiplik)-5
b(ation)41 b(b)s(ei)j(der)g(V)-8 b(ersc)m(hl)1741 2383
y(\177)1738 2379 y(usselung)44 b(sc)m(hon)h(v)m(orher)g(ausf)2844
2383 y(\177)2841 2379 y(uhren.)f(Damit)270 2492 y(erg)391
2496 y(\177)391 2492 y(ab)s(e)36 b(sic)m(h)g(eine)f(rec)m(hnerisc)m(he)
h(Besc)m(hleunigung)f(um)g(et)m(w)m(a)j(den)d(F)-8 b(aktor)38
b(10.)f(Ob)e(die)270 2605 y(V)-8 b(orb)s(erec)m(hn)m(ung)34
b(in)f(der)g(Praxis)g(jedo)s(c)m(h)h(realistisc)m(h)f(w)2236
2609 y(\177)2236 2605 y(are,)h(ist)f(nat)2680 2609 y(\177)2677
2605 y(urlic)m(h)g(eine)g(andere)270 2718 y(F)-8 b(rage.)411
2831 y(Eine)22 b(Grundv)m(oraussetzung)g(ist)h(zw)m(eifellos)f(auc)m(h)
h(eine)g(e\016zien)m(te)h(Implemen)m(tierung)270 2944
y(der)g(Multiplik)-5 b(ation)21 b(v)m(on)j(Punkten)g(auf)g(elliptisc)m
(hen)e(Kurv)m(en.)h(An)h(dieser)f(Stelle)h(sollten)270
3057 y(k)m(eine)30 b(unn)655 3061 y(\177)655 3057 y(otigen)g
(E\016zienzv)m(erluste)f(en)m(tstehen.)411 3170 y(Ein)42
b(anderer)h(Punkt,)g(der)f(no)s(c)m(h)i(angespro)s(c)m(hen)f(w)m(erden)
g(soll,)f(ist)h(die)f(generelle)270 3283 y(V)-8 b(erw)m(endung)27
b(v)m(on)h(LiDIA)e(innerhalb)f(der)h(JA)-10 b(V)g(A-An)m(w)m(endung.)28
b(En)m(tgegen)g(der)f(Erw)m(ar-)270 3395 y(tung)h(wird)e(f)716
3399 y(\177)713 3395 y(ur)h(den)h(Zugri\013)e(an)i(sic)m(h)g(relativ)f
(w)m(enig)h(Zeit)g(v)m(erbrauc)m(h)m(t.)h(Es)e(ren)m(tiert)i(sic)m(h)
270 3508 y(somit,)37 b(in)e(b)s(ezug)i(auf)g(die)f(E\016zienz,)h(auf)f
(andere)h(Programmiersprac)m(hen)f(zur)3136 3512 y(\177)3133
3508 y(uc)m(kzu-)270 3621 y(greifen,)29 b(w)m(enn)h(in)e(jenen)h(die)g
(Algorithmen)g(sc)m(hon)h(e\016zien)m(t)g(implemen)m(tiert)e(v)m
(orliegen.)p eop
%%Page: 32 33
32 32 bop eop
%%Page: 33 34
33 33 bop 270 1052 a Fr(Kapitel)66 b(6)270 1481 y Fv(Ausblic)-6
b(k)270 1940 y Fw(In)34 b(dieser)g(Arb)s(eit)g(wurde)f(ein)h(Pro)m
(vider)g(en)m(t)m(wic)m(k)m(elt,)j(der)d(un)m(ter)h(V)-8
b(erw)m(endung)34 b(ellipti-)270 2053 y(sc)m(her)27 b(Kurv)m(en)f
(Daten)h(v)m(ersc)m(hl)1358 2057 y(\177)1355 2053 y(usselt.)f(Dab)s(ei)
h(wurde)e(die)h(Theorie)g(kurz)g(angespro)s(c)m(hen,)270
2166 y(der)35 b(tec)m(hnisc)m(he)i(Hin)m(tergrund)d(nahm)h(jedo)s(c)m
(h)g(den)h(gr)2201 2170 y(\177)2201 2166 y(o\031eren)g(T)-8
b(eil)35 b(ein.)g(Ic)m(h)h(hab)s(e)f(v)m(er-)270 2279
y(suc)m(h)m(t,)i(die)e(einzelnen)g(Asp)s(ekte)h(v)m(erst)1640
2283 y(\177)1640 2279 y(andlic)m(h)f(darzustellen.)g(Es)h(wurden)e(die)
i(Kenn)m(t-)270 2392 y(nisse)28 b(v)m(ermittelt,)i(die)f(an)h(gegeb)s
(ener)g(Stelle)f(f)1907 2396 y(\177)1904 2392 y(ur)f(das)h(w)m(eitere)h
(V)-8 b(erst)2697 2396 y(\177)2697 2392 y(andnis)29 b(not)m(w)m(endig)
270 2504 y(w)m(aren.)42 b(Zur)f(V)-8 b(ertiefung)42 b(einzelner)f(F)-8
b(ragestellungen)42 b(dienen)f(die)g(Hin)m(w)m(eise)h(auf)g(die)270
2617 y(en)m(tsprec)m(hende)f(Literatur.)f(Absc)m(hlie\031end)g(soll)f
(no)s(c)m(h)h(einmal)g(v)m(ersuc)m(h)m(t)h(w)m(erden,)g(die)270
2730 y(v)m(ersc)m(hiedenen)27 b(Gesic)m(h)m(tspunkte)h(dieser)f
(Diplomarb)s(eit)e(im)h(Zusammenhang)h(zu)g(b)s(eur-)270
2843 y(teilen)i(und)g(V)-8 b(orsc)m(hl)990 2847 y(\177)990
2843 y(age)32 b(f)1183 2847 y(\177)1180 2843 y(ur)d(W)-8
b(eiteren)m(t)m(wic)m(klungen)31 b(zu)f(geb)s(en.)411
2970 y(Die)c(Arc)m(hitektur)f(der)g(JCA)f(ist)h(meiner)g(Ansic)m(h)m(t)
g(nac)m(h)h(eine)f(v)m(ern)2726 2974 y(\177)2723 2970
y(unftige)f(Struktur,)270 3083 y(um)40 b(sic)m(here)g(An)m(w)m
(endungen)g(zu)g(en)m(t)m(wic)m(k)m(eln.)i(Ob)d(diese)h(dann)g(wirklic)
m(h)e(sic)m(her)i(sind,)270 3196 y(h)321 3200 y(\177)321
3196 y(angt)32 b(nat)663 3200 y(\177)660 3196 y(urlic)m(h)e(auc)m(h)i
(v)m(on)g(den)f(v)m(erw)m(endeten)i(Algorithmen)d(ab.)i(Man)g(hat)g(ab)
s(er)f(auf)270 3309 y(jeden)25 b(F)-8 b(all)25 b(die)g(W)-8
b(ahl,)25 b(sic)m(h)g(selbst)g(um)g(Sic)m(herheitsfragen)f(zu)h(k)2518
3313 y(\177)2515 3309 y(ummern)f(und)g(m)m(u\031)i(sic)m(h)270
3422 y(dazu)41 b(nic)m(h)m(t)h(auc)m(h)g(no)s(c)m(h)f(ein)g(sinn)m(v)m
(olles)e(Strukturk)m(onzept)2461 3426 y(\177)2458 3422
y(ub)s(erlegen.)i(Sc)m(haut)g(man)270 3535 y(sic)m(h)33
b(dagegen)i(den)e(Quellco)s(de)f(in)h(den)g(Konzeptklassen)g(an,)h(so)g
(gibt)g(es)g(sc)m(hon)g(einige)270 3648 y(Absc)m(hnitte,)j(die)e(auf)i
(den)f(ersten)g(Blic)m(k)h(et)m(w)m(as)h(umst)2217 3652
y(\177)2217 3648 y(andlic)m(h)c(aussehen.)j(Der)g(Grund)270
3760 y(hierf)454 3764 y(\177)451 3760 y(ur)h(liegt)h(in)g(der)g
(Kompatibilit)1582 3764 y(\177)1582 3760 y(at)f(zu)1831
3764 y(\177)1831 3760 y(alteren)i(JDK-V)-8 b(ersionen.)40
b(Dadurc)m(h)g(m)m(u\031-)270 3873 y(ten)35 b(an)g(mehreren)f(Stellen)g
(zw)m(ei)h(leic)m(h)m(t)g(v)m(ersc)m(hiedene)g(Arc)m(hitekturen)f(b)s
(er)2969 3877 y(\177)2966 3873 y(uc)m(ksic)m(h)m(tigt)270
3986 y(w)m(erden.)26 b(T)-8 b(rotzdem)27 b(soll)e(dieser)g(Punkt)g(die)
h(Qualit)2091 3990 y(\177)2091 3986 y(at)f(des)h(Konzepts)g(nic)m(h)m
(t)h(sc)m(hm)3168 3990 y(\177)3168 3986 y(alern.)411
4113 y(W)-8 b(as)28 b(die)d(E\016zienz)h(b)s(etri\013t,)g(so)h(wurde)e
(sc)m(hon)h(in)f(Kapitel)h(5)g(einiges)g(er)2920 4117
y(\177)2920 4113 y(ortert.)h(Den-)270 4226 y(no)s(c)m(h)i(sollen)e
(hier)h(no)s(c)m(h)g(einige)g(W)-8 b(orte)31 b(darauf)d(v)m(erw)m
(endet)h(w)m(erden.)g(Eine)e(sc)m(hnelle)h(Im-)270 4339
y(plemen)m(tierung)c(der)i(Punktm)m(ultiplik)-5 b(ation)22
b(ist)j(en)m(tsc)m(heidend)g(f)2524 4343 y(\177)2521
4339 y(ur)g(einen)g(leistungsf)3240 4343 y(\177)3240
4339 y(ahi-)270 4452 y(gen)39 b(V)-8 b(ersc)m(hl)734
4456 y(\177)731 4452 y(usselungsalgorithm)m(us,)37 b(der)h(auf)h
(elliptsic)m(hen)d(Kurv)m(en)i(basiert.)h(Daraus)270
4565 y(folgt,)c(da\031)f(man)g(an)g(dieser)f(Stelle)h(jede)g(M)1810
4569 y(\177)1810 4565 y(oglic)m(hk)m(eit)h(einer)e(Besc)m(hleunigung)g
(ausn)m(ut-)270 4678 y(zen)e(sollte,)f(um)f(diese)h(dann)f(auc)m(h)i(f)
1547 4682 y(\177)1544 4678 y(ur)e(den)h(Pro)m(vider)f(v)m(erw)m(enden)i
(zu)f(k)2819 4682 y(\177)2819 4678 y(onnen.)411 4805
y(F)473 4809 y(\177)470 4805 y(ur)g(die)g(v)m(orliegende)g(An)m(w)m
(endung)g(wurde)f(JA)-10 b(V)g(A)31 b(als)f(Programmiersprac)m(he)g
(aus-)270 4918 y(gew)421 4922 y(\177)421 4918 y(ahlt.)k(Daraufhin)f
(steh)m(t)i(man)f(v)m(or)g(der)g(W)-8 b(ahl,)35 b(ob)f(f)2256
4922 y(\177)2253 4918 y(ur)f(rec)m(henin)m(tensiv)m(e)g(Op)s(eratio-)
270 5030 y(nen)28 b(auf)g(den)g(Programmco)s(de)h(anderer)f(Sprac)m
(hen)g(zur)2238 5034 y(\177)2235 5030 y(uc)m(kgegri\013en)g(w)m(erden)g
(soll.)g(Den)270 5143 y(V)-8 b(orteil)28 b(einer)g(h)838
5147 y(\177)838 5143 y(oheren)g(E\016zienz)g(erk)-5 b(auft)28
b(man)g(sic)m(h)g(dann)g(u.)15 b(a.)29 b(mit)e(dem)h(V)-8
b(erlust)28 b(der)270 5256 y(Plattform)m(unabh)904 5260
y(\177)904 5256 y(angigk)m(eit.)39 b(W)-8 b(elc)m(he)42
b(jedo)s(c)m(h)d(die)g(b)s(essere)h(V)-8 b(arian)m(te)41
b(ist,)e(h)3007 5260 y(\177)3007 5256 y(angt)h(v)m(om)270
5369 y(Einzelfall)28 b(ab.)411 5496 y(Die)23 b(V)-8 b(erw)m(endung)22
b(aktueller)g(kryptographisc)m(her)f(Standards)h(b)s(ei)f(der)h(En)m(t)
m(wic)m(klung)270 5609 y(v)m(on)j(Soft)m(w)m(are)g(f)832
5613 y(\177)829 5609 y(uhrt)e(zu)h(einheitlic)m(heren)e(An)m(w)m
(endungen.)h(Gerade)i(w)m(enn)f(viel)f(W)-8 b(ert)25
b(auf)1785 5858 y(33)p eop
%%Page: 34 35
34 34 bop 390 82 a Fw(34)1976 b(KAPITEL)29 b(6.)61 b(A)m(USBLICK)390
415 y(Erw)m(eiterbark)m(eit)36 b(und)f(Wiederv)m(erw)m(endbark)m(eit)h
(gelegt)h(wird,)e(ist)h(die)f(Anlehn)m(ung)g(an)390 528
y(Standards)827 532 y(\177)827 528 y(au\031erst)k(sinn)m(v)m(oll.)d
(Leider)g(w)m(aren)i(zum)f(Zeitpunkt)f(der)i(En)m(tsteh)m(ung)g(die-)
390 641 y(ser)e(Arb)s(eit)e(lediglic)m(h)f(V)-8 b(erfahren)36
b(zur)f(Sc)m(hl)1945 645 y(\177)1942 641 y(usselgenerierung)e(bzw.)i
(zu)h(Signaturen)e(in)390 754 y(den)28 b(Standard)f(P1363)j(in)m
(tegriert.)e(Die)g(Aufnahme)g(v)m(on)h(V)-8 b(ersc)m(hl)2704
758 y(\177)2701 754 y(usselungalgorithmen)390 867 y(ist)31
b(jedo)s(c)m(h)h(geplan)m(t.)g(In)g(der)f(Einarb)s(eitung)e(dieser)i(V)
-8 b(orgab)s(en)32 b(sehe)g(ic)m(h)f(eine)h(Erw)m(eite-)390
980 y(rungsm)685 984 y(\177)685 980 y(oglic)m(hk)m(eit)d(dieses)h(Pro)m
(viders.)531 1093 y(Es)40 b(wird)e(auc)m(h)j(sehr)f(in)m(teressan)m(t)g
(sein,)g(zu)g(v)m(erfolgen,)h(wie)e(sic)m(h)h(die)g(elliptisc)m(hen)390
1205 y(Kurv)m(en)32 b(auf)g(dem)g(Gebiet)h(der)f(Kryptographie)f(en)m
(t)m(wic)m(k)m(eln.)j(En)m(tsc)m(heidend)d(dab)s(ei)h(ist)390
1318 y(w)m(ohl)20 b(der)h(Punkt,)g(ob)g(ihre)e(V)-8 b(orteile)22
b(gegen)1879 1322 y(\177)1876 1318 y(ub)s(er)f(anderen)f(mathematisc)m
(hen)i(Strukturen)390 1431 y(b)s(estehen)32 b(bleib)s(en.)e(Des)j(w)m
(eiteren)f(bleibt)f(auc)m(h)h(abzu)m(w)m(arten,)i(w)m(elc)m(he)f(Kurv)m
(en)e(als)h(f)3427 1435 y(\177)3424 1431 y(ur)390 1544
y(die)41 b(Kryptographie)g(geeignet)i(angesehen)f(w)m(erden.)g(In)f
(jedem)h(F)-8 b(all)41 b(sind)f(elliptisc)m(he)390 1657
y(Kurv)m(en)d(ein)f(ansprec)m(hendes)h(F)-8 b(eld,)38
b(auf)g(dem)f(no)s(c)m(h)g(einige)g(En)m(t)m(wic)m(klungen)g(gemac)m(h)
m(t)390 1770 y(w)m(erden)e(k)754 1774 y(\177)754 1770
y(onnen.)g(Ic)m(h)h(ho\013e,)g(da\031)f(durc)m(h)g(diese)g(Arb)s(eit)f
(ein)g(kleiner)g(Beitrag)i(zu)g(ihrer)390 1883 y(V)-8
b(erbreitung)30 b(geleistet)h(wurde.)p eop
%%Page: 35 36
35 35 bop 270 1056 a Fr(Anhang)64 b(A)270 1489 y Fv(cdc.ec)270
1989 y Fz(A.1)135 b(Pro)l(vider)p 270 2118 3120 4 v 270
2143 4 25 v 3387 2143 V 215 2222 a Fa(1)p 270 2256 4
113 v 53 w Ff(package)46 b(cdc.ec;)p 3387 2256 V 215
2335 a Fa(2)p 270 2368 V 3387 2368 V 215 2447 a(3)p 270
2481 V 53 w Ff(import)g(cdc.ec.ecelgamal.*;)p 3387 2481
V 215 2560 a Fa(4)p 270 2594 V 53 w Ff(import)g(java.security.*;)p
3387 2594 V 215 2673 a Fa(5)p 270 2707 V 3387 2707 V
215 2786 a(6)p 270 2820 V 53 w Ff(/**)p 3387 2820 V 215
2899 a Fa(7)p 270 2933 V 101 w Ff(*)h(<code>Provider</code>)42
b(is)48 b(the)e(set)h(of)h(classes)d(that)i(implement)p
3387 2933 V 215 3012 a Fa(8)p 270 3046 V 101 w Ff(*)g(a)h(provider)d
(including)h(a)h(cipher)f(and)h(a)h(generator)d(for)i(key)g(pairs.)p
3387 3046 V 215 3125 a Fa(9)p 270 3159 V 101 w Ff(*)g(These)g(classes)f
(are)g(contained)g(in)h(the)g(subpackage)p 3387 3159
V 184 3238 a Fa(10)p 270 3272 V 102 w Ff(*)g(<code>ecelgamal</code>)42
b(which)k(is)i(also)e(the)h(provider's)e(name.)p 3387
3272 V 184 3351 a Fa(11)p 270 3385 V 102 w Ff(*)i(The)g(cipher)f(is)i
(based)e(on)h(an)g(ElGamal)f(algorithm)f(that)i(uses)p
3387 3385 V 184 3464 a Fa(12)p 270 3498 V 102 w Ff(*)g(elliptic)f
(curves)g(over)h(the)g(general)e(field)i(with)f(<code>p</code>)p
3387 3498 V 184 3577 a Fa(13)p 270 3610 V 102 w Ff(*)h(elements)f
(where)g(<code>p</code>)e(is)j(a)h(prime)e(number.)g(It)h(basically)p
3387 3610 V 184 3690 a Fa(14)p 270 3723 V 102 w Ff(*)g(maps)g(the)g
(identifiers)e(which)h(have)h(to)g(be)g(used)g(to)g(the)g(specific)p
3387 3723 V 184 3802 a Fa(15)p 270 3836 V 102 w Ff(*)g(classnames.)p
3387 3836 V 184 3915 a Fa(16)p 270 3949 V 102 w Ff(*)g(<p>)p
3387 3949 V 184 4028 a Fa(17)p 270 4062 V 102 w Ff(*)g(Details)f(to)h
(the)g(way)g(the)g(implementation)d(works)i(can)h(be)g(found)g(in)p
3387 4062 V 184 4141 a Fa(18)p 270 4175 V 102 w Ff(*)g(the)g
(description)e(of)i(the)g(classes)f(below)g(espescially)f(in)p
3387 4175 V 184 4254 a Fa(19)p 270 4288 V 102 w Ff(*)i
(<code>ECElGamalGFPCipher</)o(cod)o(e>.)p 3387 4288 V
184 4367 a Fa(20)p 270 4401 V 102 w Ff(*)g(<p>)p 3387
4401 V 184 4480 a Fa(21)p 270 4514 V 102 w Ff(*)g(The)g(C++-library)e
(for)i(calculation)e(is)i(<code>LiDIA</code>.)p 3387
4514 V 184 4593 a Fa(22)p 270 4627 V 102 w Ff(*)p 3387
4627 V 184 4706 a Fa(23)p 270 4740 V 102 w Ff(*)g(@author)f(Markus)g
(Crnjak)p 3387 4740 V 184 4819 a Fa(24)p 270 4852 V 102
w Ff(*)p 3387 4852 V 184 4932 a Fa(25)p 270 4965 V 102
w Ff(*)h(@see)g(cdc.ec.ecelgamal.ECElGam)o(alGF)o(PKey)p
3387 4965 V 184 5044 a Fa(26)p 270 5078 V 102 w Ff(*)g(@see)g
(cdc.ec.ecelgamal.ECElGam)o(alGF)o(PPri)o(vat)o(eKey)p
3387 5078 V 184 5157 a Fa(27)p 270 5191 V 102 w Ff(*)g(@see)g
(cdc.ec.ecelgamal.ECElGam)o(alGF)o(PPub)o(lic)o(Key)p
3387 5191 V 184 5270 a Fa(28)p 270 5304 V 102 w Ff(*)g(@see)g
(cdc.ec.ecelgamal.ECElGam)o(alGF)o(PKey)o(Pai)o(rGen)o(erat)o(or)p
3387 5304 V 184 5383 a Fa(29)p 270 5417 V 102 w Ff(*)g(@see)g
(cdc.ec.ecelgamal.ECElGam)o(alGF)o(PBlo)o(ckC)o(iphe)o(r)p
3387 5417 V 184 5496 a Fa(30)p 270 5530 V 102 w Ff(*)g(@see)g
(cdc.ec.ecelgamal.ECElGam)o(alGF)o(PCip)o(her)p 3387
5530 V 184 5609 a Fa(31)p 270 5643 V 102 w Ff(*)g(@see)g
(cdc.lidia.LECDomainParam)o(eter)o(sP13)o(63)p 3387 5643
V 1785 5858 a Fw(35)p eop
%%Page: 36 37
36 36 bop 390 82 a Fw(36)2080 b(ANHANG)31 b(A.)61 b(CDC.EC)304
415 y Fa(32)p 390 449 4 113 v 102 w Ff(*/)p 3507 449
V 304 528 a Fa(33)p 390 562 V 54 w Ff(public)46 b(class)h(Provider)p
3507 562 V 304 641 a Fa(34)p 390 675 V 245 w Ff(extends)f
(java.security.Provider)p 3507 675 V 304 754 a Fa(35)p
390 788 V 54 w Ff({)p 3507 788 V 304 867 a Fa(36)p 390
901 V 245 w Ff(/**)p 3507 901 V 304 980 a Fa(37)p 390
1014 V 293 w Ff(*)h(constructor,)e(that)h(makes)h(the)g
(<code>ECElGamalGFP</cod)o(e>)p 3507 1014 V 304 1093
a Fa(38)p 390 1126 V 293 w Ff(*)g(implementation)d(accessible)p
3507 1126 V 304 1205 a Fa(39)p 390 1239 V 293 w Ff(*)j(<p>)p
3507 1239 V 304 1318 a Fa(40)p 390 1352 V 293 w Ff(*)g(Accessible)e
(are)i(a)h(<code>KeyPairGenerator<)o(/cod)o(e>)41 b(and)47
b(a)p 3507 1352 V 304 1431 a Fa(41)p 390 1465 V 293 w
Ff(*)g(<code>Cipher</code>)c(both)j(specified)g(by)h("ECElGamalGFP".)p
3507 1465 V 304 1544 a Fa(42)p 390 1578 V 293 w Ff(*/)p
3507 1578 V 304 1657 a Fa(43)p 390 1691 V 245 w Ff(public)f(Provider)g
(\(\))p 3507 1691 V 304 1770 a Fa(44)p 390 1804 V 245
w Ff({)p 3507 1804 V 304 1883 a Fa(45)p 390 1917 V 436
w Ff(super)g(\("ECElGamalGFP",1.2,"Provi)o(der)41 b(for)47
b(ECElGamalGFP"\);)p 3507 1917 V 304 1996 a Fa(46)p 390
2030 V 3507 2030 V 304 2109 a(47)p 390 2143 V 436 w Ff(//)g(maps)g(the)
g(short)f(names)g(to)i(the)f(class)f(names)p 3507 2143
V 304 2222 a Fa(48)p 390 2256 V 436 w Ff(put)h
(\("KeyPairGenerator.ECElG)o(amal)o(GFP)o(",)p 3507 2256
V 304 2335 a Fa(49)p 390 2368 V 675 w Ff("cdc.ec.ecelgamal.ECElG)o
(amal)o(GFP)o(KeyP)o(airG)o(ene)o(rato)o(r"\);)p 3507
2368 V 304 2447 a Fa(50)p 390 2481 V 436 w Ff(put)g
(\("Cipher.ECElGamalGFP",)p 3507 2481 V 304 2560 a Fa(51)p
390 2594 V 675 w Ff("cdc.ec.ecelgamal.ECElG)o(amal)o(GFP)o(Ciph)o
(er"\))o(;)p 3507 2594 V 304 2673 a Fa(52)p 390 2707
V 245 w Ff(})p 3507 2707 V 304 2786 a Fa(53)p 390 2820
V 54 w Ff(})p 3507 2820 V 390 2845 4 25 v 3507 2845 V
390 2848 3120 4 v eop
%%Page: 37 38
37 37 bop 270 1051 a Fr(Anhang)64 b(B)270 1480 y Fv(cdc.ec.ecelgamal)
270 1975 y Fz(B.1)135 b(ECElGamalGFPKey)p 270 2118 3120
4 v 270 2143 4 25 v 3387 2143 V 215 2222 a Fa(1)p 270
2256 4 113 v 53 w Ff(package)46 b(cdc.ec.ecelgamal;)p
3387 2256 V 215 2335 a Fa(2)p 270 2368 V 3387 2368 V
215 2447 a(3)p 270 2481 V 53 w Ff(import)g(java.security.*;)p
3387 2481 V 215 2560 a Fa(4)p 270 2594 V 53 w Ff(import)g(cdc.lidia.*;)
p 3387 2594 V 215 2673 a Fa(5)p 270 2707 V 3387 2707
V 215 2786 a(6)p 270 2820 V 53 w Ff(/**)p 3387 2820 V
215 2899 a Fa(7)p 270 2933 V 101 w Ff(*)h(<code>ECElGamalGFPKey</cod)o
(e>)41 b(holds)47 b(the)g(common)f(key)h(parameters)p
3387 2933 V 215 3012 a Fa(8)p 270 3046 V 101 w Ff(*)g(for)g
(<code>ECElGamalGFP</code>)41 b(public)46 b(and)h(private)f(keys.)g
(These)p 3387 3046 V 215 3125 a Fa(9)p 270 3159 V 101
w Ff(*)h(are)g(called)f(domain)g(parameters)f(and)i(contain)f(the)h
(underlying)p 3387 3159 V 184 3238 a Fa(10)p 270 3272
V 102 w Ff(*)g(elliptic)f(curve)g(and)h(a)h(curve)e(point)g(that)h
(generates)e(a)j(subgroup.)p 3387 3272 V 184 3351 a Fa(11)p
270 3385 V 102 w Ff(*)f(<p>)p 3387 3385 V 184 3464 a
Fa(12)p 270 3498 V 102 w Ff(*)g(If)h(<code>G</code>)43
b(is)48 b(the)f(generator)e(and)i(<code>s</code>)d(is)j(an)p
3387 3498 V 184 3577 a Fa(13)p 270 3610 V 102 w Ff(*)g(integer)f
(number)g(with)h(<code>W)f(=)h(sG</code>,)e(then)i(<code>s</code>)p
3387 3610 V 184 3690 a Fa(14)p 270 3723 V 102 w Ff(*)g(will)g(be)g(the)
g(private)f(key)h(and)g(<code>W</code>)d(will)i(be)h(the)g(public)p
3387 3723 V 184 3802 a Fa(15)p 270 3836 V 102 w Ff(*)g(key.)p
3387 3836 V 184 3915 a Fa(16)p 270 3949 V 102 w Ff(*)g(<p>)p
3387 3949 V 184 4028 a Fa(17)p 270 4062 V 102 w Ff(*)g(The)g
(C++-library)e(for)i(calculation)e(is)i(<code>LiDIA</code>.)p
3387 4062 V 184 4141 a Fa(18)p 270 4175 V 102 w Ff(*)p
3387 4175 V 184 4254 a Fa(19)p 270 4288 V 102 w Ff(*)g(@author)f
(Markus)g(Crnjak)p 3387 4288 V 184 4367 a Fa(20)p 270
4401 V 102 w Ff(*)p 3387 4401 V 184 4480 a Fa(21)p 270
4514 V 102 w Ff(*)h(@see)g(ECElGamalGFPPrivateKey)p 3387
4514 V 184 4593 a Fa(22)p 270 4627 V 102 w Ff(*)g(@see)g
(ECElGamalGFPPublicKey)p 3387 4627 V 184 4706 a Fa(23)p
270 4740 V 102 w Ff(*)g(@see)g(ECElGamalGFPKeyPairGener)o(ator)p
3387 4740 V 184 4819 a Fa(24)p 270 4852 V 102 w Ff(*)g(@see)g
(ECElGamalGFPBlockCipher)p 3387 4852 V 184 4932 a Fa(25)p
270 4965 V 102 w Ff(*)g(@see)g(ECElGamalGFPCipher)p 3387
4965 V 184 5044 a Fa(26)p 270 5078 V 102 w Ff(*)g(@see)g
(cdc.ec.Provider)p 3387 5078 V 184 5157 a Fa(27)p 270
5191 V 102 w Ff(*)g(@see)g(cdc.lidia.LECDomainParam)o(eter)o(sP13)o(63)
p 3387 5191 V 184 5270 a Fa(28)p 270 5304 V 102 w Ff(*/)p
3387 5304 V 184 5383 a Fa(29)p 270 5417 V 54 w Ff(public)f(class)h
(ECElGamalGFPKey)p 3387 5417 V 184 5496 a Fa(30)p 270
5530 V 245 w Ff(implements)e(Key)p 3387 5530 V 184 5609
a Fa(31)p 270 5643 V 54 w Ff({)p 3387 5643 V 1785 5858
a Fw(37)p eop
%%Page: 38 39
38 38 bop 390 82 a Fw(38)1464 b(ANHANG)32 b(B.)61 b(CDC.EC.ECELGAMAL)
304 415 y Fa(32)p 390 449 4 113 v 245 w Ff(/**)p 3507
449 V 304 528 a Fa(33)p 390 562 V 293 w Ff(*)47 b(holds)g(the)g(domain)
f(parameters)p 3507 562 V 304 641 a Fa(34)p 390 675 V
293 w Ff(*)p 3507 675 V 304 754 a Fa(35)p 390 788 V 293
w Ff(*)h(@serial)p 3507 788 V 304 867 a Fa(36)p 390 901
V 293 w Ff(*/)p 3507 901 V 304 980 a Fa(37)p 390 1014
V 245 w Ff(protected)e(LECDomainParametersP1363)d(mParams;)p
3507 1014 V 304 1093 a Fa(38)p 390 1126 V 3507 1126 V
304 1205 a(39)p 390 1239 V 245 w Ff(/**)p 3507 1239 V
304 1318 a Fa(40)p 390 1352 V 293 w Ff(*)47 b(constructor,)e(that)h
(sets)h(the)g(domain)f(parameters)f(of)i(this)p 3507
1352 V 304 1431 a Fa(41)p 390 1465 V 293 w Ff(*)g
(<code>ECElGamalGFPKey</co)o(de>)p 3507 1465 V 304 1544
a Fa(42)p 390 1578 V 293 w Ff(*)p 3507 1578 V 304 1657
a Fa(43)p 390 1691 V 293 w Ff(*)g(@param)f(params)g(the)h
(lECDomainParametersP1363)42 b(contains)j(the)p 3507
1691 V 304 1770 a Fa(44)p 390 1804 V 293 w Ff(*)i(domain)f(parameters)p
3507 1804 V 304 1883 a Fa(45)p 390 1917 V 293 w Ff(*/)p
3507 1917 V 304 1996 a Fa(46)p 390 2030 V 245 w Ff(protected)f
(ECElGamalGFPKey\(LECDomainP)o(ara)o(mete)o(rsP1)o(363)c(params\))p
3507 2030 V 304 2109 a Fa(47)p 390 2143 V 245 w Ff({)p
3507 2143 V 304 2222 a Fa(48)p 390 2256 V 436 w Ff(mParams)46
b(=)h(params;)p 3507 2256 V 304 2335 a Fa(49)p 390 2368
V 245 w Ff(})p 3507 2368 V 304 2447 a Fa(50)p 390 2481
V 3507 2481 V 304 2560 a(51)p 390 2594 V 245 w Ff(/**)p
3507 2594 V 304 2673 a Fa(52)p 390 2707 V 293 w Ff(*)g(sets)g(the)g
(domain)f(parameters)f(of)i(this)p 3507 2707 V 304 2786
a Fa(53)p 390 2820 V 293 w Ff(*)g(<code>ECElGamalGFPKey</co)o(de>)p
3507 2820 V 304 2899 a Fa(54)p 390 2933 V 293 w Ff(*)p
3507 2933 V 304 3012 a Fa(55)p 390 3046 V 293 w Ff(*)g(@param)f(params)
g(the)h(lECDomainParametersP1363)42 b(contains)j(the)p
3507 3046 V 304 3125 a Fa(56)p 390 3159 V 293 w Ff(*)i(domain)f
(parameters)p 3507 3159 V 304 3238 a Fa(57)p 390 3272
V 293 w Ff(*/)p 3507 3272 V 304 3351 a Fa(58)p 390 3385
V 245 w Ff(protected)f(void)i(setParams\(LECDomainParam)o(eter)o(sP13)o
(63)41 b(params\))p 3507 3385 V 304 3464 a Fa(59)p 390
3498 V 245 w Ff({)p 3507 3498 V 304 3577 a Fa(60)p 390
3610 V 436 w Ff(mParams)46 b(=)h(params;)p 3507 3610
V 304 3689 a Fa(61)p 390 3723 V 245 w Ff(})p 3507 3723
V 304 3802 a Fa(62)p 390 3836 V 3507 3836 V 304 3915
a(63)p 390 3949 V 245 w Ff(/**)p 3507 3949 V 304 4028
a Fa(64)p 390 4062 V 293 w Ff(*)g(returns)f(the)h(domain)f(parameters)f
(of)i(this)p 3507 4062 V 304 4141 a Fa(65)p 390 4175
V 293 w Ff(*)g(<code>ECElGamalGFPKey</co)o(de>)p 3507
4175 V 304 4254 a Fa(66)p 390 4288 V 293 w Ff(*/)p 3507
4288 V 304 4367 a Fa(67)p 390 4401 V 245 w Ff(protected)e
(LECDomainParametersP1363)d(getParams\(\))p 3507 4401
V 304 4480 a Fa(68)p 390 4514 V 245 w Ff({)p 3507 4514
V 304 4593 a Fa(69)p 390 4627 V 436 w Ff(return)k(mParams;)p
3507 4627 V 304 4706 a Fa(70)p 390 4740 V 245 w Ff(})p
3507 4740 V 304 4819 a Fa(71)p 390 4852 V 3507 4852 V
304 4932 a(72)p 390 4965 V 245 w Ff(/**)p 3507 4965 V
304 5044 a Fa(73)p 390 5078 V 293 w Ff(*)h(returns)f(the)h(algorithm)e
(name,)i(that)f(is)h("ECElGamalGFP")p 3507 5078 V 304
5157 a Fa(74)p 390 5191 V 293 w Ff(*/)p 3507 5191 V 304
5270 a Fa(75)p 390 5304 V 245 w Ff(public)f(String)g(getAlgorithm\(\))p
3507 5304 V 304 5383 a Fa(76)p 390 5417 V 245 w Ff({)p
3507 5417 V 304 5496 a Fa(77)p 390 5530 V 436 w Ff(return)g
("ECElGamalGFP";)p 3507 5530 V 304 5609 a Fa(78)p 390
5643 V 245 w Ff(})p 3507 5643 V eop
%%Page: 39 40
39 39 bop 270 82 a Fw(B.1.)62 b(ECElGamalGFPKey)1978
b(39)184 415 y Fa(79)p 270 449 4 113 v 3387 449 V 184
528 a(80)p 270 562 V 245 w Ff(/**)p 3387 562 V 184 641
a Fa(81)p 270 675 V 293 w Ff(*)47 b(returns)f(<code>"NONE"</code>)d
(since)j(there)g(is)i(no)f(specific)p 3387 675 V 184
754 a Fa(82)p 270 788 V 293 w Ff(*)g(format)f(used)h(to)g(encode)f(the)
h(key)p 3387 788 V 184 867 a Fa(83)p 270 901 V 293 w
Ff(*/)p 3387 901 V 184 980 a Fa(84)p 270 1014 V 245 w
Ff(public)f(String)g(getFormat\(\))p 3387 1014 V 184
1093 a Fa(85)p 270 1126 V 245 w Ff({)p 3387 1126 V 184
1205 a Fa(86)p 270 1239 V 436 w Ff(return)g("NONE";)p
3387 1239 V 184 1318 a Fa(87)p 270 1352 V 245 w Ff(})p
3387 1352 V 184 1431 a Fa(88)p 270 1465 V 3387 1465 V
184 1544 a(89)p 270 1578 V 245 w Ff(/**)p 3387 1578 V
184 1657 a Fa(90)p 270 1691 V 293 w Ff(*)h(returns)f(<code>null</code>)
d(since)k(there)f(is)h(no)g(specific)p 3387 1691 V 184
1770 a Fa(91)p 270 1804 V 293 w Ff(*)g(format)f(used)h(to)g(encode)f
(the)h(key)p 3387 1804 V 184 1883 a Fa(92)p 270 1917
V 293 w Ff(*/)p 3387 1917 V 184 1996 a Fa(93)p 270 2030
V 245 w Ff(public)f(byte[])g(getEncoded\(\))p 3387 2030
V 184 2109 a Fa(94)p 270 2143 V 245 w Ff({)p 3387 2143
V 184 2222 a Fa(95)p 270 2256 V 436 w Ff(return)g(null;)p
3387 2256 V 184 2335 a Fa(96)p 270 2368 V 245 w Ff(})p
3387 2368 V 184 2447 a Fa(97)p 270 2481 V 54 w Ff(})p
3387 2481 V 270 2506 4 25 v 3387 2506 V 270 2509 3120
4 v eop
%%Page: 40 41
40 40 bop 390 82 a Fw(40)1464 b(ANHANG)32 b(B.)61 b(CDC.EC.ECELGAMAL)
390 415 y Fz(B.2)135 b(ECElGamalGFPPriv)-7 b(ateKey)p
390 537 3120 4 v 390 562 4 25 v 3507 562 V 335 641 a
Fa(1)p 390 675 4 113 v 53 w Ff(package)46 b(cdc.ec.ecelgamal;)p
3507 675 V 335 754 a Fa(2)p 390 788 V 3507 788 V 335
867 a(3)p 390 901 V 53 w Ff(import)g(java.math.BigInteger;)p
3507 901 V 335 980 a Fa(4)p 390 1014 V 53 w Ff(import)g
(java.security.*;)p 3507 1014 V 335 1093 a Fa(5)p 390
1126 V 53 w Ff(import)g(cdc.lidia.*;)p 3507 1126 V 335
1205 a Fa(6)p 390 1239 V 3507 1239 V 335 1318 a(7)p 390
1352 V 53 w Ff(/**)p 3507 1352 V 335 1431 a Fa(8)p 390
1465 V 101 w Ff(*)h(<code>ECElGamalGFPPrivateK)o(ey<)o(/cod)o(e>)42
b(holds)k(the)h(private)f(part)g(of)p 3507 1465 V 335
1544 a Fa(9)p 390 1578 V 101 w Ff(*)h(an)h(<code>ECElGamalGFP</cod)o
(e>)41 b(key.)p 3507 1578 V 304 1657 a Fa(10)p 390 1691
V 102 w Ff(*)47 b(<p>)p 3507 1691 V 304 1770 a Fa(11)p
390 1804 V 102 w Ff(*)g(If)h(<code>G</code>)43 b(is)48
b(the)f(generator)e(of)i(a)g(subgroup)f(of)h(an)p 3507
1804 V 304 1883 a Fa(12)p 390 1917 V 102 w Ff(*)g(elliptic)f(curve)g
(and)h(<code>s</code>)d(is)j(an)g(integer)f(number)g(with)p
3507 1917 V 304 1996 a Fa(13)p 390 2030 V 102 w Ff(*)h(<code>W)f(=)i
(sG</code>,)d(then)h(<code>s</code>)e(will)j(be)g(the)g(private)p
3507 2030 V 304 2109 a Fa(14)p 390 2143 V 102 w Ff(*)g(key)g(and)g
(<code>W</code>)d(will)j(be)g(the)g(public)f(key.)p 3507
2143 V 304 2222 a Fa(15)p 390 2256 V 102 w Ff(*)h(<p>)p
3507 2256 V 304 2335 a Fa(16)p 390 2368 V 102 w Ff(*)g(Here)g(the)g
(private)f(key)h(is)g(called)f(<code>Exponent</code>,)c(that)p
3507 2368 V 304 2447 a Fa(17)p 390 2481 V 102 w Ff(*)47
b(reminds)f(to)h(the)g(<code>Discrete)d(Logarithm)h(Problem)h
(\(DLP\)</code>.)p 3507 2481 V 304 2560 a Fa(18)p 390
2594 V 102 w Ff(*)h(<p>)p 3507 2594 V 304 2673 a Fa(19)p
390 2707 V 102 w Ff(*)g(The)g(C++-library)e(for)i(calculation)e(is)i
(<code>LiDIA</code>.)p 3507 2707 V 304 2786 a Fa(20)p
390 2820 V 102 w Ff(*)p 3507 2820 V 304 2899 a Fa(21)p
390 2933 V 102 w Ff(*)g(@author)f(Markus)g(Crnjak)p 3507
2933 V 304 3012 a Fa(22)p 390 3046 V 102 w Ff(*)p 3507
3046 V 304 3125 a Fa(23)p 390 3159 V 102 w Ff(*)h(@see)g
(ECElGamalGFPKey)p 3507 3159 V 304 3238 a Fa(24)p 390
3272 V 102 w Ff(*)g(@see)g(ECElGamalGFPPublicKey)p 3507
3272 V 304 3351 a Fa(25)p 390 3385 V 102 w Ff(*)g(@see)g
(ECElGamalGFPKeyPairGener)o(ator)p 3507 3385 V 304 3464
a Fa(26)p 390 3498 V 102 w Ff(*)g(@see)g(ECElGamalGFPBlockCipher)p
3507 3498 V 304 3577 a Fa(27)p 390 3610 V 102 w Ff(*)g(@see)g
(ECElGamalGFPCipher)p 3507 3610 V 304 3690 a Fa(28)p
390 3723 V 102 w Ff(*)g(@see)g(cdc.ec.Provider)p 3507
3723 V 304 3802 a Fa(29)p 390 3836 V 102 w Ff(*)g(@see)g
(cdc.lidia.LECDomainParam)o(eter)o(sP13)o(63)p 3507 3836
V 304 3915 a Fa(30)p 390 3949 V 102 w Ff(*/)p 3507 3949
V 304 4028 a Fa(31)p 390 4062 V 54 w Ff(public)f(class)h
(ECElGamalGFPPrivateKey)p 3507 4062 V 304 4141 a Fa(32)p
390 4175 V 245 w Ff(extends)f(ECElGamalGFPKey)p 3507
4175 V 304 4254 a Fa(33)p 390 4288 V 245 w Ff(implements)f(PrivateKey)p
3507 4288 V 304 4367 a Fa(34)p 390 4401 V 54 w Ff({)p
3507 4401 V 304 4480 a Fa(35)p 390 4514 V 245 w Ff(/**)p
3507 4514 V 304 4593 a Fa(36)p 390 4627 V 293 w Ff(*)i(holds)g(the)g
(private)e(key,)i(that)g(is)g(an)g(integer)f(number)p
3507 4627 V 304 4706 a Fa(37)p 390 4740 V 293 w Ff(*)p
3507 4740 V 304 4819 a Fa(38)p 390 4852 V 293 w Ff(*)h(@serial)p
3507 4852 V 304 4932 a Fa(39)p 390 4965 V 293 w Ff(*/)p
3507 4965 V 304 5044 a Fa(40)p 390 5078 V 245 w Ff(private)f
(BigInteger)f(mExponent;)p 3507 5078 V 304 5157 a Fa(41)p
390 5191 V 3507 5191 V 304 5270 a(42)p 390 5304 V 245
w Ff(/**)p 3507 5304 V 304 5383 a Fa(43)p 390 5417 V
293 w Ff(*)i(constructor,)e(that)h(sets)h(the)g(domain)f(parameters)f
(and)i(the)p 3507 5417 V 304 5496 a Fa(44)p 390 5530
V 293 w Ff(*)g(private)f(part)h(of)g(this)g(<code>ECElGamalGFPKey</)o
(cod)o(e>,)41 b(that)p 3507 5530 V 304 5609 a Fa(45)p
390 5643 V 293 w Ff(*)47 b(is)g(an)h(integer)d(number)p
3507 5643 V eop
%%Page: 41 42
41 41 bop 270 82 a Fw(B.2.)62 b(ECElGamalGFPPriv)-5 b(ateKey)1691
b(41)184 415 y Fa(46)p 270 449 4 113 v 293 w Ff(*)p 3387
449 V 184 528 a Fa(47)p 270 562 V 293 w Ff(*)47 b(@param)f(params)g
(the)h(lECDomainParametersP1363)42 b(contains)j(the)p
3387 562 V 184 641 a Fa(48)p 270 675 V 293 w Ff(*)i(domain)f
(parameters)p 3387 675 V 184 754 a Fa(49)p 270 788 V
293 w Ff(*)h(@param)f(exponent)g(the)h(bigInteger)e(contains)g(the)i
(private)f(key)p 3387 788 V 184 867 a Fa(50)p 270 901
V 293 w Ff(*/)p 3387 901 V 184 980 a Fa(51)p 270 1014
V 245 w Ff(protected)f(ECElGamalGFPPrivateKey\(LEC)o(Dom)o(ainP)o(aram)
o(ete)o(rsP1)o(363)p 3387 1014 V 184 1093 a Fa(52)p 270
1126 V 436 w Ff(params,BigInteger)e(exponent\))p 3387
1126 V 184 1205 a Fa(53)p 270 1239 V 245 w Ff({)p 3387
1239 V 184 1318 a Fa(54)p 270 1352 V 436 w Ff(super\(params\);)p
3387 1352 V 184 1431 a Fa(55)p 270 1465 V 436 w Ff(mExponent)i(=)j
(exponent;)p 3387 1465 V 184 1544 a Fa(56)p 270 1578
V 245 w Ff(})p 3387 1578 V 184 1657 a Fa(57)p 270 1691
V 3387 1691 V 184 1770 a(58)p 270 1804 V 245 w Ff(/**)p
3387 1804 V 184 1883 a Fa(59)p 270 1917 V 293 w Ff(*)f(returns)f(the)h
(private)f(key,)g(that)h(is)g(an)g(integer)f(number.)p
3387 1917 V 184 1996 a Fa(60)p 270 2030 V 293 w Ff(*/)p
3387 2030 V 184 2109 a Fa(61)p 270 2143 V 245 w Ff(protected)f
(BigInteger)g(getExponent\(\))p 3387 2143 V 184 2222
a Fa(62)p 270 2256 V 245 w Ff({)p 3387 2256 V 184 2335
a Fa(63)p 270 2368 V 436 w Ff(return)h(mExponent;)p 3387
2368 V 184 2447 a Fa(64)p 270 2481 V 245 w Ff(})p 3387
2481 V 184 2560 a Fa(65)p 270 2594 V 54 w Ff(})p 3387
2594 V 270 2619 4 25 v 3387 2619 V 270 2622 3120 4 v
eop
%%Page: 42 43
42 42 bop 390 82 a Fw(42)1464 b(ANHANG)32 b(B.)61 b(CDC.EC.ECELGAMAL)
390 415 y Fz(B.3)135 b(ECElGamalGFPPublicKey)p 390 537
3120 4 v 390 562 4 25 v 3507 562 V 335 641 a Fa(1)p 390
675 4 113 v 53 w Ff(package)46 b(cdc.ec.ecelgamal;)p
3507 675 V 335 754 a Fa(2)p 390 788 V 3507 788 V 335
867 a(3)p 390 901 V 53 w Ff(import)g(java.security.*;)p
3507 901 V 335 980 a Fa(4)p 390 1014 V 53 w Ff(import)g(cdc.lidia.*;)p
3507 1014 V 335 1093 a Fa(5)p 390 1126 V 3507 1126 V
335 1205 a(6)p 390 1239 V 53 w Ff(/**)p 3507 1239 V 335
1318 a Fa(7)p 390 1352 V 101 w Ff(*)h(<code>ECElGamalGFPPublicKe)o(y</)
o(code)o(>)42 b(holds)k(the)h(public)f(part)h(of)p 3507
1352 V 335 1431 a Fa(8)p 390 1465 V 101 w Ff(*)g(an)h
(<code>ECElGamalGFP</cod)o(e>)41 b(key.)p 3507 1465 V
335 1544 a Fa(9)p 390 1578 V 101 w Ff(*)47 b(<p>)p 3507
1578 V 304 1657 a Fa(10)p 390 1691 V 102 w Ff(*)g(If)h(<code>G</code>)
43 b(is)48 b(the)f(generator)e(of)i(a)g(subgroup)f(of)h(an)p
3507 1691 V 304 1770 a Fa(11)p 390 1804 V 102 w Ff(*)g(elliptic)f
(curve)g(and)h(<code>s</code>)d(is)j(an)g(integer)f(number)g(with)p
3507 1804 V 304 1883 a Fa(12)p 390 1917 V 102 w Ff(*)h(<code>W)f(=)i
(sG</code>,)d(then)h(<code>s</code>)e(will)j(be)g(the)g(private)p
3507 1917 V 304 1996 a Fa(13)p 390 2030 V 102 w Ff(*)g(key)g(and)g
(<code>W</code>)d(will)j(be)g(the)g(public)f(key.)p 3507
2030 V 304 2109 a Fa(14)p 390 2143 V 102 w Ff(*)h(<p>)p
3507 2143 V 304 2222 a Fa(15)p 390 2256 V 102 w Ff(*)g(Here)g(the)g
(public)f(key)h(is)g(called)f(<code>Power</code>,)d(that)p
3507 2256 V 304 2335 a Fa(16)p 390 2368 V 102 w Ff(*)k(reminds)f(to)h
(the)g(<code>Discrete)d(Logarithm)h(Problem)h(\(DLP\)</code>.)p
3507 2368 V 304 2447 a Fa(17)p 390 2481 V 102 w Ff(*)h(<p>)p
3507 2481 V 304 2560 a Fa(18)p 390 2594 V 102 w Ff(*)g(The)g
(C++-library)e(for)i(calculation)e(is)i(<code>LiDIA</code>.)p
3507 2594 V 304 2673 a Fa(19)p 390 2707 V 102 w Ff(*)p
3507 2707 V 304 2786 a Fa(20)p 390 2820 V 102 w Ff(*)g(@author)f
(Markus)g(Crnjak)p 3507 2820 V 304 2899 a Fa(21)p 390
2933 V 102 w Ff(*)p 3507 2933 V 304 3012 a Fa(22)p 390
3046 V 102 w Ff(*)h(@see)g(ECElGamalGFPKey)p 3507 3046
V 304 3125 a Fa(23)p 390 3159 V 102 w Ff(*)g(@see)g
(ECElGamalGFPPrivateKey)p 3507 3159 V 304 3238 a Fa(24)p
390 3272 V 102 w Ff(*)g(@see)g(ECElGamalGFPKeyPairGener)o(ator)p
3507 3272 V 304 3351 a Fa(25)p 390 3385 V 102 w Ff(*)g(@see)g
(ECElGamalGFPBlockCipher)p 3507 3385 V 304 3464 a Fa(26)p
390 3498 V 102 w Ff(*)g(@see)g(ECElGamalGFPCipher)p 3507
3498 V 304 3577 a Fa(27)p 390 3610 V 102 w Ff(*)g(@see)g
(cdc.ec.Provider)p 3507 3610 V 304 3690 a Fa(28)p 390
3723 V 102 w Ff(*)g(@see)g(cdc.lidia.LECDomainParam)o(eter)o(sP13)o(63)
p 3507 3723 V 304 3802 a Fa(29)p 390 3836 V 102 w Ff(*/)p
3507 3836 V 304 3915 a Fa(30)p 390 3949 V 54 w Ff(public)f(class)h
(ECElGamalGFPPublicKey)p 3507 3949 V 304 4028 a Fa(31)p
390 4062 V 245 w Ff(extends)f(ECElGamalGFPKey)p 3507
4062 V 304 4141 a Fa(32)p 390 4175 V 245 w Ff(implements)f(PublicKey)p
3507 4175 V 304 4254 a Fa(33)p 390 4288 V 54 w Ff({)p
3507 4288 V 304 4367 a Fa(34)p 390 4401 V 245 w Ff(/**)p
3507 4401 V 304 4480 a Fa(35)p 390 4514 V 293 w Ff(*)i(holds)g(the)g
(public)f(key,)g(that)h(is)g(an)g(elliptic)f(curve)g(point)p
3507 4514 V 304 4593 a Fa(36)p 390 4627 V 293 w Ff(*)p
3507 4627 V 304 4706 a Fa(37)p 390 4740 V 293 w Ff(*)h(@serial)p
3507 4740 V 304 4819 a Fa(38)p 390 4852 V 293 w Ff(*/)p
3507 4852 V 304 4932 a Fa(39)p 390 4965 V 245 w Ff(private)f
(LECPointGFP)f(mPower;)p 3507 4965 V 304 5044 a Fa(40)p
390 5078 V 3507 5078 V 304 5157 a(41)p 390 5191 V 245
w Ff(/**)p 3507 5191 V 304 5270 a Fa(42)p 390 5304 V
293 w Ff(*)i(constructor,)e(that)h(sets)h(the)g(domain)f(parameters)f
(and)i(the)p 3507 5304 V 304 5383 a Fa(43)p 390 5417
V 293 w Ff(*)g(public)f(part)h(of)g(this)g(<code>ECElGamalGFPKey</c)o
(ode)o(>,)42 b(that)p 3507 5417 V 304 5496 a Fa(44)p
390 5530 V 293 w Ff(*)47 b(is)g(an)h(elliptic)d(curve)i(point)p
3507 5530 V 304 5609 a Fa(45)p 390 5643 V 293 w Ff(*)p
3507 5643 V eop
%%Page: 43 44
43 43 bop 270 82 a Fw(B.3.)62 b(ECElGamalGFPPublicKey)1722
b(43)184 415 y Fa(46)p 270 449 4 113 v 293 w Ff(*)47
b(@param)f(params)g(the)h(lECDomainParametersP1363)42
b(contains)j(the)p 3387 449 V 184 528 a Fa(47)p 270 562
V 293 w Ff(*)i(domain)f(parameters)p 3387 562 V 184 641
a Fa(48)p 270 675 V 293 w Ff(*)h(@param)f(power)h(the)g(lECPointGFP)d
(contains)i(the)h(public)f(key)p 3387 675 V 184 754 a
Fa(49)p 270 788 V 293 w Ff(*/)p 3387 788 V 184 867 a
Fa(50)p 270 901 V 245 w Ff(protected)f(ECElGamalGFPPublicKey\(LECD)o
(oma)o(inPa)o(rame)o(ter)o(sP13)o(63)p 3387 901 V 184
980 a Fa(51)p 270 1014 V 436 w Ff(params,LECPointGFP)e(power\))p
3387 1014 V 184 1093 a Fa(52)p 270 1126 V 245 w Ff({)p
3387 1126 V 184 1205 a Fa(53)p 270 1239 V 436 w Ff(super\(params\);)p
3387 1239 V 184 1318 a Fa(54)p 270 1352 V 436 w Ff(mPower)j(=)i(power;)
p 3387 1352 V 184 1431 a Fa(55)p 270 1465 V 245 w Ff(})p
3387 1465 V 184 1544 a Fa(56)p 270 1578 V 3387 1578 V
184 1657 a(57)p 270 1691 V 245 w Ff(/**)p 3387 1691 V
184 1770 a Fa(58)p 270 1804 V 293 w Ff(*)f(returns)f(the)h(public)f
(key,)h(that)f(is)i(an)f(elliptic)e(curve)i(point.)p
3387 1804 V 184 1883 a Fa(59)p 270 1917 V 293 w Ff(*/)p
3387 1917 V 184 1996 a Fa(60)p 270 2030 V 245 w Ff(protected)e
(LECPointGFP)g(getPower\(\))p 3387 2030 V 184 2109 a
Fa(61)p 270 2143 V 245 w Ff({)p 3387 2143 V 184 2222
a Fa(62)p 270 2256 V 436 w Ff(return)h(mPower;)p 3387
2256 V 184 2335 a Fa(63)p 270 2368 V 245 w Ff(})p 3387
2368 V 184 2447 a Fa(64)p 270 2481 V 54 w Ff(})p 3387
2481 V 270 2506 4 25 v 3387 2506 V 270 2509 3120 4 v
eop
%%Page: 44 45
44 44 bop 390 82 a Fw(44)1464 b(ANHANG)32 b(B.)61 b(CDC.EC.ECELGAMAL)
390 415 y Fz(B.4)135 b(ECElGamalGFPKeyP)l(airGenerator)p
390 537 3120 4 v 390 562 4 25 v 3507 562 V 335 641 a
Fa(1)p 390 675 4 113 v 53 w Ff(package)46 b(cdc.ec.ecelgamal;)p
3507 675 V 335 754 a Fa(2)p 390 788 V 3507 788 V 335
867 a(3)p 390 901 V 53 w Ff(import)g(java.math.BigInteger;)p
3507 901 V 335 980 a Fa(4)p 390 1014 V 53 w Ff(import)g
(java.security.*;)p 3507 1014 V 335 1093 a Fa(5)p 390
1126 V 53 w Ff(import)g(java.security.spec.Algori)o(thmP)o(aram)o(ete)o
(rSpe)o(c;)p 3507 1126 V 335 1205 a Fa(6)p 390 1239 V
53 w Ff(import)g(cdc.lidia.*;)p 3507 1239 V 335 1318
a Fa(7)p 390 1352 V 3507 1352 V 335 1431 a(8)p 390 1465
V 53 w Ff(/**)p 3507 1465 V 335 1544 a Fa(9)p 390 1578
V 101 w Ff(*)h(<code>ECElGamalGFPKeyPairG)o(ene)o(rato)o(r</c)o(ode)o
(>)42 b(provides)j(the)p 3507 1578 V 304 1657 a Fa(10)p
390 1691 V 102 w Ff(*)i(implementation)d(to)j(generate)f(pairs)g(of)i
(<code>ECElGamalGFP</cod)o(e>)p 3507 1691 V 304 1770
a Fa(11)p 390 1804 V 102 w Ff(*)f(public)f(and)h(private)f(keys.)h
(That)f(is)h(for)g(an)g(ElGamal)f(algorithm)p 3507 1804
V 304 1883 a Fa(12)p 390 1917 V 102 w Ff(*)h(based)g(on)g(elliptic)e
(curves)i(over)f(the)h(general)f(field)g(with)p 3507
1917 V 304 1996 a Fa(13)p 390 2030 V 102 w Ff(*)h(<code>p</code>)d
(elements)i(where)g(<code>p</code>)e(is)j(a)h(prime)p
3507 2030 V 304 2109 a Fa(14)p 390 2143 V 102 w Ff(*)f(number.)p
3507 2143 V 304 2222 a Fa(15)p 390 2256 V 102 w Ff(*)g(<p>)p
3507 2256 V 304 2335 a Fa(16)p 390 2368 V 102 w Ff(*)g(If)h
(<code>G</code>)43 b(is)48 b(the)f(generator)e(of)i(a)g(subgroup)f(of)h
(an)p 3507 2368 V 304 2447 a Fa(17)p 390 2481 V 102 w
Ff(*)g(elliptic)f(curve)g(and)h(<code>s</code>)d(is)j(an)g(integer)f
(number)g(with)p 3507 2481 V 304 2560 a Fa(18)p 390 2594
V 102 w Ff(*)h(<code>W)f(=)i(sG</code>,)d(then)h(<code>s</code>)e(will)
j(be)g(the)g(private)p 3507 2594 V 304 2673 a Fa(19)p
390 2707 V 102 w Ff(*)g(key)g(and)g(<code>W</code>)d(will)j(be)g(the)g
(public)f(key.)p 3507 2707 V 304 2786 a Fa(20)p 390 2820
V 102 w Ff(*)h(<p>)p 3507 2820 V 304 2899 a Fa(21)p 390
2933 V 102 w Ff(*)g(The)g(C++-library)e(for)i(calculation)e(is)i
(<code>LiDIA</code>.)p 3507 2933 V 304 3012 a Fa(22)p
390 3046 V 102 w Ff(*)p 3507 3046 V 304 3125 a Fa(23)p
390 3159 V 102 w Ff(*)g(@author)f(Markus)g(Crnjak)p 3507
3159 V 304 3238 a Fa(24)p 390 3272 V 102 w Ff(*)p 3507
3272 V 304 3351 a Fa(25)p 390 3385 V 102 w Ff(*)h(@see)g
(ECElGamalGFPKey)p 3507 3385 V 304 3464 a Fa(26)p 390
3498 V 102 w Ff(*)g(@see)g(ECElGamalGFPPrivateKey)p 3507
3498 V 304 3577 a Fa(27)p 390 3610 V 102 w Ff(*)g(@see)g
(ECElGamalGFPPublicKey)p 3507 3610 V 304 3690 a Fa(28)p
390 3723 V 102 w Ff(*)g(@see)g(ECElGamalGFPBlockCipher)p
3507 3723 V 304 3802 a Fa(29)p 390 3836 V 102 w Ff(*)g(@see)g
(ECElGamalGFPCipher)p 3507 3836 V 304 3915 a Fa(30)p
390 3949 V 102 w Ff(*)g(@see)g(cdc.ec.Provider)p 3507
3949 V 304 4028 a Fa(31)p 390 4062 V 102 w Ff(*)g(@see)g
(cdc.lidia.LECDomainParam)o(eter)o(sP13)o(63)p 3507 4062
V 304 4141 a Fa(32)p 390 4175 V 102 w Ff(*/)p 3507 4175
V 304 4254 a Fa(33)p 390 4288 V 54 w Ff(public)f(class)h
(ECElGamalGFPKeyPairGene)o(rato)o(r)p 3507 4288 V 304
4367 a Fa(34)p 390 4401 V 245 w Ff(extends)f(KeyPairGeneratorSpi)p
3507 4401 V 304 4480 a Fa(35)p 390 4514 V 54 w Ff({)p
3507 4514 V 304 4593 a Fa(36)p 390 4627 V 245 w Ff(/**)p
3507 4627 V 304 4706 a Fa(37)p 390 4740 V 293 w Ff(*)h(holds)g(the)g
(domain)f(parameters)p 3507 4740 V 304 4819 a Fa(38)p
390 4852 V 293 w Ff(*/)p 3507 4852 V 304 4932 a Fa(39)p
390 4965 V 245 w Ff(private)g(LECDomainParametersP1363)41
b(mParams;)p 3507 4965 V 304 5044 a Fa(40)p 390 5078
V 3507 5078 V 304 5157 a(41)p 390 5191 V 245 w Ff(/**)p
3507 5191 V 304 5270 a Fa(42)p 390 5304 V 293 w Ff(*)47
b(holds)g(some)f(random)g(information)p 3507 5304 V 304
5383 a Fa(43)p 390 5417 V 293 w Ff(*/)p 3507 5417 V 304
5496 a Fa(44)p 390 5530 V 245 w Ff(private)g(SecureRandom)e
(mSecureRandom)h(=)i(null;)p 3507 5530 V 304 5609 a Fa(45)p
390 5643 V 3507 5643 V eop
%%Page: 45 46
45 45 bop 270 82 a Fw(B.4.)62 b(ECElGamalGFPKeyP)m(airGenerator)1413
b(45)184 415 y Fa(46)p 270 449 4 113 v 245 w Ff(/**)p
3387 449 V 184 528 a Fa(47)p 270 562 V 293 w Ff(*)47
b(initializes)e(this)p 3387 562 V 184 641 a Fa(48)p 270
675 V 293 w Ff(*)i(<code>ECElGamalGFPKeyPair)o(Gene)o(rato)o(r</)o
(code)o(>)42 b(with)p 3387 675 V 184 754 a Fa(49)p 270
788 V 293 w Ff(*)47 b(the)g(domain)f(parameters)f(and)i(some)g(random)f
(information.)e(This)p 3387 788 V 184 867 a Fa(50)p 270
901 V 293 w Ff(*)j(method)f(is)95 b(used)47 b(if)g(the)g(domain)f
(parameters)f(are)i(already)p 3387 901 V 184 980 a Fa(51)p
270 1014 V 293 w Ff(*)g(generated.)p 3387 1014 V 184
1093 a Fa(52)p 270 1126 V 293 w Ff(*)p 3387 1126 V 184
1205 a Fa(53)p 270 1239 V 293 w Ff(*)g(@param)f(params)g(the)h
(algorithmParameterSpec)42 b(contains)k(the)p 3387 1239
V 184 1318 a Fa(54)p 270 1352 V 293 w Ff(*)h(domain)f(parameters)p
3387 1352 V 184 1431 a Fa(55)p 270 1465 V 293 w Ff(*)h(@param)f(random)
g(the)h(secureRandom)e(contains)g(some)i(random)p 3387
1465 V 184 1544 a Fa(56)p 270 1578 V 293 w Ff(*)g(information)p
3387 1578 V 184 1657 a Fa(57)p 270 1691 V 293 w Ff(*)p
3387 1691 V 184 1770 a Fa(58)p 270 1804 V 293 w Ff(*)g(@throws)f
(InvalidAlgorithmParameter)o(Exc)o(epti)o(on)c(if)p 3387
1804 V 184 1883 a Fa(59)p 270 1917 V 293 w Ff(*)47 b
(<code>params</code>)c(is)k(not)g(an)g(instance)f(of)p
3387 1917 V 184 1996 a Fa(60)p 270 2030 V 293 w Ff(*)h
(<code>LECDomainParameters)o(P136)o(3</c)o(ode)o(>)p
3387 2030 V 184 2109 a Fa(61)p 270 2143 V 293 w Ff(*)p
3387 2143 V 184 2222 a Fa(62)p 270 2256 V 293 w Ff(*)g(@see)g
(cdc.lidia.LECDomainParam)o(eter)o(sP1)o(363)p 3387 2256
V 184 2335 a Fa(63)p 270 2368 V 293 w Ff(*/)p 3387 2368
V 184 2447 a Fa(64)p 270 2481 V 245 w Ff(public)f(void)h
(initialize\(AlgorithmPara)o(met)o(erSp)o(ec)42 b(params,)p
3387 2481 V 184 2560 a Fa(65)p 270 2594 V 341 w Ff(SecureRandom)i
(random\))p 3387 2594 V 184 2673 a Fa(66)p 270 2707 V
436 w Ff(throws)i(InvalidAlgorithmParameter)o(Exc)o(epti)o(on)p
3387 2707 V 184 2786 a Fa(67)p 270 2820 V 245 w Ff({)p
3387 2820 V 184 2899 a Fa(68)p 270 2933 V 436 w Ff(//)h(checks)f(the)h
(domain)f(parameters)p 3387 2933 V 184 3012 a Fa(69)p
270 3046 V 436 w Ff(if)h(\(!\(params)e(instanceof)g
(LECDomainParametersP1363\)\))p 3387 3046 V 184 3125
a Fa(70)p 270 3159 V 627 w Ff({)p 3387 3159 V 184 3238
a Fa(71)p 270 3272 V 818 w Ff(throw)h(new)h(InvalidAlgorithmParameter)o
(Exc)o(epti)o(on\()p 3387 3272 V 184 3351 a Fa(72)p 270
3385 V 913 w Ff("in)g(ECElGamalGFPKeyPairGenera)o(tor:)41
b(initialize:")p 3387 3385 V 184 3464 a Fa(73)p 270 3498
V 913 w Ff(+)48 b(")f(params)f(is)h(not)g(an)h(instance)d(of")p
3387 3498 V 184 3577 a Fa(74)p 270 3610 V 913 w Ff(+)j(")f
(LECDomainParametersP1363")o(\);)p 3387 3610 V 184 3689
a Fa(75)p 270 3723 V 627 w Ff(})p 3387 3723 V 184 3802
a Fa(76)p 270 3836 V 3387 3836 V 184 3915 a(77)p 270
3949 V 436 w Ff(//)g(sets)g(the)g(fields)p 3387 3949
V 184 4028 a Fa(78)p 270 4062 V 436 w Ff(mParams)f(=)h
(\(LECDomainParametersP1363)o(\)par)o(ams;)p 3387 4062
V 184 4141 a Fa(79)p 270 4175 V 436 w Ff(mSecureRandom)d(=)k(random;)p
3387 4175 V 184 4254 a Fa(80)p 270 4288 V 245 w Ff(})p
3387 4288 V 184 4367 a Fa(81)p 270 4401 V 3387 4401 V
184 4480 a(82)p 270 4514 V 245 w Ff(/**)p 3387 4514 V
184 4593 a Fa(83)p 270 4627 V 293 w Ff(*)f(initializes)e(this)p
3387 4627 V 184 4706 a Fa(84)p 270 4740 V 293 w Ff(*)i
(<code>ECElGamalGFPKeyPair)o(Gene)o(rato)o(r</)o(code)o(>)42
b(for)47 b(a)g(certain)p 3387 4740 V 184 4819 a Fa(85)p
270 4852 V 293 w Ff(*)g(strength)f(with)g(some)h(random)f(information.)
f(The)i(parameter)p 3387 4852 V 184 4932 a Fa(86)p 270
4965 V 293 w Ff(*)g(<code>strength</code>)42 b(is)47
b(interpreted)e(as)i(the)g(modulus)f(of)p 3387 4965 V
184 5044 a Fa(87)p 270 5078 V 293 w Ff(*)h(the)g(curve.)f(For)h(the)g
(cofactor)f(the)h(value)f(of)h(10)g(percent)f(is)p 3387
5078 V 184 5157 a Fa(88)p 270 5191 V 293 w Ff(*)h(set.)p
3387 5191 V 184 5270 a Fa(89)p 270 5304 V 293 w Ff(*)p
3387 5304 V 184 5383 a Fa(90)p 270 5417 V 293 w Ff(*)g(@param)f(size)h
(the)g(strength)e(of)j(the)f(keys)p 3387 5417 V 184 5496
a Fa(91)p 270 5530 V 293 w Ff(*)g(@param)f(random)g(the)h(secureRandom)
e(contains)g(random)p 3387 5530 V 184 5609 a Fa(92)p
270 5643 V 293 w Ff(*)i(information)p 3387 5643 V eop
%%Page: 46 47
46 46 bop 390 82 a Fw(46)1464 b(ANHANG)32 b(B.)61 b(CDC.EC.ECELGAMAL)
304 415 y Fa(93)p 390 449 4 113 v 293 w Ff(*/)p 3507
449 V 304 528 a Fa(94)p 390 562 V 245 w Ff(public)46
b(void)h(initialize\(int)d(size,)i(SecureRandom)f(random\))p
3507 562 V 304 641 a Fa(95)p 390 675 V 245 w Ff({)p 3507
675 V 304 754 a Fa(96)p 390 788 V 436 w Ff(try)p 3507
788 V 304 867 a Fa(97)p 390 901 V 627 w Ff({)p 3507 901
V 304 980 a Fa(98)p 390 1014 V 818 w Ff(//)i(generates)e(the)i(domain)f
(parameters)f(with)i(default)p 3507 1014 V 304 1093 a
Fa(99)p 390 1126 V 818 w Ff(//)g(values)p 3507 1126 V
274 1205 a Fa(100)p 390 1239 V 818 w Ff(LBigInt)f(field)g(=)h(new)g
(LBigInt\(1\);)p 3507 1239 V 274 1318 a Fa(101)p 390
1352 V 818 w Ff(LBigInt)f(bitsize)f(=)j(new)f(LBigInt\(size\);)p
3507 1352 V 274 1431 a Fa(102)p 390 1465 V 818 w Ff(LBigInt)f
(percentage)f(=)i(new)g(LBigInt\(10\);)p 3507 1465 V
274 1544 a Fa(103)p 390 1578 V 818 w Ff(LBigInt)f(info)g(=)i(new)f
(LBigInt\(0\);)p 3507 1578 V 274 1657 a Fa(104)p 390
1691 V 818 w Ff(mParams)f(=)h(new)g(LECDomainParametersP1363)o(\(fie)o
(ld,)p 3507 1691 V 274 1770 a Fa(105)p 390 1804 V 913
w Ff(bitsize,percentage,info\);)p 3507 1804 V 274 1883
a Fa(106)p 390 1917 V 3507 1917 V 274 1996 a(107)p 390
2030 V 818 w Ff(//)g(sets)g(the)g(remaining)e(field)p
3507 2030 V 274 2109 a Fa(108)p 390 2143 V 818 w Ff(mSecureRandom)f(=)j
(random;)p 3507 2143 V 274 2222 a Fa(109)p 390 2256 V
627 w Ff(})p 3507 2256 V 274 2335 a Fa(110)p 390 2368
V 436 w Ff(catch)f(\(LiDIAError)f(lError\))p 3507 2368
V 274 2447 a Fa(111)p 390 2481 V 627 w Ff({)p 3507 2481
V 274 2560 a Fa(112)p 390 2594 V 818 w Ff(//)i(if)g(an)g(error)g
(occurred)p 3507 2594 V 274 2673 a Fa(113)p 390 2707
V 818 w Ff(System.out.println\(lErro)o(r.t)o(oStr)o(ing\()o(\)\);)p
3507 2707 V 274 2786 a Fa(114)p 390 2820 V 818 w Ff(System.exit\(0\);)p
3507 2820 V 274 2899 a Fa(115)p 390 2933 V 627 w Ff(})p
3507 2933 V 274 3012 a Fa(116)p 390 3046 V 245 w Ff(})p
3507 3046 V 274 3125 a Fa(117)p 390 3159 V 3507 3159
V 274 3238 a(118)p 390 3272 V 245 w Ff(/**)p 3507 3272
V 274 3351 a Fa(119)p 390 3385 V 293 w Ff(*)g(generates)f(a)h
(<code>KeyPair</code>)42 b(of)47 b(an)p 3507 3385 V 274
3464 a Fa(120)p 390 3498 V 293 w Ff(*)g(<code>ECElGamalGFPPrivate)o
(Key<)o(/cod)o(e>)41 b(and)47 b(an)p 3507 3498 V 274
3577 a Fa(121)p 390 3610 V 293 w Ff(*)g(<code>ECElGamalGFPPublicK)o
(ey</)o(code)o(>)42 b(and)47 b(returns)e(it)p 3507 3610
V 274 3689 a Fa(122)p 390 3723 V 293 w Ff(*/)p 3507 3723
V 274 3802 a Fa(123)p 390 3836 V 245 w Ff(public)h(KeyPair)g
(generateKeyPair\(\))p 3507 3836 V 274 3915 a Fa(124)p
390 3949 V 245 w Ff({)p 3507 3949 V 274 4028 a Fa(125)p
390 4062 V 436 w Ff(//)h(initialization)d(if)j(necessary)p
3507 4062 V 274 4141 a Fa(126)p 390 4175 V 436 w Ff(if)g
(\(mSecureRandom)d(==)j(null\))p 3507 4175 V 274 4254
a Fa(127)p 390 4288 V 627 w Ff({)p 3507 4288 V 274 4367
a Fa(128)p 390 4401 V 818 w Ff(try)p 3507 4401 V 274
4480 a Fa(129)p 390 4514 V 1009 w Ff({)p 3507 4514 V
274 4593 a Fa(130)p 390 4627 V 1200 w Ff(//)g(generates)e(the)i(domain)
f(parameters)f(with)p 3507 4627 V 274 4706 a Fa(131)p
390 4740 V 1200 w Ff(//)i(default)f(values)p 3507 4740
V 274 4819 a Fa(132)p 390 4852 V 1200 w Ff(LBigInt)f(field)i(=)g(new)g
(LBigInt\(1\);)p 3507 4852 V 274 4932 a Fa(133)p 390
4965 V 1200 w Ff(LBigInt)e(bitsize)h(=)i(new)f(LBigInt\(200\);)p
3507 4965 V 274 5044 a Fa(134)p 390 5078 V 1200 w Ff(LBigInt)e
(percentage)g(=)j(new)f(LBigInt\(10\);)p 3507 5078 V
274 5157 a Fa(135)p 390 5191 V 1200 w Ff(LBigInt)e(info)i(=)h(new)e
(LBigInt\(0\);)p 3507 5191 V 274 5270 a Fa(136)p 390
5304 V 1200 w Ff(mParams)f(=)j(new)f(LECDomainParametersP1363)o(\()p
3507 5304 V 274 5383 a Fa(137)p 390 5417 V 1295 w Ff
(field,bitsize,percentage,)o(inf)o(o\);)p 3507 5417 V
274 5496 a Fa(138)p 390 5530 V 3507 5530 V 274 5609 a(139)p
390 5643 V 1200 w Ff(//)g(sets)f(the)h(remaining)f(field)p
3507 5643 V eop
%%Page: 47 48
47 47 bop 270 82 a Fw(B.4.)62 b(ECElGamalGFPKeyP)m(airGenerator)1413
b(47)154 415 y Fa(140)p 270 449 4 113 v 1200 w Ff(mSecureRandom)44
b(=)j(new)g(SecureRandom\(\);)p 3387 449 V 154 528 a
Fa(141)p 270 562 V 1009 w Ff(})p 3387 562 V 154 641 a
Fa(142)p 270 675 V 818 w Ff(catch)f(\(LiDIAError)f(lError\))p
3387 675 V 154 754 a Fa(143)p 270 788 V 1009 w Ff({)p
3387 788 V 154 867 a Fa(144)p 270 901 V 1200 w Ff(//)i(if)g(an)g(error)
g(occurred)p 3387 901 V 154 980 a Fa(145)p 270 1014 V
1200 w Ff(System.out.println\(lErr)o(or.t)o(oSt)o(ring)o(\(\)\);)p
3387 1014 V 154 1093 a Fa(146)p 270 1126 V 1200 w Ff(System.exit\(0\);)
p 3387 1126 V 154 1205 a Fa(147)p 270 1239 V 1009 w Ff(})p
3387 1239 V 154 1318 a Fa(148)p 270 1352 V 627 w Ff(})p
3387 1352 V 154 1431 a Fa(149)p 270 1465 V 3387 1465
V 154 1544 a(150)p 270 1578 V 436 w Ff(//)g(find)g(a)g(private)f(key)h
(s)g(with)g(0)h(<)f(s)g(<)h(order)e(of)h(the)p 3387 1578
V 154 1657 a Fa(151)p 270 1691 V 436 w Ff(//)g(generator)p
3387 1691 V 154 1770 a Fa(152)p 270 1804 V 436 w Ff(BigInteger)e
(upperBound)g(=)i(\(new)g(LBigInt\(mParams.getR\(\)\)\).)p
3387 1804 V 154 1883 a Fa(153)p 270 1917 V 531 w Ff(toBigInteger\(\);)p
3387 1917 V 154 1996 a Fa(154)p 270 2030 V 436 w Ff(BigInteger)e(s)i(=)
h(new)f(BigInteger\(upperBound.bi)o(tLe)o(ngth)o(\(\)-1)o(,)p
3387 2030 V 154 2109 a Fa(155)p 270 2143 V 531 w Ff(mSecureRandom\);)p
3387 2143 V 154 2222 a Fa(156)p 270 2256 V 3387 2256
V 154 2335 a(157)p 270 2368 V 436 w Ff(//)g(construction)e(of)i(the)g
(the)g(key)g(pair)p 3387 2368 V 154 2447 a Fa(158)p 270
2481 V 436 w Ff(ECElGamalGFPPrivateKey)42 b(privateKey)j(=)i(new)p
3387 2481 V 154 2560 a Fa(159)p 270 2594 V 531 w Ff
(ECElGamalGFPPrivateKey\(mPa)o(rams)o(,s\))o(;)p 3387
2594 V 154 2673 a Fa(160)p 270 2707 V 436 w Ff(ECElGamalGFPPublicKey)42
b(publicKey)j(=)j(new)p 3387 2707 V 154 2786 a Fa(161)p
270 2820 V 531 w Ff(ECElGamalGFPPublicKey\(mPar)o(ams,)o(mPa)o(rams)o
(.get)o(G\(\))o(.mul)o(tipl)o(y\()p 3387 2820 V 154 2899
a Fa(162)p 270 2933 V 627 w Ff(new)f(LBigInt\(s\)\)\);)p
3387 2933 V 154 3012 a Fa(163)p 270 3046 V 3387 3046
V 154 3125 a(164)p 270 3159 V 436 w Ff(//)g(and)g(back)p
3387 3159 V 154 3238 a Fa(165)p 270 3272 V 436 w Ff(return)f(new)h
(KeyPair\(publicKey,privat)o(eKey)o(\);)p 3387 3272 V
154 3351 a Fa(166)p 270 3385 V 245 w Ff(})p 3387 3385
V 154 3464 a Fa(167)p 270 3498 V 54 w Ff(})p 3387 3498
V 270 3522 4 25 v 3387 3522 V 270 3525 3120 4 v eop
%%Page: 48 49
48 48 bop 390 82 a Fw(48)1464 b(ANHANG)32 b(B.)61 b(CDC.EC.ECELGAMAL)
390 415 y Fz(B.5)135 b(ECElGamalGFPBlo)t(c)l(kCipher)p
390 537 3120 4 v 390 562 4 25 v 3507 562 V 335 641 a
Fa(1)p 390 675 4 113 v 53 w Ff(package)46 b(cdc.ec.ecelgamal;)p
3507 675 V 335 754 a Fa(2)p 390 788 V 3507 788 V 335
867 a(3)p 390 901 V 53 w Ff(import)g(java.math.BigInteger;)p
3507 901 V 335 980 a Fa(4)p 390 1014 V 53 w Ff(import)g
(java.security.*;)p 3507 1014 V 335 1093 a Fa(5)p 390
1126 V 53 w Ff(import)g(java.security.spec.*;)p 3507
1126 V 335 1205 a Fa(6)p 390 1239 V 53 w Ff(import)g(javax.crypto.*;)p
3507 1239 V 335 1318 a Fa(7)p 390 1352 V 3507 1352 V
335 1431 a(8)p 390 1465 V 53 w Ff(/**)p 3507 1465 V 335
1544 a Fa(9)p 390 1578 V 101 w Ff(*)h(<code>ECElGamalGFPBlockCip)o(her)
o(</co)o(de>)41 b(supplies)46 b(the)h(block)p 3507 1578
V 304 1657 a Fa(10)p 390 1691 V 102 w Ff(*)g(handling)f(for)h(block)f
(ciphers.)g(It)h(implements)e(a)i(buffering)f(scheme)p
3507 1691 V 304 1770 a Fa(11)p 390 1804 V 102 w Ff(*)h(by)h(storing)d
(the)i(data)g(in)g(an)g(internal)f(buffer.)p 3507 1804
V 304 1883 a Fa(12)p 390 1917 V 102 w Ff(*)h(<p>)p 3507
1917 V 304 1996 a Fa(13)p 390 2030 V 102 w Ff(*)g(The)g(two)g(abstract)
f(methods)g(<code>engineTransformBl)o(ock)o(</co)o(de>)p
3507 2030 V 304 2109 a Fa(14)p 390 2143 V 102 w Ff(*)h(and)g
(<code>engineTransformBloc)o(kFin)o(al</)o(cod)o(e>)42
b(have)k(to)h(be)p 3507 2143 V 304 2222 a Fa(15)p 390
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%%Page: 49 50
49 49 bop 270 82 a Fw(B.5.)62 b(ECElGamalGFPBlo)s(c)m(kCipher)1644
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%%Page: 50 51
50 50 bop 390 82 a Fw(50)1464 b(ANHANG)32 b(B.)61 b(CDC.EC.ECELGAMAL)
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5609 a Fa(139)p 390 5643 V 293 w Ff(*)h(start)p 3507
5643 V eop
%%Page: 51 52
51 51 bop 270 82 a Fw(B.5.)62 b(ECElGamalGFPBlo)s(c)m(kCipher)1644
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245 w Ff(})p 3387 3498 V 154 3577 a Fa(168)p 270 3610
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V eop
%%Page: 52 53
52 52 bop 390 82 a Fw(52)1464 b(ANHANG)32 b(B.)61 b(CDC.EC.ECELGAMAL)
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w Ff(int)g(bufferedLength)d(=)j(getBufferedDataLength\(\);)p
3507 1352 V 274 1431 a Fa(196)p 390 1465 V 436 w Ff(byte[])f
(totalInput)f(=)j(new)e(byte[inputLength)e(+)p 3507 1465
V 274 1544 a Fa(197)p 390 1578 V 531 w Ff(bufferedLength];)p
3507 1578 V 274 1657 a Fa(198)p 390 1691 V 436 w Ff
(getBufferedData\(totalInp)o(ut,0)o(\);)p 3507 1691 V
274 1770 a Fa(199)p 390 1804 V 436 w Ff(System.arraycopy\(input,i)o
(nput)o(Offs)o(et,)o(tota)o(lInp)o(ut,)p 3507 1804 V
274 1883 a Fa(200)p 390 1917 V 531 w Ff(bufferedLength,inputLength)o
(\);)p 3507 1917 V 274 1996 a Fa(201)p 390 2030 V 3507
2030 V 274 2109 a(202)p 390 2143 V 436 w Ff(//)j(calculates)p
3507 2143 V 274 2222 a Fa(203)p 390 2256 V 436 w Ff(int)g(blockSize)e
(=)j(engineGetBlockSize\(\);)p 3507 2256 V 274 2335 a
Fa(204)p 390 2368 V 436 w Ff(int)f(lastBlockSize)d(=)k
(totalInput.length)43 b(\045)k(blockSize;)p 3507 2368
V 274 2447 a Fa(205)p 390 2481 V 436 w Ff(int)g(lastBlockOffset)d(=)j
(totalInput.length)c(-)48 b(lastBlockSize;)p 3507 2481
V 274 2560 a Fa(206)p 390 2594 V 436 w Ff(int)f(outputLength)d(=)k(0;)p
3507 2594 V 274 2673 a Fa(207)p 390 2707 V 3507 2707
V 274 2786 a(208)p 390 2820 V 436 w Ff(//)f(processes)e(by)j(calling)d
(the)i(abstract)f(method)p 3507 2820 V 274 2899 a Fa(209)p
390 2933 V 436 w Ff(//)h(engineTransformBlock)p 3507
2933 V 274 3012 a Fa(210)p 390 3046 V 436 w Ff(for)g(\(int)g(i)g(=)g
(0;i)g(<)h(lastBlockOffset;i)43 b(+=)k(blockSize\))p
3507 3046 V 274 3125 a Fa(211)p 390 3159 V 627 w Ff({)p
3507 3159 V 274 3238 a Fa(212)p 390 3272 V 818 w Ff(outputLength)d(+=)j
(engineTransformBlock\(total)o(Inpu)o(t,)p 3507 3272
V 274 3351 a Fa(213)p 390 3385 V 913 w Ff(i,blockSize,output,output)o
(Offs)o(et)42 b(+)p 3507 3385 V 274 3464 a Fa(214)p 390
3498 V 913 w Ff(outputLength\);)p 3507 3498 V 274 3577
a Fa(215)p 390 3610 V 627 w Ff(})p 3507 3610 V 274 3689
a Fa(216)p 390 3723 V 436 w Ff(addToBufferedData\(totalI)o(nput)o(,las)
o(tBl)o(ockO)o(ffse)o(t,)p 3507 3723 V 274 3802 a Fa(217)p
390 3836 V 531 w Ff(lastBlockSize\);)p 3507 3836 V 274
3915 a Fa(218)p 390 3949 V 3507 3949 V 274 4028 a(219)p
390 4062 V 436 w Ff(//)47 b(and)g(back)p 3507 4062 V
274 4141 a Fa(220)p 390 4175 V 436 w Ff(return)f(outputLength;)p
3507 4175 V 274 4254 a Fa(221)p 390 4288 V 245 w Ff(})p
3507 4288 V 274 4367 a Fa(222)p 390 4401 V 3507 4401
V 274 4480 a(223)p 390 4514 V 245 w Ff(/**)p 3507 4514
V 274 4593 a Fa(224)p 390 4627 V 293 w Ff(*)h(adds)g(input)f(data)h(to)
g(the)g(cipher.)p 3507 4627 V 274 4706 a Fa(225)p 390
4740 V 293 w Ff(*)g(<p>)p 3507 4740 V 274 4819 a Fa(226)p
390 4852 V 293 w Ff(*)g(The)g(input)g(block)f(of)h
(<code>inputLength</code>)41 b(bytes)47 b(has)g(to)p
3507 4852 V 274 4932 a Fa(227)p 390 4965 V 293 w Ff(*)g(be)g(supplied)f
(in)h(the)g(bytearray)e(<code>input</code>,)e(starting)p
3507 4965 V 274 5044 a Fa(228)p 390 5078 V 293 w Ff(*)k(at)g
(<code>inputOffset</code>.)41 b(The)47 b(output)f(will)h(be)g(returned)
p 3507 5078 V 274 5157 a Fa(229)p 390 5191 V 293 w Ff(*)g(in)g(the)g
(function)f(value.)p 3507 5191 V 274 5270 a Fa(230)p
390 5304 V 293 w Ff(*)p 3507 5304 V 274 5383 a Fa(231)p
390 5417 V 293 w Ff(*)h(@param)f(input)h(the)g(bytearray)e(contains)g
(the)i(input)p 3507 5417 V 274 5496 a Fa(232)p 390 5530
V 293 w Ff(*)g(@param)f(inputOffset)f(the)i(position)f(where)g(the)h
(input)f(starts)p 3507 5530 V 274 5609 a Fa(233)p 390
5643 V 293 w Ff(*)h(@param)f(inputLength)f(the)i(length)f(of)h(the)g
(input)p 3507 5643 V eop
%%Page: 53 54
53 53 bop 270 82 a Fw(B.5.)62 b(ECElGamalGFPBlo)s(c)m(kCipher)1644
b(53)154 415 y Fa(234)p 270 449 4 113 v 293 w Ff(*)p
3387 449 V 154 528 a Fa(235)p 270 562 V 293 w Ff(*)47
b(@return)f(the)h(output)p 3387 562 V 154 641 a Fa(236)p
270 675 V 293 w Ff(*/)p 3387 675 V 154 754 a Fa(237)p
270 788 V 245 w Ff(protected)e(byte[])h(engineUpdate\(byte[])d
(input,int)i(inputOffset,)p 3387 788 V 154 867 a Fa(238)p
270 901 V 341 w Ff(int)h(inputLength\))p 3387 901 V 154
980 a Fa(239)p 270 1014 V 245 w Ff({)p 3387 1014 V 154
1093 a Fa(240)p 270 1126 V 436 w Ff(int)h(length)f(=)h(0;)p
3387 1126 V 154 1205 a Fa(241)p 270 1239 V 436 w Ff(byte[])f(out)h(=)g
(new)g(byte[engineGetOutputSize\(i)o(npu)o(tLen)o(gth\))o(];)p
3387 1239 V 154 1318 a Fa(242)p 270 1352 V 3387 1352
V 154 1431 a(243)p 270 1465 V 436 w Ff(//)g(calls)g(the)f(other)h
(engineUpdate)d(method)p 3387 1465 V 154 1544 a Fa(244)p
270 1578 V 436 w Ff(try)p 3387 1578 V 154 1657 a Fa(245)p
270 1691 V 627 w Ff({)p 3387 1691 V 154 1770 a Fa(246)p
270 1804 V 818 w Ff(length)i(=)h(engineUpdate\(input,inputOf)o(fse)o
(t,)p 3387 1804 V 154 1883 a Fa(247)p 270 1917 V 913
w Ff(inputLength,out,0\);)p 3387 1917 V 154 1996 a Fa(248)p
270 2030 V 627 w Ff(})p 3387 2030 V 154 2109 a Fa(249)p
270 2143 V 436 w Ff(catch)f(\(ShortBufferException)c(sbe\))47
b({})p 3387 2143 V 154 2222 a Fa(250)p 270 2256 V 3387
2256 V 154 2335 a(251)p 270 2368 V 436 w Ff(//)g(transforms)e(to)i(the)
g(right)g(size)p 3387 2368 V 154 2447 a Fa(252)p 270
2481 V 436 w Ff(if)g(\(length)f(<)h(out.length\))p 3387
2481 V 154 2560 a Fa(253)p 270 2594 V 627 w Ff({)p 3387
2594 V 154 2673 a Fa(254)p 270 2707 V 818 w Ff(byte[])f(shorter)g(=)h
(new)g(byte[length];)p 3387 2707 V 154 2786 a Fa(255)p
270 2820 V 818 w Ff(System.arraycopy\(out,0,s)o(hor)o(ter,)o(0,le)o
(ngt)o(h\);)p 3387 2820 V 154 2899 a Fa(256)p 270 2933
V 818 w Ff(out)g(=)g(shorter;)p 3387 2933 V 154 3012
a Fa(257)p 270 3046 V 627 w Ff(})p 3387 3046 V 154 3125
a Fa(258)p 270 3159 V 3387 3159 V 154 3238 a(259)p 270
3272 V 436 w Ff(//)g(and)g(back)p 3387 3272 V 154 3351
a Fa(260)p 270 3385 V 436 w Ff(return)f(out;)p 3387 3385
V 154 3464 a Fa(261)p 270 3498 V 245 w Ff(})p 3387 3498
V 154 3577 a Fa(262)p 270 3610 V 3387 3610 V 154 3689
a(263)p 270 3723 V 245 w Ff(/**)p 3387 3723 V 154 3802
a Fa(264)p 270 3836 V 293 w Ff(*)h(finishes)f(the)h(process.)e(A)j
(last)e(piece)h(of)g(input)f(data)h(can)g(be)p 3387 3836
V 154 3915 a Fa(265)p 270 3949 V 293 w Ff(*)g(added)g(to)g(the)g
(cipher.)p 3387 3949 V 154 4028 a Fa(266)p 270 4062 V
293 w Ff(*)g(<p>)p 3387 4062 V 154 4141 a Fa(267)p 270
4175 V 293 w Ff(*)g(The)g(input)g(block)f(of)h
(<code>inputLength</code>)41 b(bytes)47 b(has)g(to)p
3387 4175 V 154 4254 a Fa(268)p 270 4288 V 293 w Ff(*)g(be)g(supplied)f
(in)h(the)g(bytearray)e(<code>input</code>,)e(starting)p
3387 4288 V 154 4367 a Fa(269)p 270 4401 V 293 w Ff(*)k(at)g
(<code>inputOffset</code>.)41 b(The)47 b(output)f(will)h(be)g(written)p
3387 4401 V 154 4480 a Fa(270)p 270 4514 V 293 w Ff(*)g(in)g(the)g
(bytearray)f(<code>output</code>,)c(starting)j(at)p 3387
4514 V 154 4593 a Fa(271)p 270 4627 V 293 w Ff(*)i
(<code>outputOffset</code>)o(.)p 3387 4627 V 154 4706
a Fa(272)p 270 4740 V 293 w Ff(*)p 3387 4740 V 154 4819
a Fa(273)p 270 4852 V 293 w Ff(*)g(@param)f(input)h(the)g(bytearray)e
(contains)g(the)i(input)p 3387 4852 V 154 4932 a Fa(274)p
270 4965 V 293 w Ff(*)g(@param)f(inputOffset)f(the)i(position)f(where)g
(the)h(input)f(starts)p 3387 4965 V 154 5044 a Fa(275)p
270 5078 V 293 w Ff(*)h(@param)f(inputLength)f(the)i(length)f(of)h(the)
g(input)p 3387 5078 V 154 5157 a Fa(276)p 270 5191 V
293 w Ff(*)g(@param)f(output)g(the)h(bytearray)f(will)g(hold)h(the)g
(output)p 3387 5191 V 154 5270 a Fa(277)p 270 5304 V
293 w Ff(*)g(@param)f(outputOffset)f(the)i(position)e(where)i(the)g
(output)f(will)p 3387 5304 V 154 5383 a Fa(278)p 270
5417 V 293 w Ff(*)h(start)p 3387 5417 V 154 5496 a Fa(279)p
270 5530 V 293 w Ff(*)p 3387 5530 V 154 5609 a Fa(280)p
270 5643 V 293 w Ff(*)g(@throws)f(ShortBufferException)c(if)48
b(the)e(output)h(array)f(is)h(too)p 3387 5643 V eop
%%Page: 54 55
54 54 bop 390 82 a Fw(54)1464 b(ANHANG)32 b(B.)61 b(CDC.EC.ECELGAMAL)
274 415 y Fa(281)p 390 449 4 113 v 293 w Ff(*)47 b(small)p
3507 449 V 274 528 a Fa(282)p 390 562 V 293 w Ff(*)g(@throws)f
(IllegalBlockSizeException)41 b(will)47 b(not)g(be)g(thrown)p
3507 562 V 274 641 a Fa(283)p 390 675 V 293 w Ff(*)g(since)g(an)g
(ElGamal)f(cipher)g(does)g(not)h(mind)g(a)g(partial)f(block)p
3507 675 V 274 754 a Fa(284)p 390 788 V 293 w Ff(*)h(@throws)f
(BadPaddingException)d(will)j(not)h(be)g(thrown)f(since)h(the)p
3507 788 V 274 867 a Fa(285)p 390 901 V 293 w Ff(*)g(cipher)f(does)h
(not)g(support)f(a)h(padding)f(scheme)p 3507 901 V 274
980 a Fa(286)p 390 1014 V 293 w Ff(*)p 3507 1014 V 274
1093 a Fa(287)p 390 1126 V 293 w Ff(*)h(@return)f(the)h(number)f(of)h
(bytes)g(written)p 3507 1126 V 274 1205 a Fa(288)p 390
1239 V 293 w Ff(*/)p 3507 1239 V 274 1318 a Fa(289)p
390 1352 V 245 w Ff(protected)e(int)i(engineDoFinal\(byte[])c
(input,int)i(inputOffset,)p 3507 1352 V 274 1431 a Fa(290)p
390 1465 V 341 w Ff(int)h(inputLength,byte[])d(output,int)i
(outputOffset\))p 3507 1465 V 274 1544 a Fa(291)p 390
1578 V 436 w Ff(throws)h(ShortBufferException,Ille)o(gal)o(Bloc)o(kSiz)
o(eEx)o(cept)o(ion,)p 3507 1578 V 274 1657 a Fa(292)p
390 1691 V 531 w Ff(BadPaddingException)p 3507 1691 V
274 1770 a Fa(293)p 390 1804 V 245 w Ff({)p 3507 1804
V 274 1883 a Fa(294)p 390 1917 V 436 w Ff(//)h(concatenates)e(the)i
(data)p 3507 1917 V 274 1996 a Fa(295)p 390 2030 V 436
w Ff(int)g(bufferedLength)d(=)j(getBufferedDataLength\(\);)p
3507 2030 V 274 2109 a Fa(296)p 390 2143 V 436 w Ff(byte[])f
(totalInput)f(=)j(new)e(byte[inputLength)e(+)p 3507 2143
V 274 2222 a Fa(297)p 390 2256 V 531 w Ff(bufferedLength];)p
3507 2256 V 274 2335 a Fa(298)p 390 2368 V 436 w Ff
(getBufferedData\(totalInp)o(ut,0)o(\);)p 3507 2368 V
274 2447 a Fa(299)p 390 2481 V 436 w Ff(if)j(\(inputLength)e(>)i(0\))p
3507 2481 V 274 2560 a Fa(300)p 390 2594 V 627 w Ff({)p
3507 2594 V 274 2673 a Fa(301)p 390 2707 V 818 w Ff
(System.arraycopy\(input,i)o(npu)o(tOff)o(set,)o(tot)o(alIn)o(put,)p
3507 2707 V 274 2786 a Fa(302)p 390 2820 V 913 w Ff
(bufferedLength,inputLengt)o(h\);)p 3507 2820 V 274 2899
a Fa(303)p 390 2933 V 627 w Ff(})p 3507 2933 V 274 3012
a Fa(304)p 390 3046 V 3507 3046 V 274 3125 a(305)p 390
3159 V 436 w Ff(//)g(calculates)p 3507 3159 V 274 3238
a Fa(306)p 390 3272 V 436 w Ff(int)g(blockSize)e(=)j
(engineGetBlockSize\(\);)p 3507 3272 V 274 3351 a Fa(307)p
390 3385 V 436 w Ff(int)f(lastBlockSize)d(=)k(totalInput.length)43
b(\045)k(blockSize;)p 3507 3385 V 274 3464 a Fa(308)p
390 3498 V 436 w Ff(if)g(\(lastBlockSize)d(==)j(0)h(&&)f
(totalInput.length)c(>)k(0\))p 3507 3498 V 274 3577 a
Fa(309)p 390 3610 V 627 w Ff({)p 3507 3610 V 274 3689
a Fa(310)p 390 3723 V 818 w Ff(lastBlockSize)d(=)j(blockSize;)p
3507 3723 V 274 3802 a Fa(311)p 390 3836 V 627 w Ff(})p
3507 3836 V 274 3915 a Fa(312)p 390 3949 V 436 w Ff(int)g
(lastBlockOffset)d(=)j(totalInput.length)c(-)48 b(lastBlockSize;)p
3507 3949 V 274 4028 a Fa(313)p 390 4062 V 436 w Ff(int)f(outputLength)
d(=)k(0;)p 3507 4062 V 274 4141 a Fa(314)p 390 4175 V
3507 4175 V 274 4254 a(315)p 390 4288 V 436 w Ff(//)f(processes)e(by)j
(calling)d(the)i(abstract)f(methods)p 3507 4288 V 274
4367 a Fa(316)p 390 4401 V 436 w Ff(//)h(engineTransformBlock)42
b(and)47 b(engineTransformBlockFinal)p 3507 4401 V 274
4480 a Fa(317)p 390 4514 V 436 w Ff(for)g(\(int)g(i)g(=)g(0;i)g(<)h
(lastBlockOffset;i)43 b(+=)k(blockSize\))p 3507 4514
V 274 4593 a Fa(318)p 390 4627 V 627 w Ff({)p 3507 4627
V 274 4706 a Fa(319)p 390 4740 V 818 w Ff(outputLength)d(+=)j
(engineTransformBlock\(total)o(Inpu)o(t,)p 3507 4740
V 274 4819 a Fa(320)p 390 4852 V 913 w Ff(i,blockSize,output,output)o
(Offs)o(et)42 b(+)p 3507 4852 V 274 4932 a Fa(321)p 390
4965 V 913 w Ff(outputLength\);)p 3507 4965 V 274 5044
a Fa(322)p 390 5078 V 627 w Ff(})p 3507 5078 V 274 5157
a Fa(323)p 390 5191 V 436 w Ff(outputLength)i(+=)k
(engineTransformBlockFin)o(al\(t)o(ota)o(lInp)o(ut,)p
3507 5191 V 274 5270 a Fa(324)p 390 5304 V 531 w Ff
(lastBlockOffset,lastBlockS)o(ize,)o(out)o(put,)o(outp)o(utO)o(ffse)o
(t)42 b(+)p 3507 5304 V 274 5383 a Fa(325)p 390 5417
V 531 w Ff(outputLength\);)p 3507 5417 V 274 5496 a Fa(326)p
390 5530 V 3507 5530 V 274 5609 a(327)p 390 5643 V 436
w Ff(//)47 b(and)g(back)p 3507 5643 V eop
%%Page: 55 56
55 55 bop 270 82 a Fw(B.5.)62 b(ECElGamalGFPBlo)s(c)m(kCipher)1644
b(55)154 415 y Fa(328)p 270 449 4 113 v 436 w Ff(return)46
b(outputLength;)p 3387 449 V 154 528 a Fa(329)p 270 562
V 245 w Ff(})p 3387 562 V 154 641 a Fa(330)p 270 675
V 3387 675 V 154 754 a(331)p 270 788 V 245 w Ff(/**)p
3387 788 V 154 867 a Fa(332)p 270 901 V 293 w Ff(*)h(finishes)f(the)h
(process.)e(A)j(last)e(piece)h(of)g(input)f(data)h(can)g(be)p
3387 901 V 154 980 a Fa(333)p 270 1014 V 293 w Ff(*)g(added)g(to)g(the)
g(cipher.)p 3387 1014 V 154 1093 a Fa(334)p 270 1126
V 293 w Ff(*)g(<p>)p 3387 1126 V 154 1205 a Fa(335)p
270 1239 V 293 w Ff(*)g(The)g(input)g(block)f(of)h
(<code>inputLength</code>)41 b(bytes)47 b(has)g(to)p
3387 1239 V 154 1318 a Fa(336)p 270 1352 V 293 w Ff(*)g(be)g(supplied)f
(in)h(the)g(bytearray)e(<code>input</code>,)e(starting)p
3387 1352 V 154 1431 a Fa(337)p 270 1465 V 293 w Ff(*)k(at)g
(<code>inputOffset</code>.)41 b(The)47 b(output)f(will)h(be)g(returned)
p 3387 1465 V 154 1544 a Fa(338)p 270 1578 V 293 w Ff(*)g(in)g(the)g
(function)f(value.)p 3387 1578 V 154 1657 a Fa(339)p
270 1691 V 293 w Ff(*)p 3387 1691 V 154 1770 a Fa(340)p
270 1804 V 293 w Ff(*)h(@param)f(input)h(the)g(bytearray)e(contains)g
(the)i(input)p 3387 1804 V 154 1883 a Fa(341)p 270 1917
V 293 w Ff(*)g(@param)f(inputOffset)f(the)i(position)f(where)g(the)h
(input)f(starts)p 3387 1917 V 154 1996 a Fa(342)p 270
2030 V 293 w Ff(*)h(@param)f(inputLength)f(the)i(length)f(of)h(the)g
(input)p 3387 2030 V 154 2109 a Fa(343)p 270 2143 V 293
w Ff(*)p 3387 2143 V 154 2222 a Fa(344)p 270 2256 V 293
w Ff(*)g(@throws)f(IllegalBlockSizeException)41 b(will)47
b(not)g(be)g(thrown)p 3387 2256 V 154 2335 a Fa(345)p
270 2368 V 293 w Ff(*)g(since)g(an)g(ElGamal)f(cipher)g(does)g(not)h
(mind)g(a)g(partial)f(block)p 3387 2368 V 154 2447 a
Fa(346)p 270 2481 V 293 w Ff(*)h(@throws)f(BadPaddingException)d(will)j
(not)h(be)g(thrown)f(since)h(the)p 3387 2481 V 154 2560
a Fa(347)p 270 2594 V 293 w Ff(*)g(cipher)f(does)h(not)g(support)f(a)h
(padding)f(scheme)p 3387 2594 V 154 2673 a Fa(348)p 270
2707 V 293 w Ff(*)p 3387 2707 V 154 2786 a Fa(349)p 270
2820 V 293 w Ff(*)h(@return)f(the)h(output)p 3387 2820
V 154 2899 a Fa(350)p 270 2933 V 293 w Ff(*/)p 3387 2933
V 154 3012 a Fa(351)p 270 3046 V 245 w Ff(protected)e(byte[])h
(engineDoFinal\(byte[])d(input,int)i(inputOffset,)p 3387
3046 V 154 3125 a Fa(352)p 270 3159 V 341 w Ff(int)h(inputLength\))p
3387 3159 V 154 3238 a Fa(353)p 270 3272 V 436 w Ff(throws)g
(IllegalBlockSizeException)o(,Ba)o(dPad)o(ding)o(Exc)o(epti)o(on)p
3387 3272 V 154 3351 a Fa(354)p 270 3385 V 245 w Ff({)p
3387 3385 V 154 3464 a Fa(355)p 270 3498 V 436 w Ff(int)h(length)f(=)h
(0;)p 3387 3498 V 154 3577 a Fa(356)p 270 3610 V 436
w Ff(byte[])f(out)h(=)g(new)g(byte[engineGetOutputSize\(i)o(npu)o(tLen)
o(gth\))o(];)p 3387 3610 V 154 3689 a Fa(357)p 270 3723
V 3387 3723 V 154 3802 a(358)p 270 3836 V 436 w Ff(//)g(calls)g(the)f
(other)h(engineDoFinal)d(method)p 3387 3836 V 154 3915
a Fa(359)p 270 3949 V 436 w Ff(try)p 3387 3949 V 154
4028 a Fa(360)p 270 4062 V 627 w Ff({)p 3387 4062 V 154
4141 a Fa(361)p 270 4175 V 818 w Ff(length)i(=)h
(engineDoFinal\(input,inputO)o(ffs)o(et,)p 3387 4175
V 154 4254 a Fa(362)p 270 4288 V 913 w Ff(inputLength,out,0\);)p
3387 4288 V 154 4367 a Fa(363)p 270 4401 V 627 w Ff(})p
3387 4401 V 154 4480 a Fa(364)p 270 4514 V 436 w Ff(catch)f
(\(ShortBufferException)c(sbe\))47 b({})p 3387 4514 V
154 4593 a Fa(365)p 270 4627 V 3387 4627 V 154 4706 a(366)p
270 4740 V 436 w Ff(//)g(transforms)e(to)i(the)g(right)g(size)p
3387 4740 V 154 4819 a Fa(367)p 270 4852 V 436 w Ff(if)g(\(length)f(<)h
(out.length\))p 3387 4852 V 154 4932 a Fa(368)p 270 4965
V 627 w Ff({)p 3387 4965 V 154 5044 a Fa(369)p 270 5078
V 818 w Ff(byte[])f(shorter)g(=)h(new)g(byte[length];)p
3387 5078 V 154 5157 a Fa(370)p 270 5191 V 818 w Ff
(System.arraycopy\(out,0,s)o(hor)o(ter,)o(0,le)o(ngt)o(h\);)p
3387 5191 V 154 5270 a Fa(371)p 270 5304 V 818 w Ff(out)g(=)g(shorter;)
p 3387 5304 V 154 5383 a Fa(372)p 270 5417 V 627 w Ff(})p
3387 5417 V 154 5496 a Fa(373)p 270 5530 V 3387 5530
V 154 5609 a(374)p 270 5643 V 436 w Ff(//)g(and)g(back)p
3387 5643 V eop
%%Page: 56 57
56 56 bop 390 82 a Fw(56)1464 b(ANHANG)32 b(B.)61 b(CDC.EC.ECELGAMAL)
274 415 y Fa(375)p 390 449 4 113 v 436 w Ff(return)46
b(out;)p 3507 449 V 274 528 a Fa(376)p 390 562 V 245
w Ff(})p 3507 562 V 274 641 a Fa(377)p 390 675 V 54 w
Ff(})p 3507 675 V 390 700 4 25 v 3507 700 V 390 703 3120
4 v eop
%%Page: 57 58
57 57 bop 270 82 a Fw(B.6.)62 b(ECElGamalGFPCipher)1867
b(57)270 415 y Fz(B.6)135 b(ECElGamalGFPCipher)p 270
537 3120 4 v 270 562 4 25 v 3387 562 V 215 641 a Fa(1)p
270 675 4 113 v 53 w Ff(package)46 b(cdc.ec.ecelgamal;)p
3387 675 V 215 754 a Fa(2)p 270 788 V 3387 788 V 215
867 a(3)p 270 901 V 53 w Ff(import)g(java.math.BigInteger;)p
3387 901 V 215 980 a Fa(4)p 270 1014 V 53 w Ff(import)g
(java.security.*;)p 3387 1014 V 215 1093 a Fa(5)p 270
1126 V 53 w Ff(import)g(java.security.spec.*;)p 3387
1126 V 215 1205 a Fa(6)p 270 1239 V 53 w Ff(import)g(java.lang.Math;)p
3387 1239 V 215 1318 a Fa(7)p 270 1352 V 53 w Ff(import)g
(javax.crypto.*;)p 3387 1352 V 215 1431 a Fa(8)p 270
1465 V 53 w Ff(import)g(cdc.lidia.*;)p 3387 1465 V 215
1544 a Fa(9)p 270 1578 V 3387 1578 V 184 1657 a(10)p
270 1691 V 54 w Ff(/**)p 3387 1691 V 184 1770 a Fa(11)p
270 1804 V 102 w Ff(*)h(<code>ECElGamalGFPCipher</)o(cod)o(e>)42
b(provides)j(the)i(implementation)p 3387 1804 V 184 1883
a Fa(12)p 270 1917 V 102 w Ff(*)g(for)g(encryption)e(and)i(decryption)e
(with)i(an)g(ElGamal)f(algorithm)f(based)p 3387 1917
V 184 1996 a Fa(13)p 270 2030 V 102 w Ff(*)i(on)h(elliptic)d(curves)h
(over)h(the)g(general)f(field)g(with)h(<code>p</code>)p
3387 2030 V 184 2109 a Fa(14)p 270 2143 V 102 w Ff(*)g(elements)f
(where)g(<code>p</code>)e(is)j(a)h(prime)e(number.)p
3387 2143 V 184 2222 a Fa(15)p 270 2256 V 102 w Ff(*)h(<p>)p
3387 2256 V 184 2335 a Fa(16)p 270 2368 V 102 w Ff(*)g(The)g(security)f
(of)h(an)g(ElGamal)f(algorithm)f(depends)h(on)h(the)p
3387 2368 V 184 2447 a Fa(17)p 270 2481 V 102 w Ff(*)g(presumably)e
(difficult)h(problem)f(to)j(compute)d(discrete)h(logarithms)p
3387 2481 V 184 2560 a Fa(18)p 270 2594 V 102 w Ff(*)h(in)h(finite)e
(abelian)f(groups.)h(This)h(problem)f(is)h(\(might)f(be\))h(even)p
3387 2594 V 184 2673 a Fa(19)p 270 2707 V 102 w Ff(*)g(more)g
(difficult,)e(if)i(a)h(subgroup)d(of)i(the)g(points)f(on)i(an)f
(elliptic)p 3387 2707 V 184 2786 a Fa(20)p 270 2820 V
102 w Ff(*)g(curve)g(is)g(taken.)p 3387 2820 V 184 2899
a Fa(21)p 270 2933 V 102 w Ff(*)g(<p>)p 3387 2933 V 184
3012 a Fa(22)p 270 3046 V 102 w Ff(*)g(This)g(implementation)d(does)j
(not)f(embed)h(the)g(message)f(on)h(the)g(curve,)p 3387
3046 V 184 3125 a Fa(23)p 270 3159 V 102 w Ff(*)g(but)g(uses)g(the)g
(curve)f(to)h(encrypt)f(the)h(message.)f(It)h(works)f(as)p
3387 3159 V 184 3238 a Fa(24)p 270 3272 V 102 w Ff(*)h(follows:)p
3387 3272 V 184 3351 a Fa(25)p 270 3385 V 102 w Ff(*)g(<p>)p
3387 3385 V 184 3464 a Fa(26)p 270 3498 V 102 w Ff(*)g(Encryption:)e(A)
j(random)e(integer)f(<code>k</code>)f(is)k(generated)d(and)p
3387 3498 V 184 3577 a Fa(27)p 270 3610 V 102 w Ff(*)i(multiplied)e
(with)i(the)g(generator)e(<code>gamma</code>)e(resulting)i(in)p
3387 3610 V 184 3690 a Fa(28)p 270 3723 V 102 w Ff(*)i
(<code>beta0</code>.)c(The)k(public)f(key)h(<code>delta</code>)c(is)k
(also)p 3387 3723 V 184 3802 a Fa(29)p 270 3836 V 102
w Ff(*)g(multiplied)e(by)j(<code>k</code>)43 b(resulting)j(in)h
(<code>beta1</code>.)p 3387 3836 V 184 3915 a Fa(30)p
270 3949 V 102 w Ff(*)g(The)g(message)f(<code>m</code>)e(is)j(split)g
(into)f(two)h(halfs)p 3387 3949 V 184 4028 a Fa(31)p
270 4062 V 102 w Ff(*)g(<code>m1</code>)d(and)j(<code>m2</code>.)c(The)
k(first)g(half)f(is)p 3387 4062 V 184 4141 a Fa(32)p
270 4175 V 102 w Ff(*)h(multiplied)e(with)i(the)g(x-coordinate)d(of)k
(<code>beta0</code>)42 b(and)p 3387 4175 V 184 4254 a
Fa(33)p 270 4288 V 102 w Ff(*)47 b(the)g(second)f(half)h(with)g(the)g
(y-coordinate)d(resulting)h(in)p 3387 4288 V 184 4367
a Fa(34)p 270 4401 V 102 w Ff(*)i(<code>c1</code>)d(and)j
(<code>c2</code>.)c(The)k(ciphertext)p 3387 4401 V 184
4480 a Fa(35)p 270 4514 V 102 w Ff(*)g(eventually)e(consists)h(of)h
(<code>beta0</code>,)c(<code>c1</code>)p 3387 4514 V
184 4593 a Fa(36)p 270 4627 V 102 w Ff(*)k(and)g(<code>c2</code>.)p
3387 4627 V 184 4706 a Fa(37)p 270 4740 V 102 w Ff(*)g(<p>)p
3387 4740 V 184 4819 a Fa(38)p 270 4852 V 102 w Ff(*)g(Decryption:)e
(<code>beta0</code>)e(is)k(multiplied)e(with)i(the)g(private)p
3387 4852 V 184 4932 a Fa(39)p 270 4965 V 102 w Ff(*)g(key)g
(<code>s</code>)d(resulting)h(in)j(<code>beta1</code>.)42
b(The)p 3387 4965 V 184 5044 a Fa(40)p 270 5078 V 102
w Ff(*)47 b(inverses)f(of)h(the)g(x-)g(and)g(y-coordinates)d(of)j
(<code>beta1</code>)p 3387 5078 V 184 5157 a Fa(41)p
270 5191 V 102 w Ff(*)g(can)g(be)g(calculated)e(and)i(multiplied)e
(with)i(<code>c1</code>)d(and)p 3387 5191 V 184 5270
a Fa(42)p 270 5304 V 102 w Ff(*)j(<code>c2</code>)d(resulting)h(in)i
(<code>m1</code>)d(and)p 3387 5304 V 184 5383 a Fa(43)p
270 5417 V 102 w Ff(*)j(<code>m2</code>.)d(The)j(concatenation)d(of)j
(those)f(is)i(the)e(plaintext.)p 3387 5417 V 184 5496
a Fa(44)p 270 5530 V 102 w Ff(*)h(<p>)p 3387 5530 V 184
5609 a Fa(45)p 270 5643 V 102 w Ff(*)g(The)g(C++-library)e(for)i
(calculation)e(is)i(<code>LiDIA</code>.)p 3387 5643 V
eop
%%Page: 58 59
58 58 bop 390 82 a Fw(58)1464 b(ANHANG)32 b(B.)61 b(CDC.EC.ECELGAMAL)
304 415 y Fa(46)p 390 449 4 113 v 102 w Ff(*)47 b(<p>)p
3507 449 V 304 528 a Fa(47)p 390 562 V 102 w Ff(*)g(In)h(order)e(to)h
(implement)e(other)i(protocols)e(for)i(ElGamalGFP)p 3507
562 V 304 641 a Fa(48)p 390 675 V 102 w Ff(*)g(encryption)e(and)i
(decryption)e(just)i(the)g(two)g(methods)p 3507 675 V
304 754 a Fa(49)p 390 788 V 102 w Ff(*)g(<code>encryptBlock</code>)41
b(and)47 b(<code>decryptBlock</code>)41 b(have)p 3507
788 V 304 867 a Fa(50)p 390 901 V 102 w Ff(*)47 b(to)h(be)f
(redesigned.)p 3507 901 V 304 980 a Fa(51)p 390 1014
V 102 w Ff(*)p 3507 1014 V 304 1093 a Fa(52)p 390 1126
V 102 w Ff(*)g(@author)f(Markus)g(Crnjak)p 3507 1126
V 304 1205 a Fa(53)p 390 1239 V 102 w Ff(*)p 3507 1239
V 304 1318 a Fa(54)p 390 1352 V 102 w Ff(*)h(@see)g(ECElGamalGFPKey)p
3507 1352 V 304 1431 a Fa(55)p 390 1465 V 102 w Ff(*)g(@see)g
(ECElGamalGFPPrivateKey)p 3507 1465 V 304 1544 a Fa(56)p
390 1578 V 102 w Ff(*)g(@see)g(ECElGamalGFPPublicKey)p
3507 1578 V 304 1657 a Fa(57)p 390 1691 V 102 w Ff(*)g(@see)g
(ECElGamalGFPKeyPairGener)o(ator)p 3507 1691 V 304 1770
a Fa(58)p 390 1804 V 102 w Ff(*)g(@see)g(ECElGamalGFPBlockCipher)p
3507 1804 V 304 1883 a Fa(59)p 390 1917 V 102 w Ff(*)g(@see)g
(cdc.ec.Provider)p 3507 1917 V 304 1996 a Fa(60)p 390
2030 V 102 w Ff(*)g(@see)g(cdc.lidia.LECDomainParam)o(eter)o(sP13)o(63)
p 3507 2030 V 304 2109 a Fa(61)p 390 2143 V 102 w Ff(*/)p
3507 2143 V 304 2222 a Fa(62)p 390 2256 V 54 w Ff(public)f(class)h
(ECElGamalGFPCipher)p 3507 2256 V 304 2335 a Fa(63)p
390 2368 V 245 w Ff(extends)f(ECElGamalGFPBlockCipher)p
3507 2368 V 304 2447 a Fa(64)p 390 2481 V 54 w Ff({)p
3507 2481 V 304 2560 a Fa(65)p 390 2594 V 245 w Ff(/**)p
3507 2594 V 304 2673 a Fa(66)p 390 2707 V 293 w Ff(*)h(holds)g(the)g
(domain)f(parameters)p 3507 2707 V 304 2786 a Fa(67)p
390 2820 V 293 w Ff(*/)p 3507 2820 V 304 2899 a Fa(68)p
390 2933 V 245 w Ff(protected)f(LECDomainParametersP1363)d(mParams;)p
3507 2933 V 304 3012 a Fa(69)p 390 3046 V 3507 3046 V
304 3125 a(70)p 390 3159 V 245 w Ff(/**)p 3507 3159 V
304 3238 a Fa(71)p 390 3272 V 293 w Ff(*)47 b(holds)g(the)g(state)f(of)
h(the)g(cipher)f(which)g(is)i(either)p 3507 3272 V 304
3351 a Fa(72)p 390 3385 V 293 w Ff(*)f(<code>Cipher.ENCRYPT_MODE)o
(</co)o(de>)41 b(or)p 3507 3385 V 304 3464 a Fa(73)p
390 3498 V 293 w Ff(*)47 b(<code>Cipher.DECRYPT_MODE)o(</co)o(de>.)p
3507 3498 V 304 3577 a Fa(74)p 390 3610 V 293 w Ff(*/)p
3507 3610 V 304 3689 a Fa(75)p 390 3723 V 245 w Ff(protected)e(int)i
(mState;)p 3507 3723 V 304 3802 a Fa(76)p 390 3836 V
3507 3836 V 304 3915 a(77)p 390 3949 V 245 w Ff(/**)p
3507 3949 V 304 4028 a Fa(78)p 390 4062 V 293 w Ff(*)g(holds)g(the)g
(key)f(of)i(the)f(cipher)f(which)g(is)h(either)f(an)p
3507 4062 V 304 4141 a Fa(79)p 390 4175 V 293 w Ff(*)h
(<code>ECElGamalGFPPrivate)o(Key<)o(/cod)o(e>)41 b(or)48
b(an)p 3507 4175 V 304 4254 a Fa(80)p 390 4288 V 293
w Ff(*)f(<code>ECElGamalGFPPublicK)o(ey</)o(code)o(>.)p
3507 4288 V 304 4367 a Fa(81)p 390 4401 V 293 w Ff(*/)p
3507 4401 V 304 4480 a Fa(82)p 390 4514 V 245 w Ff(protected)e(Key)i
(mKey;)p 3507 4514 V 304 4593 a Fa(83)p 390 4627 V 3507
4627 V 304 4706 a(84)p 390 4740 V 245 w Ff(/**)p 3507
4740 V 304 4819 a Fa(85)p 390 4852 V 293 w Ff(*)g(holds)g(some)f
(random)g(information)p 3507 4852 V 304 4932 a Fa(86)p
390 4965 V 293 w Ff(*/)p 3507 4965 V 304 5044 a Fa(87)p
390 5078 V 245 w Ff(protected)f(SecureRandom)g(mSecureRandom;)p
3507 5078 V 304 5157 a Fa(88)p 390 5191 V 3507 5191 V
304 5270 a(89)p 390 5304 V 245 w Ff(/**)p 3507 5304 V
304 5383 a Fa(90)p 390 5417 V 293 w Ff(*)i(holds)g(the)g(block)f(size)h
(for)f(the)h(plaintext)p 3507 5417 V 304 5496 a Fa(91)p
390 5530 V 293 w Ff(*/)p 3507 5530 V 304 5609 a Fa(92)p
390 5643 V 245 w Ff(protected)e(int)i(mPlainBlockSize;)p
3507 5643 V eop
%%Page: 59 60
59 59 bop 270 82 a Fw(B.6.)62 b(ECElGamalGFPCipher)1867
b(59)184 415 y Fa(93)p 270 449 4 113 v 3387 449 V 184
528 a(94)p 270 562 V 245 w Ff(/**)p 3387 562 V 184 641
a Fa(95)p 270 675 V 293 w Ff(*)47 b(holds)g(the)g(block)f(size)h(for)f
(the)h(ciphertext)p 3387 675 V 184 754 a Fa(96)p 270
788 V 293 w Ff(*/)p 3387 788 V 184 867 a Fa(97)p 270
901 V 245 w Ff(protected)e(int)i(mCipherBlockSize;)p
3387 901 V 184 980 a Fa(98)p 270 1014 V 3387 1014 V 184
1093 a(99)p 270 1126 V 245 w Ff(/**)p 3387 1126 V 154
1205 a Fa(100)p 270 1239 V 293 w Ff(*)g(returns)f(the)h(block)f(size)h
(depending)e(on)i(the)g(current)f(state)h(of)p 3387 1239
V 154 1318 a Fa(101)p 270 1352 V 293 w Ff(*)g(the)g(cipher)p
3387 1352 V 154 1431 a Fa(102)p 270 1465 V 293 w Ff(*)p
3387 1465 V 154 1544 a Fa(103)p 270 1578 V 293 w Ff(*)g(@see)g(#mState)
p 3387 1578 V 154 1657 a Fa(104)p 270 1691 V 293 w Ff(*/)p
3387 1691 V 154 1770 a Fa(105)p 270 1804 V 245 w Ff(protected)e(int)i
(engineGetBlockSize\(\))p 3387 1804 V 154 1883 a Fa(106)p
270 1917 V 245 w Ff({)p 3387 1917 V 154 1996 a Fa(107)p
270 2030 V 436 w Ff(//)g(checks)f(the)h(state)p 3387
2030 V 154 2109 a Fa(108)p 270 2143 V 436 w Ff(if)g(\(mState)f(==)h
(Cipher.ENCRYPT_MODE\))p 3387 2143 V 154 2222 a Fa(109)p
270 2256 V 627 w Ff({)p 3387 2256 V 154 2335 a Fa(110)p
270 2368 V 818 w Ff(return)f(mPlainBlockSize;)p 3387
2368 V 154 2447 a Fa(111)p 270 2481 V 627 w Ff(})p 3387
2481 V 154 2560 a Fa(112)p 270 2594 V 436 w Ff(else)p
3387 2594 V 154 2673 a Fa(113)p 270 2707 V 627 w Ff({)p
3387 2707 V 154 2786 a Fa(114)p 270 2820 V 818 w Ff(return)g
(mCipherBlockSize;)p 3387 2820 V 154 2899 a Fa(115)p
270 2933 V 627 w Ff(})p 3387 2933 V 154 3012 a Fa(116)p
270 3046 V 245 w Ff(})p 3387 3046 V 154 3125 a Fa(117)p
270 3159 V 3387 3159 V 154 3238 a(118)p 270 3272 V 245
w Ff(/**)p 3387 3272 V 154 3351 a Fa(119)p 270 3385 V
293 w Ff(*)h(returns)f(the)h(maximum)f(output)g(size)h(for)f(the)h
(next)g(call)g(to)g(an)p 3387 3385 V 154 3464 a Fa(120)p
270 3498 V 293 w Ff(*)g(<code>update</code>)c(or)k
(<code>doFinal</code>)42 b(method)k(with)p 3387 3498
V 154 3577 a Fa(121)p 270 3610 V 293 w Ff(*)h(an)g(input)g(of)g(length)
f(<code>inputLength</code>)p 3387 3610 V 154 3689 a Fa(122)p
270 3723 V 293 w Ff(*)p 3387 3723 V 154 3802 a Fa(123)p
270 3836 V 293 w Ff(*)h(@param)f(inputLength)f(possible)h(length)g(for)
h(the)g(next)f(piece)h(of)p 3387 3836 V 154 3915 a Fa(124)p
270 3949 V 293 w Ff(*)g(input)p 3387 3949 V 154 4028
a Fa(125)p 270 4062 V 293 w Ff(*/)p 3387 4062 V 154 4141
a Fa(126)p 270 4175 V 245 w Ff(protected)e(int)i
(engineGetOutputSize\(int)42 b(inputLength\))p 3387 4175
V 154 4254 a Fa(127)p 270 4288 V 245 w Ff({)p 3387 4288
V 154 4367 a Fa(128)p 270 4401 V 436 w Ff(int)47 b(inBlocks;)p
3387 4401 V 154 4480 a Fa(129)p 270 4514 V 436 w Ff(int)g(outLength;)p
3387 4514 V 154 4593 a Fa(130)p 270 4627 V 3387 4627
V 154 4706 a(131)p 270 4740 V 436 w Ff(//)g(checks)f(the)h(state)p
3387 4740 V 154 4819 a Fa(132)p 270 4852 V 436 w Ff(if)g(\(mState)f(==)
h(Cipher.ENCRYPT_MODE\))p 3387 4852 V 154 4932 a Fa(133)p
270 4965 V 627 w Ff(//)g(calculates)e(here)p 3387 4965
V 154 5044 a Fa(134)p 270 5078 V 627 w Ff({)p 3387 5078
V 154 5157 a Fa(135)p 270 5191 V 818 w Ff(inBlocks)g(=)j(\(inputLength)
c(+)k(getBufferedDataLength\(\))p 3392 5191 V 154 5270
a Fa(136)p 270 5304 V 913 w Ff(+)g(mPlainBlockSize)43
b(-)48 b(1\))f(/)g(mPlainBlockSize;)p 3387 5304 V 154
5383 a Fa(137)p 270 5417 V 818 w Ff(outLength)e(=)j(inBlocks)d(*)j
(mCipherBlockSize;)p 3387 5417 V 154 5496 a Fa(138)p
270 5530 V 627 w Ff(})p 3387 5530 V 154 5609 a Fa(139)p
270 5643 V 436 w Ff(else)p 3387 5643 V eop
%%Page: 60 61
60 60 bop 390 82 a Fw(60)1464 b(ANHANG)32 b(B.)61 b(CDC.EC.ECELGAMAL)
274 415 y Fa(140)p 390 449 4 113 v 627 w Ff(//)47 b(or)g(here)p
3507 449 V 274 528 a Fa(141)p 390 562 V 627 w Ff({)p
3507 562 V 274 641 a Fa(142)p 390 675 V 818 w Ff(inBlocks)e(=)j
(\(inputLength)c(+)k(getBufferedDataLength\(\))p 3512
675 V 274 754 a Fa(143)p 390 788 V 913 w Ff(+)g(mCipherBlockSize)43
b(-)48 b(1\))f(/)g(mCipherBlockSize;)p 3507 788 V 274
867 a Fa(144)p 390 901 V 818 w Ff(outLength)e(=)j(inBlocks)d(*)j
(mPlainBlockSize;)p 3507 901 V 274 980 a Fa(145)p 390
1014 V 627 w Ff(})p 3507 1014 V 274 1093 a Fa(146)p 390
1126 V 3507 1126 V 274 1205 a(147)p 390 1239 V 436 w
Ff(//)f(and)g(back)p 3507 1239 V 274 1318 a Fa(148)p
390 1352 V 436 w Ff(return)f(outLength;)p 3507 1352 V
274 1431 a Fa(149)p 390 1465 V 245 w Ff(})p 3507 1465
V 274 1544 a Fa(150)p 390 1578 V 3507 1578 V 274 1657
a(151)p 390 1691 V 245 w Ff(/**)p 3507 1691 V 274 1770
a Fa(152)p 390 1804 V 293 w Ff(*)h(initializes)e(this)i
(<code>ECElGamalGFPCiphe)o(r</c)o(ode)o(>)42 b(with)47
b(a)p 3507 1804 V 274 1883 a Fa(153)p 390 1917 V 293
w Ff(*)g(key)g(and)95 b(some)46 b(random)h(information)d(for)j
(encryption)e(or)p 3507 1917 V 274 1996 a Fa(154)p 390
2030 V 293 w Ff(*)i(decryption)p 3507 2030 V 274 2109
a Fa(155)p 390 2143 V 293 w Ff(*)p 3507 2143 V 274 2222
a Fa(156)p 390 2256 V 293 w Ff(*)g(@param)f(opmode)g(specifies)g(the)h
(mode)f(of)h(the)g(cipher)f(which)h(is)p 3507 2256 V
274 2335 a Fa(157)p 390 2368 V 293 w Ff(*)g(either)f
(<code>Cipher.ENCRYPT_MODE<)o(/co)o(de>)41 b(or)p 3507
2368 V 274 2447 a Fa(158)p 390 2481 V 293 w Ff(*)47 b
(<code>Cipher.DECRYPT_MODE)o(</co)o(de>)p 3507 2481 V
274 2560 a Fa(159)p 390 2594 V 293 w Ff(*)g(@param)f(key)h(the)g(key)g
(is)g(either)f(a)i(private)e(or)h(a)g(public)f(key)p
3507 2594 V 274 2673 a Fa(160)p 390 2707 V 293 w Ff(*)h(depending)f(on)
h(<code>opmode</code>)p 3507 2707 V 274 2786 a Fa(161)p
390 2820 V 293 w Ff(*)g(@param)f(random)g(the)h(secureRandom)e(holds)h
(the)h(random)p 3507 2820 V 274 2899 a Fa(162)p 390 2933
V 293 w Ff(*)g(information)p 3507 2933 V 274 3012 a Fa(163)p
390 3046 V 293 w Ff(*)p 3507 3046 V 274 3125 a Fa(164)p
390 3159 V 293 w Ff(*)g(@throws)f(InvalidKeyException)d(if)k
(<code>key</code>)c(is)p 3507 3159 V 274 3238 a Fa(165)p
390 3272 V 293 w Ff(*)k(neither)f(an)h(instance)f(of)h
(<code>ECElGamalGFPPublic)o(Key<)o(/cod)o(e>)p 3507 3272
V 274 3351 a Fa(166)p 390 3385 V 293 w Ff(*)g(nor)g(of)g
(<code>ECElGamalGFPPrivateK)o(ey<)o(/cod)o(e>)p 3507
3385 V 274 3464 a Fa(167)p 390 3498 V 293 w Ff(*)p 3507
3498 V 274 3577 a Fa(168)p 390 3610 V 293 w Ff(*)g(@see)g
(cdc.lidia.LECDomainParam)o(eter)o(sP1)o(363)p 3507 3610
V 274 3689 a Fa(169)p 390 3723 V 293 w Ff(*/)p 3507 3723
V 274 3802 a Fa(170)p 390 3836 V 245 w Ff(protected)e(void)i
(engineInit\(int)d(opmode,Key)h(key,)p 3507 3836 V 274
3915 a Fa(171)p 390 3949 V 341 w Ff(SecureRandom)f(random\))p
3507 3949 V 274 4028 a Fa(172)p 390 4062 V 436 w Ff(throws)i
(InvalidKeyException)p 3507 4062 V 274 4141 a Fa(173)p
390 4175 V 245 w Ff({)p 3507 4175 V 274 4254 a Fa(174)p
390 4288 V 436 w Ff(//)h(checks)f(the)h(key)p 3507 4288
V 274 4367 a Fa(175)p 390 4401 V 436 w Ff(if)g(\(opmode)f(==)h
(Cipher.ENCRYPT_MODE\))p 3507 4401 V 274 4480 a Fa(176)p
390 4514 V 627 w Ff(if)g(\(!\(key)f(instanceof)f
(ECElGamalGFPPublicKey\)\))p 3507 4514 V 274 4593 a Fa(177)p
390 4627 V 818 w Ff(throw)h(new)h(InvalidKeyException\("I)42
b(didn't)k(get)h(an)g(")p 3507 4627 V 274 4706 a Fa(178)p
390 4740 V 913 w Ff(+)h("ECElGamalPublicKey"\);)p 3507
4740 V 274 4819 a Fa(179)p 390 4852 V 436 w Ff(else)f(if)g(\(opmode)f
(==)h(Cipher.DECRYPT_MODE\))p 3507 4852 V 274 4932 a
Fa(180)p 390 4965 V 627 w Ff(if)g(\(!\(key)f(instanceof)f
(ECElGamalGFPPrivateKey\)\))p 3507 4965 V 274 5044 a
Fa(181)p 390 5078 V 818 w Ff(throw)h(new)h(InvalidKeyException\("I)42
b(didn't)k(get)h(an)g(")p 3507 5078 V 274 5157 a Fa(182)p
390 5191 V 913 w Ff(+)h("ECElGamalPrivateKey"\);)p 3507
5191 V 274 5270 a Fa(183)p 390 5304 V 436 w Ff(else)f(throw)f(new)h
(IllegalArgumentException)o(\("Ba)o(d)42 b(mode:)k(")i(+)p
3507 5304 V 274 5383 a Fa(184)p 390 5417 V 531 w Ff(opmode\);)p
3507 5417 V 274 5496 a Fa(185)p 390 5530 V 3507 5530
V 274 5609 a(186)p 390 5643 V 436 w Ff(//)f(sets)g(the)g(fields)p
3507 5643 V eop
%%Page: 61 62
61 61 bop 270 82 a Fw(B.6.)62 b(ECElGamalGFPCipher)1867
b(61)154 415 y Fa(187)p 270 449 4 113 v 436 w Ff(mState)46
b(=)i(opmode;)p 3387 449 V 154 528 a Fa(188)p 270 562
V 436 w Ff(mKey)f(=)g(key;)p 3387 562 V 154 641 a Fa(189)p
270 675 V 436 w Ff(mParams)f(=)h(\(\(ECElGamalGFPKey\)key\).ge)o(tPar)o
(ams\()o(\);)p 3387 675 V 154 754 a Fa(190)p 270 788
V 436 w Ff(mSecureRandom)d(=)k(random;)p 3387 788 V 154
867 a Fa(191)p 270 901 V 436 w Ff(calculateBlockSizes\(mKey)o(\);)p
3387 901 V 154 980 a Fa(192)p 270 1014 V 245 w Ff(})p
3387 1014 V 154 1093 a Fa(193)p 270 1126 V 3387 1126
V 154 1205 a(194)p 270 1239 V 245 w Ff(/**)p 3387 1239
V 154 1318 a Fa(195)p 270 1352 V 293 w Ff(*)f(initializes)e(this)i
(<code>ECElGamalGFPCiphe)o(r</c)o(ode)o(>)42 b(with)47
b(a)p 3387 1352 V 154 1431 a Fa(196)p 270 1465 V 293
w Ff(*)g(key)g(and)g(the)g(domain)f(parameters)p 3387
1465 V 154 1544 a Fa(197)p 270 1578 V 293 w Ff(*)p 3387
1578 V 154 1657 a Fa(198)p 270 1691 V 293 w Ff(*)h(@param)f(opmode)g
(specifies)g(the)h(mode)f(of)h(the)g(cipher)f(which)h(is)p
3387 1691 V 154 1770 a Fa(199)p 270 1804 V 293 w Ff(*)g(either)f
(<code>Cipher.ENCRYPT_MODE<)o(/co)o(de>)41 b(or)p 3387
1804 V 154 1883 a Fa(200)p 270 1917 V 293 w Ff(*)47 b
(<code>Cipher.DECRYPT_MODE)o(</co)o(de>)p 3387 1917 V
154 1996 a Fa(201)p 270 2030 V 293 w Ff(*)g(@param)f(key)h(the)g(key)g
(is)g(either)f(a)i(private)e(or)h(a)g(public)f(key)p
3387 2030 V 154 2109 a Fa(202)p 270 2143 V 293 w Ff(*)h(depending)f(on)
h(<code>opmode</code>)p 3387 2143 V 154 2222 a Fa(203)p
270 2256 V 293 w Ff(*)g(@param)f(params)g(the)h(algorithmParameterSpec)
42 b(holds)k(the)h(domain)p 3387 2256 V 154 2335 a Fa(204)p
270 2368 V 293 w Ff(*)g(parameters)p 3387 2368 V 154
2447 a Fa(205)p 270 2481 V 293 w Ff(*)p 3387 2481 V 154
2560 a Fa(206)p 270 2594 V 293 w Ff(*)g(@throws)f(InvalidKeyException)d
(if)k(<code>params</code>)42 b(is)48 b(not)p 3387 2594
V 154 2673 a Fa(207)p 270 2707 V 293 w Ff(*)f(the)g(same)g(object)f(as)
h(the)g(params)f(contained)f(in)p 3387 2707 V 154 2786
a Fa(208)p 270 2820 V 293 w Ff(*)i(<code>key</code>)p
3387 2820 V 154 2899 a Fa(209)p 270 2933 V 293 w Ff(*)p
3387 2933 V 154 3012 a Fa(210)p 270 3046 V 293 w Ff(*)g(@see)g
(cdc.lidia.LECDomainParam)o(eter)o(sP1)o(363)p 3387 3046
V 154 3125 a Fa(211)p 270 3159 V 293 w Ff(*/)p 3387 3159
V 154 3238 a Fa(212)p 270 3272 V 245 w Ff(protected)e(void)i
(engineInit\(int)d(opmode,Key)h(key,)p 3387 3272 V 154
3351 a Fa(213)p 270 3385 V 1486 w Ff(AlgorithmParameterSpec)d(params,)p
3387 3385 V 154 3464 a Fa(214)p 270 3498 V 1486 w Ff(SecureRandom)i
(random\))p 3387 3498 V 154 3577 a Fa(215)p 270 3610
V 436 w Ff(throws)i(InvalidKeyException)p 3387 3610 V
154 3689 a Fa(216)p 270 3723 V 245 w Ff({)p 3387 3723
V 154 3802 a Fa(217)p 270 3836 V 436 w Ff(//)h(checks)f(the)h(domain)f
(parameters)f(of)i(the)g(key)g(and)g(the)p 3387 3836
V 154 3915 a Fa(218)p 270 3949 V 436 w Ff(//)g(argument)p
3387 3949 V 154 4028 a Fa(219)p 270 4062 V 436 w Ff(if)g
(\(!\(\(ECElGamalGFPKey\)key\).)o(getP)o(ara)o(ms\(\))o(.equ)o(als)o
(\(par)o(ams\))o(\))p 3387 4062 V 154 4141 a Fa(220)p
270 4175 V 627 w Ff({)p 3387 4175 V 154 4254 a Fa(221)p
270 4288 V 818 w Ff(throw)f(new)h(InvalidKeyException\("The)41
b(domain)46 b(")p 3387 4288 V 154 4367 a Fa(222)p 270
4401 V 913 w Ff(+)i("parameters)c(of)k(the)f(key)f(and)h(the)g
(argument")p 3387 4401 V 154 4480 a Fa(223)p 270 4514
V 913 w Ff(+)h("params)e(are)g(not)h(identical!"\);)p
3387 4514 V 154 4593 a Fa(224)p 270 4627 V 627 w Ff(})p
3387 4627 V 154 4706 a Fa(225)p 270 4740 V 3387 4740
V 154 4819 a(226)p 270 4852 V 436 w Ff(//)g(sets)g(the)g(fields)p
3387 4852 V 154 4932 a Fa(227)p 270 4965 V 436 w Ff(mState)f(=)i
(opmode;)p 3387 4965 V 154 5044 a Fa(228)p 270 5078 V
436 w Ff(mKey)f(=)g(key;)p 3387 5078 V 154 5157 a Fa(229)p
270 5191 V 436 w Ff(mParams)f(=)h(\(LECDomainParametersP1363)o(\)par)o
(ams;)p 3387 5191 V 154 5270 a Fa(230)p 270 5304 V 436
w Ff(mSecureRandom)d(=)k(new)f(SecureRandom\(\);;)p 3387
5304 V 154 5383 a Fa(231)p 270 5417 V 436 w Ff
(calculateBlockSizes\(mKey)o(\);)p 3387 5417 V 154 5496
a Fa(232)p 270 5530 V 245 w Ff(})p 3387 5530 V 154 5609
a Fa(233)p 270 5643 V 3387 5643 V eop
%%Page: 62 63
62 62 bop 390 82 a Fw(62)1464 b(ANHANG)32 b(B.)61 b(CDC.EC.ECELGAMAL)
274 415 y Fa(234)p 390 449 4 113 v 245 w Ff(/**)p 3507
449 V 274 528 a Fa(235)p 390 562 V 293 w Ff(*)47 b(calculates)e(the)i
(block)g(sizes)f(of)h(this)p 3507 562 V 274 641 a Fa(236)p
390 675 V 293 w Ff(*)g(<code>ECElGamalGFPCipher<)o(/cod)o(e>)42
b(depending)j(on)i(the)g(size)g(of)p 3507 675 V 274 754
a Fa(237)p 390 788 V 293 w Ff(*)g(the)g(key)p 3507 788
V 274 867 a Fa(238)p 390 901 V 293 w Ff(*)p 3507 901
V 274 980 a Fa(239)p 390 1014 V 293 w Ff(*)g(@param)f(key)h(the)g(key)g
(is)g(an)g(<code>ECElGamalGFPKey</cod)o(e>)p 3507 1014
V 274 1093 a Fa(240)p 390 1126 V 293 w Ff(*)p 3507 1126
V 274 1205 a Fa(241)p 390 1239 V 293 w Ff(*)g(@see)g(#mPlainBlockSize)p
3507 1239 V 274 1318 a Fa(242)p 390 1352 V 293 w Ff(*)g(@see)g
(#mCipherBlockSize)p 3507 1352 V 274 1431 a Fa(243)p
390 1465 V 293 w Ff(*/)p 3507 1465 V 274 1544 a Fa(244)p
390 1578 V 245 w Ff(protected)e(void)i(calculateBlockSizes\(Key)41
b(key\))p 3507 1578 V 274 1657 a Fa(245)p 390 1691 V
245 w Ff({)p 3507 1691 V 274 1770 a Fa(246)p 390 1804
V 436 w Ff(//)47 b(extracts)f(the)h(modulus)e(and)i(calculates)p
3507 1804 V 274 1883 a Fa(247)p 390 1917 V 436 w Ff(BigInteger)e
(biModulusLength)f(=)j(\(\(ECElGamalGFPKey\)key\).)p
3507 1917 V 274 1996 a Fa(248)p 390 2030 V 531 w Ff
(getParams\(\).getQ\(\);)p 3507 2030 V 274 2109 a Fa(249)p
390 2143 V 436 w Ff(int)g(modulusLength)d(=)k(biModulusLength.bitLeng)o
(th\()o(\);)p 3507 2143 V 274 2222 a Fa(250)p 390 2256
V 436 w Ff(mPlainBlockSize)c(=)j(\(modulusLength)d(-)j(1\))h(/)f(8)h(*)
f(2;)p 3507 2256 V 274 2335 a Fa(251)p 390 2368 V 436
w Ff(mCipherBlockSize)c(=)48 b(\(\(modulusLength)43 b(+)48
b(7\))f(/)h(8\))f(*)g(4;)p 3507 2368 V 274 2447 a Fa(252)p
390 2481 V 245 w Ff(})p 3507 2481 V 274 2560 a Fa(253)p
390 2594 V 3507 2594 V 274 2673 a(254)p 390 2707 V 245
w Ff(/**)p 3507 2707 V 274 2786 a Fa(255)p 390 2820 V
293 w Ff(*)g(returns)f(the)h(number)f(converted)f(into)i(a)g
(bytearray.)e(This)i(is)p 3507 2820 V 274 2899 a Fa(256)p
390 2933 V 293 w Ff(*)g(just)g(a)g(helper)f(method)h(because)e(of)j
(the)e(sign)h(of)g(the)g(number.)p 3507 2933 V 274 3012
a Fa(257)p 390 3046 V 293 w Ff(*)p 3507 3046 V 274 3125
a Fa(258)p 390 3159 V 293 w Ff(*)g(@param)f(big)h(the)g(bigInteger)e
(holds)i(the)f(number)h(which)f(has)h(to)p 3507 3159
V 274 3238 a Fa(259)p 390 3272 V 293 w Ff(*)g(be)g(transformed)p
3507 3272 V 274 3351 a Fa(260)p 390 3385 V 293 w Ff(*/)p
3507 3385 V 274 3464 a Fa(261)p 390 3498 V 245 w Ff(protected)e(byte[])
h(getBytes\(BigInteger)d(big\))p 3507 3498 V 274 3577
a Fa(262)p 390 3610 V 245 w Ff({)p 3507 3610 V 274 3689
a Fa(263)p 390 3723 V 436 w Ff(//)k(transforms)e(the)i(number)p
3507 3723 V 274 3802 a Fa(264)p 390 3836 V 436 w Ff(byte[])f(bigBytes)g
(=)h(big.toByteArray\(\);)p 3507 3836 V 274 3915 a Fa(265)p
390 3949 V 3507 3949 V 274 4028 a(266)p 390 4062 V 436
w Ff(if)g(\(\(big.bitLength\(\))c(\045)48 b(8\))f(!=)g(0\))p
3507 4062 V 274 4141 a Fa(267)p 390 4175 V 627 w Ff({)p
3507 4175 V 274 4254 a Fa(268)p 390 4288 V 818 w Ff(return)f(bigBytes;)
p 3507 4288 V 274 4367 a Fa(269)p 390 4401 V 627 w Ff(})p
3507 4401 V 274 4480 a Fa(270)p 390 4514 V 436 w Ff(else)p
3507 4514 V 274 4593 a Fa(271)p 390 4627 V 627 w Ff({)p
3507 4627 V 274 4706 a Fa(272)p 390 4740 V 818 w Ff(//)h(transforms)e
(to)i(the)g(right)f(size)p 3507 4740 V 274 4819 a Fa(273)p
390 4852 V 818 w Ff(byte[])g(smallerBytes)e(=)k(new)f
(byte[big.bitLength\(\))p 3507 4852 V 274 4932 a Fa(274)p
390 4965 V 913 w Ff(/)h(8];)p 3507 4965 V 274 5044 a
Fa(275)p 390 5078 V 818 w Ff(System.arraycopy\(bigByte)o(s,1)o(,sma)o
(ller)o(Byt)o(es,0)o(,)p 3507 5078 V 274 5157 a Fa(276)p
390 5191 V 913 w Ff(smallerBytes.length\);)p 3507 5191
V 274 5270 a Fa(277)p 390 5304 V 818 w Ff(return)e(smallerBytes;)p
3507 5304 V 274 5383 a Fa(278)p 390 5417 V 627 w Ff(})p
3507 5417 V 274 5496 a Fa(279)p 390 5530 V 245 w Ff(})p
3507 5530 V 274 5609 a Fa(280)p 390 5643 V 3507 5643
V eop
%%Page: 63 64
63 63 bop 270 82 a Fw(B.6.)62 b(ECElGamalGFPCipher)1867
b(63)154 415 y Fa(281)p 270 449 4 113 v 245 w Ff(/**)p
3387 449 V 154 528 a Fa(282)p 270 562 V 293 w Ff(*)47
b(encrypts)f(one)h(block)f(of)h(data.)p 3387 562 V 154
641 a Fa(283)p 270 675 V 293 w Ff(*)g(<p>)p 3387 675
V 154 754 a Fa(284)p 270 788 V 293 w Ff(*)g(The)g(input)g(block)f(of)h
(<code>inputLength</code>)41 b(bytes)47 b(has)g(to)p
3387 788 V 154 867 a Fa(285)p 270 901 V 293 w Ff(*)g(be)g(supplied)f
(in)h(the)g(bytearray)e(<code>input</code>,)e(starting)p
3387 901 V 154 980 a Fa(286)p 270 1014 V 293 w Ff(*)k(at)g
(<code>inputOffset</code>.)41 b(The)47 b(output)f(should)g(be)i
(written)p 3387 1014 V 154 1093 a Fa(287)p 270 1126 V
293 w Ff(*)f(in)g(the)g(bytearray)f(<code>output</code>,)c(starting)j
(at)p 3387 1126 V 154 1205 a Fa(288)p 270 1239 V 293
w Ff(*)i(<code>outputOffset</code>)o(.)p 3387 1239 V
154 1318 a Fa(289)p 270 1352 V 293 w Ff(*)p 3387 1352
V 154 1431 a Fa(290)p 270 1465 V 293 w Ff(*)g(@param)f(input)h(the)g
(bytearray)e(holds)h(the)h(input)p 3387 1465 V 154 1544
a Fa(291)p 270 1578 V 293 w Ff(*)g(@param)f(inputOffset)f(the)i
(position)f(where)g(the)h(input)f(starts)p 3387 1578
V 154 1657 a Fa(292)p 270 1691 V 293 w Ff(*)h(@param)f(inputLength)f
(the)i(length)f(of)h(the)g(input)p 3387 1691 V 154 1770
a Fa(293)p 270 1804 V 293 w Ff(*)g(@param)f(output)g(the)h(bytearray)f
(will)g(hold)h(the)g(output)p 3387 1804 V 154 1883 a
Fa(294)p 270 1917 V 293 w Ff(*)g(@param)f(outputOffset)f(the)i
(position)e(where)i(the)g(output)f(starts)p 3387 1917
V 154 1996 a Fa(295)p 270 2030 V 293 w Ff(*)p 3387 2030
V 154 2109 a Fa(296)p 270 2143 V 293 w Ff(*)h(@return)f(the)h(number)f
(of)h(bytes)g(written)p 3387 2143 V 154 2222 a Fa(297)p
270 2256 V 293 w Ff(*/)p 3387 2256 V 154 2335 a Fa(298)p
270 2368 V 245 w Ff(protected)e(int)i(encryptBlock\(byte[])c(in,int)j
(inOff,int)f(inLen,)p 3387 2368 V 154 2447 a Fa(299)p
270 2481 V 341 w Ff(byte[])h(out,int)f(outOff\))p 3387
2481 V 154 2560 a Fa(300)p 270 2594 V 245 w Ff({)p 3387
2594 V 154 2673 a Fa(301)p 270 2707 V 436 w Ff(//)i(extracts)f(the)h
(data)p 3387 2707 V 154 2786 a Fa(302)p 270 2820 V 436
w Ff(byte[])f(messageBytes)f(=)i(new)g(byte[mPlainBlockSize];)p
3387 2820 V 154 2899 a Fa(303)p 270 2933 V 436 w Ff(byte[])f(firstHalf)
f(=)j(new)f(byte[mPlainBlockSize/2])o(;)p 3387 2933 V
154 3012 a Fa(304)p 270 3046 V 436 w Ff(byte[])f(secondHalf)f(=)j(new)e
(byte[mPlainBlockSize/2];)p 3387 3046 V 154 3125 a Fa(305)p
270 3159 V 436 w Ff(int)h(inputLength)e(=)i(Math.min\(mPlainBlockSize,)
o(inL)o(en\);)p 3387 3159 V 154 3238 a Fa(306)p 270 3272
V 436 w Ff(System.arraycopy\(in,inOf)o(f,me)o(ssag)o(eBy)o(tes,)o(0,in)
o(put)o(Leng)o(th\);)p 3387 3272 V 154 3351 a Fa(307)p
270 3385 V 436 w Ff(if)g(\(inputLength)e(<=)i(\(mPlainBlockSize/2\)\))p
3387 3385 V 154 3464 a Fa(308)p 270 3498 V 627 w Ff({)p
3387 3498 V 154 3577 a Fa(309)p 270 3610 V 818 w Ff
(System.arraycopy\(message)o(Byt)o(es,0)o(,fir)o(stH)o(alf,)o(0,)p
3387 3610 V 154 3689 a Fa(310)p 270 3723 V 913 w Ff(inputLength/2\);)p
3387 3723 V 154 3802 a Fa(311)p 270 3836 V 627 w Ff(})p
3387 3836 V 154 3915 a Fa(312)p 270 3949 V 436 w Ff(else)p
3387 3949 V 154 4028 a Fa(313)p 270 4062 V 627 w Ff({)p
3387 4062 V 154 4141 a Fa(314)p 270 4175 V 818 w Ff
(System.arraycopy\(message)o(Byt)o(es,0)o(,fir)o(stH)o(alf,)o(0,)p
3387 4175 V 154 4254 a Fa(315)p 270 4288 V 913 w Ff
(mPlainBlockSize/2\);)p 3387 4288 V 154 4367 a Fa(316)p
270 4401 V 818 w Ff(System.arraycopy\(message)o(Byt)o(es,m)o(Plai)o
(nBl)o(ockS)o(ize/)o(2,)p 3387 4401 V 154 4480 a Fa(317)p
270 4514 V 913 w Ff(secondHalf,0,inputLength-)o(mPla)o(inBl)o(ock)o
(Size)o(/2\);)p 3387 4514 V 154 4593 a Fa(318)p 270 4627
V 627 w Ff(})p 3387 4627 V 154 4706 a Fa(319)p 270 4740
V 3387 4740 V 154 4819 a(320)p 270 4852 V 436 w Ff(//)g(gets)g(the)g
(necessary)e(objects)h(for)h(calculation)p 3387 4852
V 154 4932 a Fa(321)p 270 4965 V 436 w Ff(ECElGamalGFPPublicKey)42
b(key)47 b(=)g(\(ECElGamalGFPPublicKey\)mKe)o(y;)p 3387
4965 V 154 5044 a Fa(322)p 270 5078 V 436 w Ff(BigInteger)e(q)i(=)h
(mParams.getQ\(\);)p 3387 5078 V 154 5157 a Fa(323)p
270 5191 V 436 w Ff(LECPointGFP)d(generator)g(=)i(mParams.getG\(\);)p
3387 5191 V 154 5270 a Fa(324)p 270 5304 V 436 w Ff(BigInteger)e(k;)p
3387 5304 V 154 5383 a Fa(325)p 270 5417 V 436 w Ff(LECPointGFP)g
(betaOne;)p 3387 5417 V 154 5496 a Fa(326)p 270 5530
V 436 w Ff(BigInteger)g(betaOneX;)p 3387 5530 V 154 5609
a Fa(327)p 270 5643 V 436 w Ff(BigInteger)g(betaOneY;)p
3387 5643 V eop
%%Page: 64 65
64 64 bop 390 82 a Fw(64)1464 b(ANHANG)32 b(B.)61 b(CDC.EC.ECELGAMAL)
274 415 y Fa(328)p 390 449 4 113 v 3507 449 V 274 528
a(329)p 390 562 V 436 w Ff(//)47 b(creates)f(the)h("secret")p
3507 562 V 274 641 a Fa(330)p 390 675 V 436 w Ff(do)p
3507 675 V 274 754 a Fa(331)p 390 788 V 627 w Ff({)p
3507 788 V 274 867 a Fa(332)p 390 901 V 818 w Ff(k)g(=)h(new)f
(BigInteger\(\(q.bitLength)o(\(\)\))41 b(-)48 b(1,)p
3507 901 V 274 980 a Fa(333)p 390 1014 V 913 w Ff(mSecureRandom\);)p
3507 1014 V 274 1093 a Fa(334)p 390 1126 V 818 w Ff(betaOne)e(=)h
(key.getPower\(\).multiply\()p 3507 1126 V 274 1205 a
Fa(335)p 390 1239 V 913 w Ff(new)g(LBigInt\(k\)\);)p
3507 1239 V 274 1318 a Fa(336)p 390 1352 V 818 w Ff(betaOneX)e(=)j
(betaOne.getX\(\).toBigInte)o(ger)o(\(\);)p 3507 1352
V 274 1431 a Fa(337)p 390 1465 V 818 w Ff(betaOneY)d(=)j
(betaOne.getY\(\).toBigInte)o(ger)o(\(\);)p 3507 1465
V 274 1544 a Fa(338)p 390 1578 V 627 w Ff(})f(while)g
(\(!\(betaOneX.gcd\(q\).equa)o(ls\(B)o(igIn)o(teg)o(er.O)o(NE\)\))p
3507 1578 V 274 1657 a Fa(339)p 390 1691 V 722 w Ff(||)h
(!\(betaOneY.gcd\(q\).equal)o(s\(B)o(igIn)o(tege)o(r.O)o(NE\)\))o(\);)p
3507 1691 V 274 1770 a Fa(340)p 390 1804 V 3507 1804
V 274 1883 a(341)p 390 1917 V 436 w Ff(//)95 b(calculates)45
b(the)i(ciphertext)p 3507 1917 V 274 1996 a Fa(342)p
390 2030 V 436 w Ff(LECPointGFP)e(betaZero)g(=)j(generator.multiply\()p
3507 2030 V 274 2109 a Fa(343)p 390 2143 V 531 w Ff(new)f
(LBigInt\(k\)\);)p 3507 2143 V 274 2222 a Fa(344)p 390
2256 V 436 w Ff(BigInteger)e(mOne)i(=)g(new)g(BigInteger\(firstHalf\);)
p 3507 2256 V 274 2335 a Fa(345)p 390 2368 V 436 w Ff(BigInteger)e
(mTwo)i(=)g(new)g(BigInteger\(secondHalf\);)p 3507 2368
V 274 2447 a Fa(346)p 390 2481 V 436 w Ff(BigInteger)e(cOne)i(=)g
(betaOneX.multiply\(mOne\).m)o(od\()o(q\);)p 3507 2481
V 274 2560 a Fa(347)p 390 2594 V 436 w Ff(BigInteger)e(cTwo)i(=)g
(betaOneY.multiply\(mTwo\).m)o(od\()o(q\);)p 3507 2594
V 274 2673 a Fa(348)p 390 2707 V 436 w Ff(BigInteger)e(betaZeroX)g(=)j
(betaZero.getX\(\).toBigIn)o(tege)o(r\(\);)p 3507 2707
V 274 2786 a Fa(349)p 390 2820 V 436 w Ff(BigInteger)d(betaZeroY)g(=)j
(betaZero.getY\(\).toBigIn)o(tege)o(r\(\);)p 3507 2820
V 274 2899 a Fa(350)p 390 2933 V 436 w Ff(byte[])e(betaZeroXBytes)e(=)j
(getBytes\(betaZeroX\);)p 3507 2933 V 274 3012 a Fa(351)p
390 3046 V 436 w Ff(byte[])f(betaZeroYBytes)e(=)j
(getBytes\(betaZeroY\);)p 3507 3046 V 274 3125 a Fa(352)p
390 3159 V 436 w Ff(byte[])f(cOneBytes)f(=)j(getBytes\(cOne\);)p
3507 3159 V 274 3238 a Fa(353)p 390 3272 V 436 w Ff(byte[])e(cTwoBytes)
f(=)j(getBytes\(cTwo\);)p 3507 3272 V 274 3351 a Fa(354)p
390 3385 V 3507 3385 V 274 3464 a(355)p 390 3498 V 436
w Ff(//)f(transforms)e(the)i(ciphertext)e(to)i(the)g(right)g(format)p
3507 3498 V 274 3577 a Fa(356)p 390 3610 V 436 w Ff
(System.arraycopy\(cOneByt)o(es,0)o(,out)o(,ou)o(tOff)o(+)p
3507 3610 V 274 3689 a Fa(357)p 390 3723 V 531 w Ff
(mCipherBlockSize/4-cOneByt)o(es.l)o(eng)o(th,c)o(OneB)o(yte)o(s.le)o
(ngth)o(\);)p 3507 3723 V 274 3802 a Fa(358)p 390 3836
V 436 w Ff(System.arraycopy\(cTwoByt)o(es,0)o(,out)o(,ou)o(tOff)o(+)p
3507 3836 V 274 3915 a Fa(359)p 390 3949 V 531 w Ff
(mCipherBlockSize/2-cTwoByt)o(es.l)o(eng)o(th,c)o(TwoB)o(yte)o(s.le)o
(ngth)o(\);)p 3507 3949 V 274 4028 a Fa(360)p 390 4062
V 436 w Ff(System.arraycopy\(betaZer)o(oXBy)o(tes,)o(0,o)o(ut,o)o(utOf)
o(f+)p 3507 4062 V 274 4141 a Fa(361)p 390 4175 V 531
w Ff(mCipherBlockSize/4*3-betaZ)o(eroX)o(Byt)o(es.l)o(engt)o(h,)p
3507 4175 V 274 4254 a Fa(362)p 390 4288 V 531 w Ff
(betaZeroXBytes.length\);)p 3507 4288 V 274 4367 a Fa(363)p
390 4401 V 436 w Ff(System.arraycopy\(betaZer)o(oYBy)o(tes,)o(0,o)o
(ut,o)o(utOf)o(f+)p 3507 4401 V 274 4480 a Fa(364)p 390
4514 V 531 w Ff(mCipherBlockSize-betaZeroY)o(Byte)o(s.l)o(engt)o(h,)p
3507 4514 V 274 4593 a Fa(365)p 390 4627 V 531 w Ff
(betaZeroYBytes.length\);)p 3507 4627 V 274 4706 a Fa(366)p
390 4740 V 3507 4740 V 274 4819 a(367)p 390 4852 V 436
w Ff(//)g(and)g(back)p 3507 4852 V 274 4932 a Fa(368)p
390 4965 V 436 w Ff(return)f(mCipherBlockSize;)p 3507
4965 V 274 5044 a Fa(369)p 390 5078 V 245 w Ff(})p 3507
5078 V 274 5157 a Fa(370)p 390 5191 V 3507 5191 V 274
5270 a(371)p 390 5304 V 245 w Ff(/**)p 3507 5304 V 274
5383 a Fa(372)p 390 5417 V 293 w Ff(*)h(decrypts)f(one)h(block)f(of)h
(data.)p 3507 5417 V 274 5496 a Fa(373)p 390 5530 V 293
w Ff(*)g(<p>)p 3507 5530 V 274 5609 a Fa(374)p 390 5643
V 293 w Ff(*)g(The)g(input)g(block)f(of)h(<code>inputLength</code>)41
b(bytes)47 b(has)g(to)p 3507 5643 V eop
%%Page: 65 66
65 65 bop 270 82 a Fw(B.6.)62 b(ECElGamalGFPCipher)1867
b(65)154 415 y Fa(375)p 270 449 4 113 v 293 w Ff(*)47
b(be)g(supplied)f(in)h(the)g(bytearray)e(<code>input</code>,)e
(starting)p 3387 449 V 154 528 a Fa(376)p 270 562 V 293
w Ff(*)k(at)g(<code>inputOffset</code>.)41 b(The)47 b(output)f(should)g
(be)i(written)p 3387 562 V 154 641 a Fa(377)p 270 675
V 293 w Ff(*)f(in)g(the)g(bytearray)f(<code>output</code>,)c(starting)j
(at)p 3387 675 V 154 754 a Fa(378)p 270 788 V 293 w Ff(*)i
(<code>outputOffset</code>)o(.)p 3387 788 V 154 867 a
Fa(379)p 270 901 V 293 w Ff(*)p 3387 901 V 154 980 a
Fa(380)p 270 1014 V 293 w Ff(*)g(@param)f(input)h(the)g(bytearray)e
(holds)h(the)h(input)p 3387 1014 V 154 1093 a Fa(381)p
270 1126 V 293 w Ff(*)g(@param)f(inputOffset)f(the)i(position)f(where)g
(the)h(input)f(starts)p 3387 1126 V 154 1205 a Fa(382)p
270 1239 V 293 w Ff(*)h(@param)f(inputLength)f(the)i(length)f(of)h(the)
g(input)p 3387 1239 V 154 1318 a Fa(383)p 270 1352 V
293 w Ff(*)g(@param)f(output)g(the)h(bytearray)f(will)g(hold)h(the)g
(output)p 3387 1352 V 154 1431 a Fa(384)p 270 1465 V
293 w Ff(*)g(@param)f(outputOffset)f(the)i(position)e(where)i(the)g
(output)f(starts)p 3387 1465 V 154 1544 a Fa(385)p 270
1578 V 293 w Ff(*)p 3387 1578 V 154 1657 a Fa(386)p 270
1691 V 293 w Ff(*)h(@return)f(the)h(number)f(of)h(bytes)g(written)p
3387 1691 V 154 1770 a Fa(387)p 270 1804 V 293 w Ff(*/)p
3387 1804 V 154 1883 a Fa(388)p 270 1917 V 245 w Ff(protected)e(int)i
(decryptBlock\(byte[])c(in,int)j(inOff,int)f(inLen,)p
3387 1917 V 154 1996 a Fa(389)p 270 2030 V 341 w Ff(byte[])h(out,int)f
(outOff\))p 3387 2030 V 154 2109 a Fa(390)p 270 2143
V 245 w Ff({)p 3387 2143 V 154 2222 a Fa(391)p 270 2256
V 436 w Ff(//)i(extracts)f(the)h(data)p 3387 2256 V 154
2335 a Fa(392)p 270 2368 V 436 w Ff(byte[])f(cipherBytes)f(=)i(new)g
(byte[mCipherBlockSize];)p 3387 2368 V 154 2447 a Fa(393)p
270 2481 V 436 w Ff(byte[])f(cOneBytes)f(=)j(new)f
(byte[mCipherBlockSize/4)o(];)p 3387 2481 V 154 2560
a Fa(394)p 270 2594 V 436 w Ff(byte[])f(cTwoBytes)f(=)j(new)f
(byte[mCipherBlockSize/4)o(];)p 3387 2594 V 154 2673
a Fa(395)p 270 2707 V 436 w Ff(byte[])f(betaZeroXBytes)e(=)j(new)g
(byte[mCipherBlockSize/4];)p 3387 2707 V 154 2786 a Fa(396)p
270 2820 V 436 w Ff(byte[])f(betaZeroYBytes)e(=)j(new)g
(byte[mCipherBlockSize/4];)p 3387 2820 V 154 2899 a Fa(397)p
270 2933 V 436 w Ff(System.arraycopy\(in,inOf)o(f,ci)o(pher)o(Byt)o
(es,0)o(,)p 3387 2933 V 154 3012 a Fa(398)p 270 3046
V 531 w Ff(mCipherBlockSize\);)p 3387 3046 V 154 3125
a Fa(399)p 270 3159 V 436 w Ff(System.arraycopy\(cipherB)o(ytes)o(,0,c)
o(One)o(Byte)o(s,0,)p 3387 3159 V 154 3238 a Fa(400)p
270 3272 V 531 w Ff(mCipherBlockSize/4\);)p 3387 3272
V 154 3351 a Fa(401)p 270 3385 V 436 w Ff(System.arraycopy\(cipherB)o
(ytes)o(,mCi)o(phe)o(rBlo)o(ckSi)o(ze/)o(4,)p 3387 3385
V 154 3464 a Fa(402)p 270 3498 V 531 w Ff(cTwoBytes,0,mCipherBlockSi)o
(ze/4)o(\);)p 3387 3498 V 154 3577 a Fa(403)p 270 3610
V 436 w Ff(System.arraycopy\(cipherB)o(ytes)o(,mCi)o(phe)o(rBlo)o(ckSi)
o(ze/)o(2,)p 3387 3610 V 154 3689 a Fa(404)p 270 3723
V 531 w Ff(betaZeroXBytes,0,mCipherBl)o(ockS)o(ize)o(/4\);)p
3387 3723 V 154 3802 a Fa(405)p 270 3836 V 436 w Ff
(System.arraycopy\(cipherB)o(ytes)o(,mCi)o(phe)o(rBlo)o(ckSi)o(ze/)o
(4*3,)p 3387 3836 V 154 3915 a Fa(406)p 270 3949 V 531
w Ff(betaZeroYBytes,0,mCipherBl)o(ockS)o(ize)o(/4\);)p
3387 3949 V 154 4028 a Fa(407)p 270 4062 V 3387 4062
V 154 4141 a(408)p 270 4175 V 436 w Ff(//)g(gets)g(the)g(necessary)e
(objects)h(for)h(calculation)p 3387 4175 V 154 4254 a
Fa(409)p 270 4288 V 436 w Ff(ECElGamalGFPPrivateKey)42
b(key)47 b(=)g(\(ECElGamalGFPPrivateKey\))p 3387 4288
V 154 4367 a Fa(410)p 270 4401 V 531 w Ff(mKey;)p 3387
4401 V 154 4480 a Fa(411)p 270 4514 V 436 w Ff(BigInteger)e(q)i(=)h
(key.getParams\(\).getQ\(\);)p 3387 4514 V 154 4593 a
Fa(412)p 270 4627 V 436 w Ff(BigInteger)d(exponent)h(=)h
(key.getExponent\(\);)p 3387 4627 V 154 4706 a Fa(413)p
270 4740 V 436 w Ff(LECPointGFP)e(betaZero)g(=)j(new)f
(LECPointGFP\(\);)p 3387 4740 V 154 4819 a Fa(414)p 270
4852 V 436 w Ff(try)p 3387 4852 V 154 4932 a Fa(415)p
270 4965 V 627 w Ff({)p 3387 4965 V 154 5044 a Fa(416)p
270 5078 V 818 w Ff(//)95 b(calculates)45 b(the)i(plaintext)p
3387 5078 V 154 5157 a Fa(417)p 270 5191 V 818 w Ff(betaZero)e(=)j(new)
f(LECPointGFP\(new)c(LBigInt\()p 3387 5191 V 154 5270
a Fa(418)p 270 5304 V 913 w Ff(new)k(BigInteger\(betaZeroXBytes)o
(\)\),n)o(ew)41 b(LBigInt\()p 3387 5304 V 154 5383 a
Fa(419)p 270 5417 V 913 w Ff(new)47 b(BigInteger\(betaZeroYBytes)o
(\)\),k)o(ey.)p 3387 5417 V 154 5496 a Fa(420)p 270 5530
V 913 w Ff(getParams\(\).getCurve\(\),ne)o(w)42 b(LBigInt\(q\)\);)p
3387 5530 V 154 5609 a Fa(421)p 270 5643 V 818 w Ff(LECPointGFP)j
(betaOne)g(=)j(betaZero.multiply\()p 3387 5643 V eop
%%Page: 66 67
66 66 bop 390 82 a Fw(66)1464 b(ANHANG)32 b(B.)61 b(CDC.EC.ECELGAMAL)
274 415 y Fa(422)p 390 449 4 113 v 913 w Ff(new)47 b
(LBigInt\(exponent\)\);)p 3507 449 V 274 528 a Fa(423)p
390 562 V 818 w Ff(BigInteger)e(betaOneX)g(=)j(betaOne.getX\(\).)p
3507 562 V 274 641 a Fa(424)p 390 675 V 913 w Ff(toBigInteger\(\);)p
3507 675 V 274 754 a Fa(425)p 390 788 V 818 w Ff(BigInteger)d(betaOneY)
g(=)j(betaOne.getY\(\).)p 3507 788 V 274 867 a Fa(426)p
390 901 V 913 w Ff(toBigInteger\(\);)p 3507 901 V 274
980 a Fa(427)p 390 1014 V 818 w Ff(BigInteger)d(cOne)h(=)i(new)f
(BigInteger\(cOneBytes\);)p 3507 1014 V 274 1093 a Fa(428)p
390 1126 V 818 w Ff(BigInteger)e(cTwo)h(=)i(new)f
(BigInteger\(cTwoBytes\);)p 3507 1126 V 274 1205 a Fa(429)p
390 1239 V 818 w Ff(BigInteger)e(mOne)h(=)i(BigInteger.ZERO;)p
3507 1239 V 274 1318 a Fa(430)p 390 1352 V 818 w Ff(BigInteger)d(mTwo)h
(=)i(BigInteger.ZERO;)p 3507 1352 V 274 1431 a Fa(431)p
390 1465 V 818 w Ff(if)f(\(!cOne.equals\(BigInteger)o(.ZER)o(O\)\))p
3507 1465 V 274 1544 a Fa(432)p 390 1578 V 1009 w Ff({)p
3507 1578 V 274 1657 a Fa(433)p 390 1691 V 1200 w Ff(mOne)f(=)i
(\(cOne.multiply\(betaOneX)o(.)p 3507 1691 V 274 1770
a Fa(434)p 390 1804 V 1295 w Ff(modInverse\(q\)\).mod\(q\)\);)p
3507 1804 V 274 1883 a Fa(435)p 390 1917 V 1009 w Ff(})p
3507 1917 V 274 1996 a Fa(436)p 390 2030 V 818 w Ff(if)f
(\(!cTwo.equals\(BigInteger)o(.ZER)o(O\)\))p 3507 2030
V 274 2109 a Fa(437)p 390 2143 V 1009 w Ff({)p 3507 2143
V 274 2222 a Fa(438)p 390 2256 V 1200 w Ff(mTwo)f(=)i
(\(cTwo.multiply\(betaOneY)o(.)p 3507 2256 V 274 2335
a Fa(439)p 390 2368 V 1295 w Ff(modInverse\(q\)\).mod\(q\)\);)p
3507 2368 V 274 2447 a Fa(440)p 390 2481 V 1009 w Ff(})p
3507 2481 V 274 2560 a Fa(441)p 390 2594 V 818 w Ff(byte[])e(mOneBytes)
f(=)j(getBytes\(mOne\);)p 3507 2594 V 274 2673 a Fa(442)p
390 2707 V 818 w Ff(byte[])e(mTwoBytes)f(=)j(getBytes\(mTwo\);)p
3507 2707 V 274 2786 a Fa(443)p 390 2820 V 3507 2820
V 274 2899 a(444)p 390 2933 V 818 w Ff(//)f(transforms)e(the)i
(plaintext)e(to)i(the)g(right)g(format)p 3507 2933 V
274 3012 a Fa(445)p 390 3046 V 818 w Ff(System.arraycopy\(mOneByt)o
(es,)o(0,ou)o(t,ou)o(tOf)o(f+)p 3507 3046 V 274 3125
a Fa(446)p 390 3159 V 913 w Ff(mPlainBlockSize/2-mOneByt)o(es.l)o(engt)
o(h,m)o(OneB)o(ytes)o(.)p 3507 3159 V 274 3238 a Fa(447)p
390 3272 V 913 w Ff(length\);)p 3507 3272 V 274 3351
a Fa(448)p 390 3385 V 818 w Ff(System.arraycopy\(mTwoByt)o(es,)o(0,ou)o
(t,ou)o(tOf)o(f+)p 3507 3385 V 274 3464 a Fa(449)p 390
3498 V 913 w Ff(mPlainBlockSize-mTwoBytes)o(.len)o(gth,)o(mTw)o(oByt)o
(es.)p 3507 3498 V 274 3577 a Fa(450)p 390 3610 V 913
w Ff(length\);)p 3507 3610 V 274 3689 a Fa(451)p 390
3723 V 627 w Ff(})p 3507 3723 V 274 3802 a Fa(452)p 390
3836 V 436 w Ff(catch)f(\(LiDIAError)f(le\))p 3507 3836
V 274 3915 a Fa(453)p 390 3949 V 627 w Ff({)p 3507 3949
V 274 4028 a Fa(454)p 390 4062 V 818 w Ff(System.out.println\(le.to)o
(Str)o(ing\()o(\)\);)p 3507 4062 V 274 4141 a Fa(455)p
390 4175 V 627 w Ff(})p 3507 4175 V 274 4254 a Fa(456)p
390 4288 V 3507 4288 V 274 4367 a(457)p 390 4401 V 436
w Ff(//)i(and)g(back)p 3507 4401 V 274 4480 a Fa(458)p
390 4514 V 436 w Ff(return)f(mPlainBlockSize;)p 3507
4514 V 274 4593 a Fa(459)p 390 4627 V 245 w Ff(})p 3507
4627 V 274 4706 a Fa(460)p 390 4740 V 3507 4740 V 274
4819 a(461)p 390 4852 V 245 w Ff(/**)p 3507 4852 V 274
4932 a Fa(462)p 390 4965 V 293 w Ff(*)h(returns)f(<code>null</code>)d
(since)k(there)f(is)h(no)g(initial)p 3507 4965 V 274
5044 a Fa(463)p 390 5078 V 293 w Ff(*)g(vector)f(required)g(for)h(this)
f(implementation)p 3507 5078 V 274 5157 a Fa(464)p 390
5191 V 293 w Ff(*/)p 3507 5191 V 274 5270 a Fa(465)p
390 5304 V 245 w Ff(protected)f(byte[])h(engineGetIV\(\))p
3507 5304 V 274 5383 a Fa(466)p 390 5417 V 245 w Ff({)p
3507 5417 V 274 5496 a Fa(467)p 390 5530 V 436 w Ff(return)g(null;)p
3507 5530 V 274 5609 a Fa(468)p 390 5643 V 245 w Ff(})p
3507 5643 V eop
%%Page: 67 68
67 67 bop 270 82 a Fw(B.6.)62 b(ECElGamalGFPCipher)1867
b(67)154 415 y Fa(469)p 270 449 4 113 v 3387 449 V 154
528 a(470)p 270 562 V 245 w Ff(/**)p 3387 562 V 154 641
a Fa(471)p 270 675 V 293 w Ff(*)47 b(@throws)f
(NoSuchAlgorithmException)41 b(always)46 b(since)p 3387
675 V 154 754 a Fa(472)p 270 788 V 293 w Ff(*)h
(<code>ECElGamalGFPCipher<)o(/cod)o(e>)42 b(supports)j(no)i(modes)p
3387 788 V 154 867 a Fa(473)p 270 901 V 293 w Ff(*/)p
3387 901 V 154 980 a Fa(474)p 270 1014 V 245 w Ff(protected)e(void)i
(engineSetMode\(String)42 b(mode\))p 3387 1014 V 154
1093 a Fa(475)p 270 1126 V 436 w Ff(throws)k(NoSuchAlgorithmException)p
3387 1126 V 154 1205 a Fa(476)p 270 1239 V 245 w Ff({)p
3387 1239 V 154 1318 a Fa(477)p 270 1352 V 436 w Ff(throw)g(new)h
(NoSuchAlgorithmException\()o("ECE)o(lGam)o(alG)o(FPCi)o(pher)41
b(")p 3387 1352 V 154 1431 a Fa(478)p 270 1465 V 531
w Ff(+)48 b("supports)d(no)i(modes."\);)p 3387 1465 V
154 1544 a Fa(479)p 270 1578 V 245 w Ff(})p 3387 1578
V 154 1657 a Fa(480)p 270 1691 V 3387 1691 V 154 1770
a(481)p 270 1804 V 245 w Ff(/**)p 3387 1804 V 154 1883
a Fa(482)p 270 1917 V 293 w Ff(*)g(@throws)f(NoSuchPaddingException)c
(always)k(since)p 3387 1917 V 154 1996 a Fa(483)p 270
2030 V 293 w Ff(*)h(<code>ECElGamalGFPCipher<)o(/cod)o(e>)42
b(supports)j(no)i(padding)p 3387 2030 V 154 2109 a Fa(484)p
270 2143 V 293 w Ff(*/)p 3387 2143 V 154 2222 a Fa(485)p
270 2256 V 245 w Ff(protected)e(void)i(engineSetPadding\(String)41
b(padding\))p 3387 2256 V 154 2335 a Fa(486)p 270 2368
V 436 w Ff(throws)46 b(NoSuchPaddingException)p 3387
2368 V 154 2447 a Fa(487)p 270 2481 V 245 w Ff({)p 3387
2481 V 154 2560 a Fa(488)p 270 2594 V 436 w Ff(throw)g(new)h
(NoSuchPaddingException\("E)o(CElG)o(amal)o(GFP)o(Ciph)o(er)42
b(")p 3387 2594 V 154 2673 a Fa(489)p 270 2707 V 531
w Ff(+)48 b("supports)d(no)i(padding."\);)p 3387 2707
V 154 2786 a Fa(490)p 270 2820 V 245 w Ff(})p 3387 2820
V 154 2899 a Fa(491)p 270 2933 V 3387 2933 V 154 3012
a(492)p 270 3046 V 245 w Ff(/**)p 3387 3046 V 154 3125
a Fa(493)p 270 3159 V 293 w Ff(*)g(This)g(method)f(is)h(called)f
(whenever)g(a)h(full)g(block)f(needs)h(to)g(be)p 3387
3159 V 154 3238 a Fa(494)p 270 3272 V 293 w Ff(*)g(encrypted)f(or)h
(decrypted.)e(It)i(simply)f(calls)g(one)h(of)g(the)p
3387 3272 V 154 3351 a Fa(495)p 270 3385 V 293 w Ff(*)g(methods)f
(<code>encryptBlock</code>)41 b(or)p 3387 3385 V 154
3464 a Fa(496)p 270 3498 V 293 w Ff(*)47 b(<code>decryptBlock</code>)41
b(depending)k(on)j(the)f(current)e(state)p 3387 3498
V 154 3577 a Fa(497)p 270 3610 V 293 w Ff(*)i(of)g(the)g(cipher.)p
3387 3610 V 154 3689 a Fa(498)p 270 3723 V 293 w Ff(*)g(<p>)p
3387 3723 V 154 3802 a Fa(499)p 270 3836 V 293 w Ff(*)g(The)g(input)g
(block)f(of)h(<code>inputLength</code>)41 b(bytes)47
b(has)g(to)p 3387 3836 V 154 3915 a Fa(500)p 270 3949
V 293 w Ff(*)g(be)g(supplied)f(in)h(the)g(bytearray)e
(<code>input</code>,)e(starting)p 3387 3949 V 154 4028
a Fa(501)p 270 4062 V 293 w Ff(*)k(in)g(the)g(at)h
(<code>inputOffset</code)o(>.)41 b(The)47 b(output)f(should)h(be)p
3387 4062 V 154 4141 a Fa(502)p 270 4175 V 293 w Ff(*)g(written)f
(bytearray)f(<code>output</code>,)e(starting)i(at)p 3387
4175 V 154 4254 a Fa(503)p 270 4288 V 293 w Ff(*)i
(<code>outputOffset</code>)o(.)p 3387 4288 V 154 4367
a Fa(504)p 270 4401 V 293 w Ff(*)p 3387 4401 V 154 4480
a Fa(505)p 270 4514 V 293 w Ff(*)g(@param)f(input)h(the)g(bytearray)e
(holds)h(the)h(input)p 3387 4514 V 154 4593 a Fa(506)p
270 4627 V 293 w Ff(*)g(@param)f(inputOffset)f(the)i(position)f(where)g
(the)h(input)f(starts)p 3387 4627 V 154 4706 a Fa(507)p
270 4740 V 293 w Ff(*)h(@param)f(inputLength)f(the)i(length)f(of)h(the)
g(input)p 3387 4740 V 154 4819 a Fa(508)p 270 4852 V
293 w Ff(*)g(@param)f(output)g(the)h(bytearray)f(will)g(hold)h(the)g
(output)p 3387 4852 V 154 4932 a Fa(509)p 270 4965 V
293 w Ff(*)g(@param)f(outputOffset)f(the)i(position)e(where)i(the)g
(output)f(starts)p 3387 4965 V 154 5044 a Fa(510)p 270
5078 V 293 w Ff(*)p 3387 5078 V 154 5157 a Fa(511)p 270
5191 V 293 w Ff(*)h(@throws)f(ShortBufferException)c(if)48
b(the)e(output)h(array)f(is)h(too)p 3387 5191 V 154 5270
a Fa(512)p 270 5304 V 293 w Ff(*)g(small)p 3387 5304
V 154 5383 a Fa(513)p 270 5417 V 293 w Ff(*)p 3387 5417
V 154 5496 a Fa(514)p 270 5530 V 293 w Ff(*)g(@return)f(the)h(number)f
(of)h(bytes)g(written)p 3387 5530 V 154 5609 a Fa(515)p
270 5643 V 293 w Ff(*)p 3387 5643 V eop
%%Page: 68 69
68 68 bop 390 82 a Fw(68)1464 b(ANHANG)32 b(B.)61 b(CDC.EC.ECELGAMAL)
274 415 y Fa(516)p 390 449 4 113 v 293 w Ff(*)47 b(@see)g(#mState)p
3507 449 V 274 528 a Fa(517)p 390 562 V 293 w Ff(*/)p
3507 562 V 274 641 a Fa(518)p 390 675 V 245 w Ff(protected)e(int)i
(engineTransformBlock\(byte)o([])42 b(input,)p 3507 675
V 274 754 a Fa(519)p 390 788 V 341 w Ff(int)k(inputOffset,int)e
(inputLength,byte[])f(output,)p 3507 788 V 274 867 a
Fa(520)p 390 901 V 341 w Ff(int)j(outputOffset\))p 3507
901 V 274 980 a Fa(521)p 390 1014 V 436 w Ff(throws)g
(ShortBufferException)p 3507 1014 V 274 1093 a Fa(522)p
390 1126 V 245 w Ff({)p 3507 1126 V 274 1205 a Fa(523)p
390 1239 V 436 w Ff(//)h(depending)e(on)j(the)f(state,)f(the)h
(correspondant)d(method)i(is)p 3507 1239 V 274 1318 a
Fa(524)p 390 1352 V 436 w Ff(//)h(called)p 3507 1352
V 274 1431 a Fa(525)p 390 1465 V 436 w Ff(if)g(\(mState)f(==)h
(Cipher.ENCRYPT_MODE\))p 3507 1465 V 274 1544 a Fa(526)p
390 1578 V 627 w Ff(return)f(encryptBlock\(input,input)o(Offs)o(et,i)o
(npu)o(tLen)o(gth,)p 3507 1578 V 274 1657 a Fa(527)p
390 1691 V 722 w Ff(output,outputOffset\);)p 3507 1691
V 274 1770 a Fa(528)p 390 1804 V 436 w Ff(else)h(if)g(\(mState)f(==)h
(Cipher.DECRYPT_MODE\))p 3507 1804 V 274 1883 a Fa(529)p
390 1917 V 627 w Ff(return)f(decryptBlock\(input,input)o(Offs)o(et,i)o
(npu)o(tLen)o(gth,)p 3507 1917 V 274 1996 a Fa(530)p
390 2030 V 722 w Ff(output,outputOffset\);)p 3507 2030
V 274 2109 a Fa(531)p 390 2143 V 436 w Ff(else)h(return)f(0;)p
3507 2143 V 274 2222 a Fa(532)p 390 2256 V 245 w Ff(})p
3507 2256 V 274 2335 a Fa(533)p 390 2368 V 3507 2368
V 274 2447 a(534)p 390 2481 V 245 w Ff(/**)p 3507 2481
V 274 2560 a Fa(535)p 390 2594 V 293 w Ff(*)h(This)g(method)f(is)h
(called)f(whenever)g(the)h(last)f(block)h(needs)f(to)h(be)p
3507 2594 V 274 2673 a Fa(536)p 390 2707 V 293 w Ff(*)g(encrypted)f(or)
h(decrypted.)e(The)i(input)f(can)h(be)g(shorter)f(than)h(one)p
3507 2707 V 274 2786 a Fa(537)p 390 2820 V 293 w Ff(*)g(block.)f(It)i
(basically)d(calls)h(the)h(method)p 3507 2820 V 274 2899
a Fa(538)p 390 2933 V 293 w Ff(*)g(<code>engineTransformBloc)o(k</c)o
(ode>)o(.)p 3507 2933 V 274 3012 a Fa(539)p 390 3046
V 293 w Ff(*)g(<p>)p 3507 3046 V 274 3125 a Fa(540)p
390 3159 V 293 w Ff(*)g(The)g(input)g(of)g(<code>inputLength</code)o(>)
42 b(bytes)k(has)h(to)g(be)p 3507 3159 V 274 3238 a Fa(541)p
390 3272 V 293 w Ff(*)g(supplied)f(in)h(the)g(bytearray)e
(<code>input</code>,)e(starting)i(at)p 3507 3272 V 274
3351 a Fa(542)p 390 3385 V 293 w Ff(*)i(<code>inputOffset</code>.)41
b(The)47 b(output)f(should)g(be)h(written)f(in)p 3507
3385 V 274 3464 a Fa(543)p 390 3498 V 293 w Ff(*)h(the)g(bytearray)e
(<code>output</code>,)e(starting)i(at)p 3507 3498 V 274
3577 a Fa(544)p 390 3610 V 293 w Ff(*)i(<code>outputOffset</code>)o(.)p
3507 3610 V 274 3689 a Fa(545)p 390 3723 V 293 w Ff(*)p
3507 3723 V 274 3802 a Fa(546)p 390 3836 V 293 w Ff(*)g(@param)f(input)
h(the)g(bytearray)e(holds)h(the)h(input)p 3507 3836 V
274 3915 a Fa(547)p 390 3949 V 293 w Ff(*)g(@param)f(inputOffset)f(the)
i(position)f(where)g(the)h(input)f(starts)p 3507 3949
V 274 4028 a Fa(548)p 390 4062 V 293 w Ff(*)h(@param)f(inputLength)f
(the)i(length)f(of)h(the)g(input)p 3507 4062 V 274 4141
a Fa(549)p 390 4175 V 293 w Ff(*)g(@param)f(output)g(the)h(bytearray)f
(will)g(hold)h(the)g(output)p 3507 4175 V 274 4254 a
Fa(550)p 390 4288 V 293 w Ff(*)g(@param)f(outputOffset)f(the)i
(position)e(where)i(the)g(output)f(starts)p 3507 4288
V 274 4367 a Fa(551)p 390 4401 V 293 w Ff(*)p 3507 4401
V 274 4480 a Fa(552)p 390 4514 V 293 w Ff(*)h(@throws)f
(ShortBufferException)c(if)48 b(the)e(output)h(array)f(is)h(too)p
3507 4514 V 274 4593 a Fa(553)p 390 4627 V 293 w Ff(*)g(small)p
3507 4627 V 274 4706 a Fa(554)p 390 4740 V 293 w Ff(*)p
3507 4740 V 274 4819 a Fa(555)p 390 4852 V 293 w Ff(*)g(@return)f(the)h
(number)f(of)h(bytes)g(written)p 3507 4852 V 274 4932
a Fa(556)p 390 4965 V 293 w Ff(*/)p 3507 4965 V 274 5044
a Fa(557)p 390 5078 V 245 w Ff(protected)e(int)i
(engineTransformBlockFinal)o(\(byt)o(e[])41 b(input,)p
3507 5078 V 274 5157 a Fa(558)p 390 5191 V 341 w Ff(int)46
b(inputOffset,int)e(inputLength,byte[])f(output,)p 3507
5191 V 274 5270 a Fa(559)p 390 5304 V 341 w Ff(int)j(outputOffset\))p
3507 5304 V 274 5383 a Fa(560)p 390 5417 V 436 w Ff(throws)g
(ShortBufferException)p 3507 5417 V 274 5496 a Fa(561)p
390 5530 V 245 w Ff({)p 3507 5530 V 274 5609 a Fa(562)p
390 5643 V 436 w Ff(//)h(checks)f(the)h(input)g(length)p
3507 5643 V eop
%%Page: 69 70
69 69 bop 270 82 a Fw(B.6.)62 b(ECElGamalGFPCipher)1867
b(69)154 415 y Fa(563)p 270 449 4 113 v 436 w Ff(if)47
b(\(inputLength)e(==)i(0\))p 3387 449 V 154 528 a Fa(564)p
270 562 V 627 w Ff({)p 3387 562 V 154 641 a Fa(565)p
270 675 V 818 w Ff(return)f(0;)p 3387 675 V 154 754 a
Fa(566)p 270 788 V 627 w Ff(})p 3387 788 V 154 867 a
Fa(567)p 270 901 V 3387 901 V 154 980 a(568)p 270 1014
V 436 w Ff(//)h(calls)g(the)f(engineTransformBlock)d(method)p
3387 1014 V 154 1093 a Fa(569)p 270 1126 V 436 w Ff(return)j
(engineTransformBlock\(inpu)o(t,i)o(nput)o(Offs)o(et,)p
3387 1126 V 154 1205 a Fa(570)p 270 1239 V 531 w Ff
(inputLength,output,outputO)o(ffse)o(t\);)p 3387 1239
V 154 1318 a Fa(571)p 270 1352 V 245 w Ff(})p 3387 1352
V 154 1431 a Fa(572)p 270 1465 V 54 w Ff(})p 3387 1465
V 270 1490 4 25 v 3387 1490 V 270 1493 3120 4 v eop
%%Page: 70 71
70 70 bop eop
%%Page: 71 72
71 71 bop 270 1050 a Fr(Anhang)64 b(C)270 1477 y Fv(cdc.lidia)270
1971 y Fz(C.1)135 b(LECDomainP)l(arametersP1363)49 b(\(JA)-15
b(V)g(A-Co)t(de\))p 270 2118 3120 4 v 270 2143 4 25 v
3387 2143 V 215 2222 a Fa(1)p 270 2256 4 113 v 53 w Ff(package)46
b(cdc.lidia;)p 3387 2256 V 215 2335 a Fa(2)p 270 2368
V 3387 2368 V 215 2447 a(3)p 270 2481 V 53 w Ff(import)g(java.math.*;)p
3387 2481 V 215 2560 a Fa(4)p 270 2594 V 53 w Ff(import)g
(java.security.*;)p 3387 2594 V 215 2673 a Fa(5)p 270
2707 V 53 w Ff(import)g(cdc.lidia.*;)p 3387 2707 V 215
2786 a Fa(6)p 270 2820 V 3387 2820 V 215 2899 a(7)p 270
2933 V 53 w Ff(/**)p 3387 2933 V 215 3012 a Fa(8)p 270
3046 V 101 w Ff(*)h(<code>LECDomainParametersP)o(136)o(3</c)o(ode>)41
b(specifies)k(the)i(elliptic)p 3387 3046 V 215 3125 a
Fa(9)p 270 3159 V 101 w Ff(*)g(curve)g(parameters)e(for)i(the)g
(standard)e(<code>IEEE)g(P1363/D8</code>.)p 3387 3159
V 184 3238 a Fa(10)p 270 3272 V 102 w Ff(*)i(These)g(are)g(an)g
(elliptic)e(curve)p 3387 3272 V 184 3351 a Fa(11)p 270
3385 V 102 w Ff(*)i(<code>E:)f(y^2)h(=)g(x^3)g(+)h(ax)f(+)g(b</code>)f
(with)g(its)h(prime)g(modulus)p 3387 3385 V 184 3464
a Fa(12)p 270 3498 V 102 w Ff(*)g(<code>q</code>)d(and)j(its)g
(coefficients)e(<code>a</code>)e(and)p 3387 3498 V 184
3577 a Fa(13)p 270 3610 V 102 w Ff(*)k(<code>b</code>.)d(Further)i(a)h
(point)g(<code>G</code>)d(on)j(the)g(curve,)p 3387 3610
V 184 3690 a Fa(14)p 270 3723 V 102 w Ff(*)g(its)g(order)g(\(that)f(is)
h(the)g(order)f(of)i(the)f(generated)e(subgroup\))g(and)p
3387 3723 V 184 3802 a Fa(15)p 270 3836 V 102 w Ff(*)i(the)g(cofactor)f
(with)g(regard)h(to)g(the)g(number)f(of)h(points)f(on)p
3387 3836 V 184 3915 a Fa(16)p 270 3949 V 102 w Ff(*)h(<code>E</code>.)
d(The)j(field)f(will)h(further)f(be)h(called)f(GF\(q\).)p
3387 3949 V 184 4028 a Fa(17)p 270 4062 V 102 w Ff(*)h(<p>)p
3387 4062 V 184 4141 a Fa(18)p 270 4175 V 102 w Ff(*)g(This)g(class)f
(contains)g(<code>native</code>-meth)o(ods,)41 b(which)46
b(means)p 3387 4175 V 184 4254 a Fa(19)p 270 4288 V 102
w Ff(*)h(that)g(they)g(are)g(implemented)d(in)j(an)h(other)e
(programming)f(language)p 3387 4288 V 184 4367 a Fa(20)p
270 4401 V 102 w Ff(*)i(than)g(<code>JAVA</code>.)c(In)k(this)g(case)f
(it)h(is)h(realized)d(in)p 3387 4401 V 184 4480 a Fa(21)p
270 4514 V 102 w Ff(*)i(<code>C++</code>)d(by)j(using)f(the)p
3387 4514 V 184 4593 a Fa(22)p 270 4627 V 102 w Ff(*)h(<code>JAVA)e
(native)h(interface)g(\(JNI\)</code>.)p 3387 4627 V 184
4706 a Fa(23)p 270 4740 V 102 w Ff(*)p 3387 4740 V 184
4819 a Fa(24)p 270 4852 V 102 w Ff(*)h(@author)f(Birgit)g(Henhapl)p
3387 4852 V 184 4932 a Fa(25)p 270 4965 V 102 w Ff(*)h(@author)f
(Markus)g(Crnjak)p 3387 4965 V 184 5044 a Fa(26)p 270
5078 V 102 w Ff(*/)p 3387 5078 V 184 5157 a Fa(27)p 270
5191 V 54 w Ff(public)g(class)h(LECDomainParametersP136)o(3)p
3387 5191 V 184 5270 a Fa(28)p 270 5304 V 245 w Ff(extends)f
(LiDIAObject)p 3387 5304 V 184 5383 a Fa(29)p 270 5417
V 245 w Ff(implements)f(java.security.spec.Algori)o(thm)o(Para)o(mete)o
(rSp)o(ec)p 3387 5417 V 184 5496 a Fa(30)p 270 5530 V
54 w Ff({)p 3387 5530 V 184 5609 a Fa(31)p 270 5643 V
245 w Ff(/**)p 3387 5643 V 1785 5858 a Fw(71)p eop
%%Page: 72 73
72 72 bop 390 82 a Fw(72)1950 b(ANHANG)31 b(C.)61 b(CDC.LIDIA)304
415 y Fa(32)p 390 449 4 113 v 293 w Ff(*)47 b(holds)g(the)g(value)f(of)
h(the)g(modulus)f(q,)h(that)g(is)g(a)g(prime)g(integer)p
3507 449 V 304 528 a Fa(33)p 390 562 V 293 w Ff(*)p 3507
562 V 304 641 a Fa(34)p 390 675 V 293 w Ff(*)g(@serial)p
3507 675 V 304 754 a Fa(35)p 390 788 V 293 w Ff(*/)p
3507 788 V 304 867 a Fa(36)p 390 901 V 245 w Ff(protected)e(BigInteger)
g(mQ;)p 3507 901 V 304 980 a Fa(37)p 390 1014 V 3507
1014 V 304 1093 a(38)p 390 1126 V 245 w Ff(/**)p 3507
1126 V 304 1205 a Fa(39)p 390 1239 V 293 w Ff(*)i(holds)g(the)g
(coefficient)d(a)k(of)f(the)g(elliptic)e(curve)p 3507
1239 V 304 1318 a Fa(40)p 390 1352 V 293 w Ff(*)i(<code>mE</code>)p
3507 1352 V 304 1431 a Fa(41)p 390 1465 V 293 w Ff(*)p
3507 1465 V 304 1544 a Fa(42)p 390 1578 V 293 w Ff(*)g(@serial)p
3507 1578 V 304 1657 a Fa(43)p 390 1691 V 293 w Ff(*/)p
3507 1691 V 304 1770 a Fa(44)p 390 1804 V 245 w Ff(protected)e
(BigInteger)g(mA;)p 3507 1804 V 304 1883 a Fa(45)p 390
1917 V 3507 1917 V 304 1996 a(46)p 390 2030 V 245 w Ff(/**)p
3507 2030 V 304 2109 a Fa(47)p 390 2143 V 293 w Ff(*)i(holds)g(the)g
(coefficient)d(b)k(of)f(the)g(elliptic)e(curve)p 3507
2143 V 304 2222 a Fa(48)p 390 2256 V 293 w Ff(*)i(<code>mE</code>)p
3507 2256 V 304 2335 a Fa(49)p 390 2368 V 293 w Ff(*)p
3507 2368 V 304 2447 a Fa(50)p 390 2481 V 293 w Ff(*)g(@serial)p
3507 2481 V 304 2560 a Fa(51)p 390 2594 V 293 w Ff(*/)p
3507 2594 V 304 2673 a Fa(52)p 390 2707 V 245 w Ff(protected)e
(BigInteger)g(mB;)p 3507 2707 V 304 2786 a Fa(53)p 390
2820 V 3507 2820 V 304 2899 a(54)p 390 2933 V 245 w Ff(/**)p
3507 2933 V 304 3012 a Fa(55)p 390 3046 V 293 w Ff(*)i(holds)g(the)g
(elliptic)e(curve)h(<code>mE</code>)p 3507 3046 V 304
3125 a Fa(56)p 390 3159 V 293 w Ff(*)p 3507 3159 V 304
3238 a Fa(57)p 390 3272 V 293 w Ff(*)h(@serial)p 3507
3272 V 304 3351 a Fa(58)p 390 3385 V 293 w Ff(*/)p 3507
3385 V 304 3464 a Fa(59)p 390 3498 V 245 w Ff(protected)e(LECCurveGFP)g
(mE;)p 3507 3498 V 304 3577 a Fa(60)p 390 3610 V 3507
3610 V 304 3689 a(61)p 390 3723 V 245 w Ff(/**)p 3507
3723 V 304 3802 a Fa(62)p 390 3836 V 293 w Ff(*)i(holds)g(a)g(curve)f
(point)h(G)g(that)g(generates)e(a)j(subgroup)d(on)p 3507
3836 V 304 3915 a Fa(63)p 390 3949 V 293 w Ff(*)i(<code>mE</code>)d(of)
j(order)f(<code>mR</code>)p 3507 3949 V 304 4028 a Fa(64)p
390 4062 V 293 w Ff(*)p 3507 4062 V 304 4141 a Fa(65)p
390 4175 V 293 w Ff(*)h(@serial)p 3507 4175 V 304 4254
a Fa(66)p 390 4288 V 293 w Ff(*/)p 3507 4288 V 304 4367
a Fa(67)p 390 4401 V 245 w Ff(protected)e(LECPointGFP)g(mG;)p
3507 4401 V 304 4480 a Fa(68)p 390 4514 V 3507 4514 V
304 4593 a(69)p 390 4627 V 245 w Ff(/**)p 3507 4627 V
304 4706 a Fa(70)p 390 4740 V 293 w Ff(*)i(holds)g(a)g(positive)f
(prime)g(integer)g(r,)h(the)g(order)f(of)p 3507 4740
V 304 4819 a Fa(71)p 390 4852 V 293 w Ff(*)h(<code>mG</code>,)d(which)i
(divides)g(the)h(number)f(of)h(points)f(of)p 3507 4852
V 304 4932 a Fa(72)p 390 4965 V 293 w Ff(*)h(<code>mE</code>)p
3507 4965 V 304 5044 a Fa(73)p 390 5078 V 293 w Ff(*)p
3507 5078 V 304 5157 a Fa(74)p 390 5191 V 293 w Ff(*)g(@serial)p
3507 5191 V 304 5270 a Fa(75)p 390 5304 V 293 w Ff(*/)p
3507 5304 V 304 5383 a Fa(76)p 390 5417 V 245 w Ff(protected)e
(BigInteger)g(mR;)p 3507 5417 V 304 5496 a Fa(77)p 390
5530 V 3507 5530 V 304 5609 a(78)p 390 5643 V 245 w Ff(/**)p
3507 5643 V eop
%%Page: 73 74
73 73 bop 270 82 a Fw(C.1.)61 b(LECDomainP)m(arametersP1363)33
b(\(JA)-10 b(V)g(A-Co)s(de\))1051 b(73)184 415 y Fa(79)p
270 449 4 113 v 293 w Ff(*)47 b(holds)g(the)g(cofactor)e(k,)i(where)g
(the)g(product)e(of)j(k)f(and)g(mR)g(is)p 3387 449 V
184 528 a Fa(80)p 270 562 V 293 w Ff(*)g(the)g(number)f(of)h(points)g
(on)g(<code>mE</code>)p 3387 562 V 184 641 a Fa(81)p
270 675 V 293 w Ff(*)p 3387 675 V 184 754 a Fa(82)p 270
788 V 293 w Ff(*)g(@serial)p 3387 788 V 184 867 a Fa(83)p
270 901 V 293 w Ff(*/)p 3387 901 V 184 980 a Fa(84)p
270 1014 V 245 w Ff(protected)e(BigInteger)g(mK;)p 3387
1014 V 184 1093 a Fa(85)p 270 1126 V 3387 1126 V 184
1205 a(86)p 270 1239 V 245 w Ff(/**)p 3387 1239 V 184
1318 a Fa(87)p 270 1352 V 293 w Ff(*)i(holds)g(the)g(first)f
(coordinate)f(of)i(the)g(generator)e(mG)p 3387 1352 V
184 1431 a Fa(88)p 270 1465 V 293 w Ff(*)p 3387 1465
V 184 1544 a Fa(89)p 270 1578 V 293 w Ff(*)i(@serial)p
3387 1578 V 184 1657 a Fa(90)p 270 1691 V 293 w Ff(*/)p
3387 1691 V 184 1770 a Fa(91)p 270 1804 V 245 w Ff(protected)e
(BigInteger)g(mX;)p 3387 1804 V 184 1883 a Fa(92)p 270
1917 V 3387 1917 V 184 1996 a(93)p 270 2030 V 245 w Ff(/**)p
3387 2030 V 184 2109 a Fa(94)p 270 2143 V 293 w Ff(*)i(holds)g(the)g
(second)f(coordinate)f(of)i(the)g(generator)e(mG)p 3387
2143 V 184 2222 a Fa(95)p 270 2256 V 293 w Ff(*)p 3387
2256 V 184 2335 a Fa(96)p 270 2368 V 293 w Ff(*)i(@serial)p
3387 2368 V 184 2447 a Fa(97)p 270 2481 V 293 w Ff(*/)p
3387 2481 V 184 2560 a Fa(98)p 270 2594 V 245 w Ff(protected)e
(BigInteger)g(mY;)p 3387 2594 V 184 2673 a Fa(99)p 270
2707 V 3387 2707 V 154 2786 a(100)p 270 2820 V 245 w
Ff(/**)p 3387 2820 V 154 2899 a Fa(101)p 270 2933 V 293
w Ff(*)i(constructor,)e(that)h(initializes)f(the)p 3387
2933 V 154 3012 a Fa(102)p 270 3046 V 293 w Ff(*)i
(<code>LECDomainParameters)o(P136)o(3</c)o(ode)o(>)42
b(object)k(with)h(the)p 3387 3046 V 154 3125 a Fa(103)p
270 3159 V 293 w Ff(*)g(specified)f(parameter)f(values.)h(At)h(the)g
(moment)f(there)g(is)h(just)p 3387 3159 V 154 3238 a
Fa(104)p 270 3272 V 293 w Ff(*)g(the)g(case)g(of)g(GF\(q\))f
(implemented,)f(where)h(q)i(is)f(a)g(prime)g(number.)p
3387 3272 V 154 3351 a Fa(105)p 270 3385 V 293 w Ff(*)p
3387 3385 V 154 3464 a Fa(106)p 270 3498 V 293 w Ff(*)g(@param)f(q)i
(the)f(bigInteger)e(contains)g(the)i(value)g(for)f(the)h(field)p
3387 3498 V 154 3577 a Fa(107)p 270 3610 V 293 w Ff(*)g(GF\(q\))p
3387 3610 V 154 3689 a Fa(108)p 270 3723 V 293 w Ff(*)g(@param)f(a)i
(the)f(bigInteger)e(contains)g(an)i(elliptic)f(curve)p
3387 3723 V 154 3802 a Fa(109)p 270 3836 V 293 w Ff(*)h(coefficient)p
3387 3836 V 154 3915 a Fa(110)p 270 3949 V 293 w Ff(*)g(@param)f(b)i
(the)f(bigInteger)e(contains)g(an)i(elliptic)f(curve)p
3387 3949 V 154 4028 a Fa(111)p 270 4062 V 293 w Ff(*)h(coefficient)p
3387 4062 V 154 4141 a Fa(112)p 270 4175 V 293 w Ff(*)g(@param)f(g)i
(the)f(lECPointGFP)d(contains)i(a)h(point)g(that)f(generates)p
3387 4175 V 154 4254 a Fa(113)p 270 4288 V 293 w Ff(*)h(a)h(subgroup)d
(on)i(the)g(curve)p 3387 4288 V 154 4367 a Fa(114)p 270
4401 V 293 w Ff(*)g(@param)f(r)i(the)f(bigInteger)e(contains)g(the)i
(order)g(of)g(the)p 3387 4401 V 154 4480 a Fa(115)p 270
4514 V 293 w Ff(*)g(generator)p 3387 4514 V 154 4593
a Fa(116)p 270 4627 V 293 w Ff(*)g(@param)f(k)i(the)f(bigInteger)e
(contains)g(the)i(cofactor)f(of)h(the)p 3387 4627 V 154
4706 a Fa(117)p 270 4740 V 293 w Ff(*)g(generator's)e(order)p
3387 4740 V 154 4819 a Fa(118)p 270 4852 V 293 w Ff(*)p
3387 4852 V 154 4932 a Fa(119)p 270 4965 V 293 w Ff(*)i(@throws)f
(LiDIAError)p 3387 4965 V 154 5044 a Fa(120)p 270 5078
V 293 w Ff(*/)p 3387 5078 V 154 5157 a Fa(121)p 270 5191
V 245 w Ff(public)g(LECDomainParametersP1363\()o(BigI)o(nte)o(ger)41
b(q,BigInteger)k(a,)p 3387 5191 V 154 5270 a Fa(122)p
270 5304 V 341 w Ff(BigInteger)g(b,LECPointGFP)f(g,BigInteger)g
(r,BigInteger)h(k\))p 3387 5304 V 154 5383 a Fa(123)p
270 5417 V 436 w Ff(throws)h(LiDIAError)p 3387 5417 V
154 5496 a Fa(124)p 270 5530 V 245 w Ff({)p 3387 5530
V 154 5609 a Fa(125)p 270 5643 V 436 w Ff(assign\(q,a,b,g,r,k\);)p
3387 5643 V eop
%%Page: 74 75
74 74 bop 390 82 a Fw(74)1950 b(ANHANG)31 b(C.)61 b(CDC.LIDIA)274
415 y Fa(126)p 390 449 4 113 v 245 w Ff(})p 3507 449
V 274 528 a Fa(127)p 390 562 V 3507 562 V 274 641 a(128)p
390 675 V 245 w Ff(/**)p 3507 675 V 274 754 a Fa(129)p
390 788 V 293 w Ff(*)47 b(constructor,)e(that)h(initializes)f(the)p
3507 788 V 274 867 a Fa(130)p 390 901 V 293 w Ff(*)i
(<code>LECDomainParameters)o(P136)o(3</c)o(ode)o(>)42
b(object)k(for)h(a)p 3507 901 V 274 980 a Fa(131)p 390
1014 V 293 w Ff(*)g(certain)f(strength.)f(At)j(the)e(moment)h(there)f
(is)h(just)g(the)g(case)f(of)p 3507 1014 V 274 1093 a
Fa(132)p 390 1126 V 293 w Ff(*)h(GF\(q\))g(implemented,)d(where)i(q)i
(is)f(a)h(prime)e(number.)p 3507 1126 V 274 1205 a Fa(133)p
390 1239 V 293 w Ff(*)p 3507 1239 V 274 1318 a Fa(134)p
390 1352 V 293 w Ff(*)h(@param)f(field)h(the)g(lBigInt)e(specifies)h
(the)h(field)f(\(1)h(for)p 3507 1352 V 274 1431 a Fa(135)p
390 1465 V 293 w Ff(*)g(GF\(q\)\))p 3507 1465 V 274 1544
a Fa(136)p 390 1578 V 293 w Ff(*)g(@param)f(bitsize)g(the)h(lBigInt)f
(specifies)f(the)i(bitsize)f(of)h(the)p 3507 1578 V 274
1657 a Fa(137)p 390 1691 V 293 w Ff(*)g(order)g(of)g(the)g(subgroup)p
3507 1691 V 274 1770 a Fa(138)p 390 1804 V 293 w Ff(*)g(@param)f
(percentage)f(the)i(lBigInt)f(specifies)f(the)i(percentage)e(of)p
3507 1804 V 274 1883 a Fa(139)p 390 1917 V 293 w Ff(*)i(the)g(cofactor)
p 3507 1917 V 274 1996 a Fa(140)p 390 2030 V 293 w Ff(*)g(@param)f
(info)h(the)g(lBigInt)f(specifies)f(the)i(amount)f(of)h(verbosed)p
3507 2030 V 274 2109 a Fa(141)p 390 2143 V 293 w Ff(*)g(messages)f(\(1)
h(for)g(nothing,)e(10)j(for)f(some)f(messages)g(and)h(100)g(for)p
3507 2143 V 274 2222 a Fa(142)p 390 2256 V 293 w Ff(*)g(all)g(possible)
f(ones\))p 3507 2256 V 274 2335 a Fa(143)p 390 2368 V
293 w Ff(*)p 3507 2368 V 274 2447 a Fa(144)p 390 2481
V 293 w Ff(*)h(@throws)f(LiDIAError)p 3507 2481 V 274
2560 a Fa(145)p 390 2594 V 293 w Ff(*/)p 3507 2594 V
274 2673 a Fa(146)p 390 2707 V 245 w Ff(public)g
(LECDomainParametersP1363\()o(LBig)o(Int)41 b(field,)p
3507 2707 V 274 2786 a Fa(147)p 390 2820 V 341 w Ff(LBigInt)k
(bitsize,LBigInt)f(percentage,LBigInt)f(info\))p 3507
2820 V 274 2899 a Fa(148)p 390 2933 V 436 w Ff(throws)j(LiDIAError)p
3507 2933 V 274 3012 a Fa(149)p 390 3046 V 245 w Ff({)p
3507 3046 V 274 3125 a Fa(150)p 390 3159 V 436 w Ff
(assign\(field,bitsize,per)o(cent)o(age,)o(inf)o(o\);)p
3507 3159 V 274 3238 a Fa(151)p 390 3272 V 245 w Ff(})p
3507 3272 V 274 3351 a Fa(152)p 390 3385 V 3507 3385
V 274 3464 a(153)p 390 3498 V 245 w Ff(/**)p 3507 3498
V 274 3577 a Fa(154)p 390 3610 V 293 w Ff(*)h(sets)g(the)g(domain)f
(parameters.)f(At)i(the)g(moment)f(there)g(is)h(just)p
3507 3610 V 274 3689 a Fa(155)p 390 3723 V 293 w Ff(*)g(the)g(case)g
(of)g(GF\(q\))f(implemented,)f(where)h(q)i(is)f(a)g(prime)g(number.)p
3507 3723 V 274 3802 a Fa(156)p 390 3836 V 293 w Ff(*)p
3507 3836 V 274 3915 a Fa(157)p 390 3949 V 293 w Ff(*)g(@param)f(q)i
(the)f(bigInteger)e(contains)g(the)i(value)g(for)f(the)h(field)p
3507 3949 V 274 4028 a Fa(158)p 390 4062 V 293 w Ff(*)g(GF\(q\))p
3507 4062 V 274 4141 a Fa(159)p 390 4175 V 293 w Ff(*)g(@param)f(a)i
(the)f(bigInteger)e(contains)g(an)i(elliptic)f(curve)p
3507 4175 V 274 4254 a Fa(160)p 390 4288 V 293 w Ff(*)h(coefficient)p
3507 4288 V 274 4367 a Fa(161)p 390 4401 V 293 w Ff(*)g(@param)f(b)i
(the)f(bigInteger)e(contains)g(an)i(elliptic)f(curve)p
3507 4401 V 274 4480 a Fa(162)p 390 4514 V 293 w Ff(*)h(coefficient)p
3507 4514 V 274 4593 a Fa(163)p 390 4627 V 293 w Ff(*)g(@param)f(g)i
(the)f(lECPointGFP)d(contains)i(a)h(point)g(that)f(generates)p
3507 4627 V 274 4706 a Fa(164)p 390 4740 V 293 w Ff(*)h(a)h(subgroup)d
(on)i(the)g(curve)p 3507 4740 V 274 4819 a Fa(165)p 390
4852 V 293 w Ff(*)g(@param)f(r)i(the)f(bigInteger)e(contains)g(the)i
(order)g(of)g(the)p 3507 4852 V 274 4932 a Fa(166)p 390
4965 V 293 w Ff(*)g(generator)p 3507 4965 V 274 5044
a Fa(167)p 390 5078 V 293 w Ff(*)g(@param)f(k)i(the)f(bigInteger)e
(contains)g(the)i(cofactor)f(of)h(the)p 3507 5078 V 274
5157 a Fa(168)p 390 5191 V 293 w Ff(*)g(generator's)e(order)p
3507 5191 V 274 5270 a Fa(169)p 390 5304 V 293 w Ff(*)p
3507 5304 V 274 5383 a Fa(170)p 390 5417 V 293 w Ff(*)i(@throws)f
(LiDIAError)p 3507 5417 V 274 5496 a Fa(171)p 390 5530
V 293 w Ff(*/)p 3507 5530 V 274 5609 a Fa(172)p 390 5643
V 245 w Ff(public)g(void)h(assign\(BigInteger)c(q,BigInteger)h
(a,BigInteger)h(b,)p 3507 5643 V eop
%%Page: 75 76
75 75 bop 270 82 a Fw(C.1.)61 b(LECDomainP)m(arametersP1363)33
b(\(JA)-10 b(V)g(A-Co)s(de\))1051 b(75)154 415 y Fa(173)p
270 449 4 113 v 341 w Ff(LECPointGFP)44 b(g,BigInteger)h(r,BigInteger)f
(k\))p 3387 449 V 154 528 a Fa(174)p 270 562 V 436 w
Ff(throws)i(LiDIAError)p 3387 562 V 154 641 a Fa(175)p
270 675 V 245 w Ff({)p 3387 675 V 154 754 a Fa(176)p
270 788 V 436 w Ff(//)h(sets)g(the)g(fields)p 3387 788
V 154 867 a Fa(177)p 270 901 V 436 w Ff(mQ)g(=)h(q;)p
3387 901 V 154 980 a Fa(178)p 270 1014 V 436 w Ff(mA)f(=)h(a;)p
3387 1014 V 154 1093 a Fa(179)p 270 1126 V 436 w Ff(mB)f(=)h(b;)p
3387 1126 V 154 1205 a Fa(180)p 270 1239 V 436 w Ff(mG)f(=)h(g;)p
3387 1239 V 154 1318 a Fa(181)p 270 1352 V 436 w Ff(mR)f(=)h(r;)p
3387 1352 V 154 1431 a Fa(182)p 270 1465 V 436 w Ff(mK)f(=)h(k;)p
3387 1465 V 154 1544 a Fa(183)p 270 1578 V 436 w Ff(mE)f(=)h(new)f
(LECCurveGFP\(new)c(LBigInt\(a\),new)h(LBigInt\(b\),)p
3387 1578 V 154 1657 a Fa(184)p 270 1691 V 531 w Ff(new)j
(LBigInt\(q\)\);)p 3387 1691 V 154 1770 a Fa(185)p 270
1804 V 245 w Ff(})p 3387 1804 V 154 1883 a Fa(186)p 270
1917 V 3387 1917 V 154 1996 a(187)p 270 2030 V 245 w
Ff(/**)p 3387 2030 V 154 2109 a Fa(188)p 270 2143 V 293
w Ff(*)g(sets)g(the)g(domain)f(parameters,)f(by)i(using)f(the)h
(specified)p 3387 2143 V 154 2222 a Fa(189)p 270 2256
V 293 w Ff(*)g(arguments)f(for)g(their)h(generation.)e(After)h(the)h
(usage)f(of)h(the)p 3387 2256 V 154 2335 a Fa(190)p 270
2368 V 293 w Ff(*)g(corresponding)d(LiDIA-class)h(that)i(one)g(will)f
(be)i(destroyed.)d(At)p 3387 2368 V 154 2447 a Fa(191)p
270 2481 V 293 w Ff(*)i(the)g(moment)f(there)h(is)g(just)f(the)h(case)g
(of)g(GF\(q\))g(implemented,)p 3387 2481 V 154 2560 a
Fa(192)p 270 2594 V 293 w Ff(*)g(where)g(q)g(is)g(a)h(prime)e(number.)p
3387 2594 V 154 2673 a Fa(193)p 270 2707 V 293 w Ff(*)p
3387 2707 V 154 2786 a Fa(194)p 270 2820 V 293 w Ff(*)h(@param)f(field)
h(the)g(lBigInt)e(specifies)h(the)h(field)f(\(1)h(for)p
3387 2820 V 154 2899 a Fa(195)p 270 2933 V 293 w Ff(*)g(GF\(q\)\))p
3387 2933 V 154 3012 a Fa(196)p 270 3046 V 293 w Ff(*)g(@param)f
(bitsize)g(the)h(lBigInt)f(specifies)f(the)i(bitsize)f(of)h(the)p
3387 3046 V 154 3125 a Fa(197)p 270 3159 V 293 w Ff(*)g(order)g(of)g
(the)g(subgroup)p 3387 3159 V 154 3238 a Fa(198)p 270
3272 V 293 w Ff(*)g(@param)f(percentage)f(the)i(lBigInt)f(specifies)f
(the)i(percentage)e(of)p 3387 3272 V 154 3351 a Fa(199)p
270 3385 V 293 w Ff(*)i(the)g(cofactor)p 3387 3385 V
154 3464 a Fa(200)p 270 3498 V 293 w Ff(*)g(@param)f(info)h(the)g
(lBigInt)f(specifies)f(the)i(amount)f(of)h(verbosed)p
3387 3498 V 154 3577 a Fa(201)p 270 3610 V 293 w Ff(*)g(messages)f(\(1)
h(for)g(nothing,)e(10)j(for)f(some)f(messages)g(and)h(100)g(for)p
3387 3610 V 154 3689 a Fa(202)p 270 3723 V 293 w Ff(*)g(all)g(possible)
f(ones\))p 3387 3723 V 154 3802 a Fa(203)p 270 3836 V
293 w Ff(*)p 3387 3836 V 154 3915 a Fa(204)p 270 3949
V 293 w Ff(*)h(@throws)f(LiDIAError)p 3387 3949 V 154
4028 a Fa(205)p 270 4062 V 293 w Ff(*/)p 3387 4062 V
154 4141 a Fa(206)p 270 4175 V 245 w Ff(public)g(void)h
(assign\(LBigInt)d(field,LBigInt)g(bitsize,)p 3387 4175
V 154 4254 a Fa(207)p 270 4288 V 341 w Ff(LBigInt)h(percentage,LBigInt)
e(info\))p 3387 4288 V 154 4367 a Fa(208)p 270 4401 V
436 w Ff(throws)j(LiDIAError)p 3387 4401 V 154 4480 a
Fa(209)p 270 4514 V 245 w Ff({)p 3387 4514 V 154 4593
a Fa(210)p 270 4627 V 436 w Ff(//)h(creates)f(the)h(parameters)p
3387 4627 V 154 4706 a Fa(211)p 270 4740 V 436 w Ff
(generateParameters\(field)o(,bit)o(size)o(,pe)o(rcen)o(tage)o(,in)o
(fo\);)p 3387 4740 V 154 4819 a Fa(212)p 270 4852 V 3387
4852 V 154 4932 a(213)p 270 4965 V 436 w Ff(//)g(sets)g(the)g(fields)f
(with)g(the)h(accessed)f(parameters)p 3387 4965 V 154
5044 a Fa(214)p 270 5078 V 436 w Ff(mQ)h(=)h
(LgetQ\(\).toBigInteger\(\);)p 3387 5078 V 154 5157 a
Fa(215)p 270 5191 V 436 w Ff(mA)f(=)h(LgetA\(\).toBigInteger\(\);)p
3387 5191 V 154 5270 a Fa(216)p 270 5304 V 436 w Ff(mB)f(=)h
(LgetB\(\).toBigInteger\(\);)p 3387 5304 V 154 5383 a
Fa(217)p 270 5417 V 436 w Ff(mR)f(=)h(LgetR\(\).toBigInteger\(\);)p
3387 5417 V 154 5496 a Fa(218)p 270 5530 V 436 w Ff(mK)f(=)h
(LgetK\(\).toBigInteger\(\);)p 3387 5530 V 154 5609 a
Fa(219)p 270 5643 V 436 w Ff(mX)f(=)h(LgetX\(\).toBigInteger\(\);)p
3387 5643 V eop
%%Page: 76 77
76 76 bop 390 82 a Fw(76)1950 b(ANHANG)31 b(C.)61 b(CDC.LIDIA)274
415 y Fa(220)p 390 449 4 113 v 436 w Ff(mY)47 b(=)h
(LgetY\(\).toBigInteger\(\);)p 3507 449 V 274 528 a Fa(221)p
390 562 V 436 w Ff(mE)f(=)h(new)f(LECCurveGFP\(new)c(LBigInt\(mA\),new)
h(LBigInt\(mB\),)p 3507 562 V 274 641 a Fa(222)p 390
675 V 531 w Ff(new)j(LBigInt\(mQ\)\);)p 3507 675 V 274
754 a Fa(223)p 390 788 V 436 w Ff(mG)g(=)h(new)f(LECPointGFP\(new)c
(LBigInt\(mX\),new)h(LBigInt\(mY\),mE,)p 3507 788 V 274
867 a Fa(224)p 390 901 V 531 w Ff(new)j(LBigInt\(mQ\)\);)p
3507 901 V 274 980 a Fa(225)p 390 1014 V 3507 1014 V
274 1093 a(226)p 390 1126 V 436 w Ff(//)g(destroys)f(the)h
(corresponding)d(LiDIA-object)p 3507 1126 V 274 1205
a Fa(227)p 390 1239 V 436 w Ff(finalize\(\);)p 3507 1239
V 274 1318 a Fa(228)p 390 1352 V 245 w Ff(})p 3507 1352
V 274 1431 a Fa(229)p 390 1465 V 3507 1465 V 274 1544
a(230)p 390 1578 V 245 w Ff(/**)p 3507 1578 V 274 1657
a Fa(231)p 390 1691 V 293 w Ff(*)j(returns)f(the)h(value)f(of)i(the)e
(modulus)g(q,)h(that)g(is)g(a)h(prime)p 3507 1691 V 274
1770 a Fa(232)p 390 1804 V 293 w Ff(*)f(integer)p 3507
1804 V 274 1883 a Fa(233)p 390 1917 V 293 w Ff(*/)p 3507
1917 V 274 1996 a Fa(234)p 390 2030 V 245 w Ff(public)f(BigInteger)f
(getQ\(\){)p 3507 2030 V 274 2109 a Fa(235)p 390 2143
V 436 w Ff(return)h(mQ;)p 3507 2143 V 274 2222 a Fa(236)p
390 2256 V 245 w Ff(})p 3507 2256 V 274 2335 a Fa(237)p
390 2368 V 3507 2368 V 274 2447 a(238)p 390 2481 V 245
w Ff(/**)p 3507 2481 V 274 2560 a Fa(239)p 390 2594 V
293 w Ff(*)h(returns)f(the)h(coefficient)e(a)i(of)g(the)g(curve)p
3507 2594 V 274 2673 a Fa(240)p 390 2707 V 293 w Ff(*/)p
3507 2707 V 274 2786 a Fa(241)p 390 2820 V 245 w Ff(public)f
(BigInteger)f(getA\(\){)p 3507 2820 V 274 2899 a Fa(242)p
390 2933 V 436 w Ff(return)h(mA;)p 3507 2933 V 274 3012
a Fa(243)p 390 3046 V 245 w Ff(})p 3507 3046 V 274 3125
a Fa(244)p 390 3159 V 3507 3159 V 274 3238 a(245)p 390
3272 V 245 w Ff(/**)p 3507 3272 V 274 3351 a Fa(246)p
390 3385 V 293 w Ff(*)h(returns)f(the)h(coefficient)e(b)i(of)g(the)g
(curve)p 3507 3385 V 274 3464 a Fa(247)p 390 3498 V 293
w Ff(*/)p 3507 3498 V 274 3577 a Fa(248)p 390 3610 V
245 w Ff(public)f(BigInteger)f(getB\(\){)p 3507 3610
V 274 3689 a Fa(249)p 390 3723 V 436 w Ff(return)h(mB;)p
3507 3723 V 274 3802 a Fa(250)p 390 3836 V 245 w Ff(})p
3507 3836 V 274 3915 a Fa(251)p 390 3949 V 3507 3949
V 274 4028 a(252)p 390 4062 V 245 w Ff(/**)p 3507 4062
V 274 4141 a Fa(253)p 390 4175 V 293 w Ff(*)h(returns)f(the)h(curve)f
(point)h(that)f(generates)g(the)h(subgroup)p 3507 4175
V 274 4254 a Fa(254)p 390 4288 V 293 w Ff(*/)p 3507 4288
V 274 4367 a Fa(255)p 390 4401 V 245 w Ff(public)f(LECPointGFP)f
(getG\(\){)p 3507 4401 V 274 4480 a Fa(256)p 390 4514
V 436 w Ff(return)h(mG;)p 3507 4514 V 274 4593 a Fa(257)p
390 4627 V 245 w Ff(})p 3507 4627 V 274 4706 a Fa(258)p
390 4740 V 3507 4740 V 274 4819 a(259)p 390 4852 V 245
w Ff(/**)p 3507 4852 V 274 4932 a Fa(260)p 390 4965 V
293 w Ff(*)h(returns)f(the)h(order)f(of)i(the)e(generator)p
3507 4965 V 274 5044 a Fa(261)p 390 5078 V 293 w Ff(*/)p
3507 5078 V 274 5157 a Fa(262)p 390 5191 V 245 w Ff(public)g
(BigInteger)f(getR\(\){)p 3507 5191 V 274 5270 a Fa(263)p
390 5304 V 436 w Ff(return)h(mR;)p 3507 5304 V 274 5383
a Fa(264)p 390 5417 V 245 w Ff(})p 3507 5417 V 274 5496
a Fa(265)p 390 5530 V 3507 5530 V 274 5609 a(266)p 390
5643 V 245 w Ff(/**)p 3507 5643 V eop
%%Page: 77 78
77 77 bop 270 82 a Fw(C.1.)61 b(LECDomainP)m(arametersP1363)33
b(\(JA)-10 b(V)g(A-Co)s(de\))1051 b(77)154 415 y Fa(267)p
270 449 4 113 v 293 w Ff(*)47 b(returns)f(the)h(cofactor)f(of)h(the)g
(generator's)d(order)p 3387 449 V 154 528 a Fa(268)p
270 562 V 293 w Ff(*/)p 3387 562 V 154 641 a Fa(269)p
270 675 V 293 w Ff(public)i(BigInteger)f(getK\(\){)p
3387 675 V 154 754 a Fa(270)p 270 788 V 436 w Ff(return)h(mK;)p
3387 788 V 154 867 a Fa(271)p 270 901 V 245 w Ff(})p
3387 901 V 154 980 a Fa(272)p 270 1014 V 3387 1014 V
154 1093 a(273)p 270 1126 V 245 w Ff(/**)p 3387 1126
V 154 1205 a Fa(274)p 270 1239 V 293 w Ff(*)h(returns)f(the)h(elliptic)
f(curve)p 3387 1239 V 154 1318 a Fa(275)p 270 1352 V
293 w Ff(*/)p 3387 1352 V 154 1431 a Fa(276)p 270 1465
V 245 w Ff(public)g(LECCurveGFP)f(getCurve\(\))g({)p
3387 1465 V 154 1544 a Fa(277)p 270 1578 V 436 w Ff(return)h(mE;)p
3387 1578 V 154 1657 a Fa(278)p 270 1691 V 245 w Ff(})p
3387 1691 V 154 1770 a Fa(279)p 270 1804 V 3387 1804
V 154 1883 a(280)p 270 1917 V 245 w Ff(/**)p 3387 1917
V 154 1996 a Fa(281)p 270 2030 V 293 w Ff(*)h(initializes)e(static)h
(native)g(resources.)f(The)i(method)f(is)h(called)p 3387
2030 V 154 2109 a Fa(282)p 270 2143 V 293 w Ff(*)g(automatically)d
(before)j(the)f(constructor.)p 3387 2143 V 154 2222 a
Fa(283)p 270 2256 V 293 w Ff(*/)p 3387 2256 V 154 2335
a Fa(284)p 270 2368 V 245 w Ff(public)g(static)g(native)g(void)h
(staticDPInit\(\);)p 3387 2368 V 154 2447 a Fa(285)p
270 2481 V 3387 2481 V 154 2560 a(286)p 270 2594 V 245
w Ff(/**)p 3387 2594 V 154 2673 a Fa(287)p 270 2707 V
293 w Ff(*)g(initializes)e(nonstatic)g(native)h(resources.)f(The)i
(method)f(is)p 3387 2707 V 154 2786 a Fa(288)p 270 2820
V 293 w Ff(*)h(called)f(automatically)f(before)h(the)h(constructor.)p
3387 2820 V 154 2899 a Fa(289)p 270 2933 V 293 w Ff(*/)p
3387 2933 V 154 3012 a Fa(290)p 270 3046 V 245 w Ff(protected)e(native)
h(void)h(init\(\);)p 3387 3046 V 154 3125 a Fa(291)p
270 3159 V 3387 3159 V 154 3238 a(292)p 270 3272 V 245
w Ff(/**)p 3387 3272 V 154 3351 a Fa(293)p 270 3385 V
293 w Ff(*)g(frees)g(the)g(native)f(resources.)f(The)i(method)f(is)h
(called)p 3387 3385 V 154 3464 a Fa(294)p 270 3498 V
293 w Ff(*)g(automatically)d(by)k(the)f(garbage)e(collector.)p
3387 3498 V 154 3577 a Fa(295)p 270 3610 V 293 w Ff(*/)p
3387 3610 V 154 3689 a Fa(296)p 270 3723 V 245 w Ff(protected)g(native)
h(void)h(finalize\(\);)p 3387 3723 V 154 3802 a Fa(297)p
270 3836 V 3387 3836 V 154 3915 a(298)p 270 3949 V 245
w Ff(/**)p 3387 3949 V 154 4028 a Fa(299)p 270 4062 V
293 w Ff(*)g(generates)f(the)g(domain)h(parameters,)d(by)j(using)g(the)
g(specified)p 3387 4062 V 154 4141 a Fa(300)p 270 4175
V 293 w Ff(*)g(arguments)f(for)g(their)h(generation.)e(At)i(the)g
(moment)f(there)g(is)p 3387 4175 V 154 4254 a Fa(301)p
270 4288 V 293 w Ff(*)h(just)g(the)g(case)f(of)i(GF\(q\))e
(implemented,)e(where)j(q)g(is)g(a)h(prime)p 3387 4288
V 154 4367 a Fa(302)p 270 4401 V 293 w Ff(*)f(number.)p
3387 4401 V 154 4480 a Fa(303)p 270 4514 V 293 w Ff(*)p
3387 4514 V 154 4593 a Fa(304)p 270 4627 V 293 w Ff(*)g(@param)f(field)
h(the)g(lBigInt)e(specifies)h(the)h(field)f(\(1)h(for)p
3387 4627 V 154 4706 a Fa(305)p 270 4740 V 293 w Ff(*)g(GF\(q\)\))p
3387 4740 V 154 4819 a Fa(306)p 270 4852 V 293 w Ff(*)g(@param)f
(bitsize)g(the)h(lBigInt)f(specifies)f(the)i(bitsize)f(of)h(the)p
3387 4852 V 154 4932 a Fa(307)p 270 4965 V 293 w Ff(*)g(order)g(of)g
(the)g(subgroup)p 3387 4965 V 154 5044 a Fa(308)p 270
5078 V 293 w Ff(*)g(@param)f(percentage)f(the)i(lBigInt)f(specifies)f
(the)i(percentage)e(of)p 3387 5078 V 154 5157 a Fa(309)p
270 5191 V 293 w Ff(*)i(the)g(cofactor)p 3387 5191 V
154 5270 a Fa(310)p 270 5304 V 293 w Ff(*)g(@param)f(info)h(the)g
(lBigInt)f(specifies)f(the)i(amount)f(of)h(verbosed)p
3387 5304 V 154 5383 a Fa(311)p 270 5417 V 293 w Ff(*)g(messages)f(\(1)
h(for)g(nothing,)e(10)j(for)f(some)f(messages)g(and)h(100)g(for)p
3387 5417 V 154 5496 a Fa(312)p 270 5530 V 293 w Ff(*)g(all)g(possible)
f(ones\))p 3387 5530 V 154 5609 a Fa(313)p 270 5643 V
293 w Ff(*/)p 3387 5643 V eop
%%Page: 78 79
78 78 bop 390 82 a Fw(78)1950 b(ANHANG)31 b(C.)61 b(CDC.LIDIA)274
415 y Fa(314)p 390 449 4 113 v 245 w Ff(public)46 b(native)g(void)h
(generateParameters\(LBigI)o(nt)42 b(field,)p 3507 449
V 274 528 a Fa(315)p 390 562 V 341 w Ff(LBigInt)j(bitsize,LBigInt)f
(percentage,LBigInt)f(info\);)p 3507 562 V 274 641 a
Fa(316)p 390 675 V 3507 675 V 274 754 a(317)p 390 788
V 245 w Ff(/**)p 3507 788 V 274 867 a Fa(318)p 390 901
V 293 w Ff(*)k(accesses)f(the)h(value)f(of)h(the)g(modulus)f(q,)h(that)
g(is)g(a)g(prime)p 3507 901 V 274 980 a Fa(319)p 390
1014 V 293 w Ff(*)g(integer)p 3507 1014 V 274 1093 a
Fa(320)p 390 1126 V 293 w Ff(*/)p 3507 1126 V 274 1205
a Fa(321)p 390 1239 V 245 w Ff(public)f(native)g(LBigInt)g(LgetQ\(\);)p
3507 1239 V 274 1318 a Fa(322)p 390 1352 V 3507 1352
V 274 1431 a(323)p 390 1465 V 245 w Ff(/**)p 3507 1465
V 274 1544 a Fa(324)p 390 1578 V 293 w Ff(*)h(accesses)f(the)h
(coefficient)d(a)k(of)f(the)g(curve)p 3507 1578 V 274
1657 a Fa(325)p 390 1691 V 293 w Ff(*/)p 3507 1691 V
274 1770 a Fa(326)p 390 1804 V 245 w Ff(public)f(native)g(LBigInt)g
(LgetA\(\);)p 3507 1804 V 274 1883 a Fa(327)p 390 1917
V 3507 1917 V 274 1996 a(328)p 390 2030 V 245 w Ff(/**)p
3507 2030 V 274 2109 a Fa(329)p 390 2143 V 293 w Ff(*)h(accesses)f(the)
h(coefficient)d(b)k(of)f(the)g(curve)p 3507 2143 V 274
2222 a Fa(330)p 390 2256 V 293 w Ff(*/)p 3507 2256 V
274 2335 a Fa(331)p 390 2368 V 245 w Ff(public)f(native)g(LBigInt)g
(LgetB\(\);)p 3507 2368 V 274 2447 a Fa(332)p 390 2481
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%%Page: 79 80
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%%Page: 80 81
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%%Page: 81 82
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%%Page: 82 83
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(jpResult;)p 3507 3385 V 274 3464 a Fa(354)p 390 3498
V 150 w Ff(bigint)h(*b_Result)f(=)i(new)g(bigint)f
(\(\(\(EC_domain_parameters_P1)o(363)41 b(*\))p 3507
3498 V 274 3577 a Fa(355)p 390 3610 V 245 w Ff
(nThis.p\)->get_y\(\).mantiss)o(a\(\))o(\);)p 3507 3610
V 274 3689 a Fa(356)p 390 3723 V 150 w Ff(jpResult.p)k(=)i(b_Result;)p
3507 3723 V 274 3802 a Fa(357)p 390 3836 V 3507 3836
V 274 3915 a(358)p 390 3949 V 150 w Ff(//)g(creates)f(the)h(return)f
(object)p 3507 3949 V 274 4028 a Fa(359)p 390 4062 V
150 w Ff(Result)g(=)h(env->NewObject)d(\(LBigInt,LBigInt_LBigInt,)o
(jpR)o(esul)o(t.l,)p 3507 4062 V 274 4141 a Fa(360)p
390 4175 V 245 w Ff(JNI_TRUE\);)p 3507 4175 V 274 4254
a Fa(361)p 390 4288 V 150 w Ff(if)j(\(handleJavaException)42
b(\(env\)\))p 3507 4288 V 274 4367 a Fa(362)p 390 4401
V 245 w Ff(return)k(NULL;)p 3507 4401 V 274 4480 a Fa(363)p
390 4514 V 3507 4514 V 274 4593 a(364)p 390 4627 V 150
w Ff(//)h(and)g(back)p 3507 4627 V 274 4706 a Fa(365)p
390 4740 V 150 w Ff(return)f(Result;)p 3507 4740 V 274
4819 a Fa(366)p 390 4852 V 54 w Ff(})p 3507 4852 V 390
4877 4 25 v 3507 4877 V 390 4880 3120 4 v eop
%%Page: 87 88
87 87 bop 270 1084 a Fv(Literaturv)-6 b(erzeic)g(hnis)270
1534 y Fw([Buc99])81 b Ft(Buchmann,)38 b(Johannes)p Fw(:)33
b Fu(Einf)1839 1538 y(\177)1839 1534 y(uhrung)k(in)g(die)h(Krypto)-5
b(gr)g(aphie)p Fw(.)58 b(Springer)646 1647 y(V)-8 b(erlag,)32
b(Berlin,)d(1999.)270 1853 y([Bur94])85 b Ft(Bur)-6 b(ton,)35
b(D)m(a)-8 b(vid)36 b(M.)p Fw(:)c Fu(Elementary)j(Numb)-5
b(er)33 b(The)-5 b(ory)p Fw(.)47 b(W)m(CB)32 b(Publishers,)646
1966 y(Dubuque,)e(USA,)g(3.)h(Auflage,)g(1994.)270 2172
y([Ham98])47 b Ft(Hamd)n(y,)28 b(Safua)-6 b(t)p Fw(:)23
b Fu(A)n(nnotationes)28 b(De)f(R)-5 b(ationibus)29 b(Cryptolo)-5
b(giae)p Fw(.)34 b(Diplom-)646 2285 y(arb)s(eit,)21 b(Univ)m(ersit)1291
2289 y(\177)1291 2285 y(at)g(Ham)m(burg,)g(F)-8 b(ac)m(h)m(b)s(ereic)m
(h)22 b(Informatik,)f(No)m(v)m(em)m(b)s(er)h(1998.)270
2491 y([Hen99])77 b Ft(Henhapl,)39 b(Bir)n(git)p Fw(:)d
Fu(Digitales)i(Signier)-5 b(en)38 b(mit)f(el)5 b(liptischen)39
b(Kurven)3224 2495 y(\177)3224 2491 y(ub)-5 b(er)646
2604 y(Primk)889 2608 y(\177)890 2604 y(orp)g(ern)p Fw(.)32
b(Diplomarb)s(eit,)22 b(T)-8 b(ec)m(hnisc)m(he)24 b(Univ)m(ersit)2593
2608 y(\177)2593 2604 y(at)f(Darmstadt,)i(F)-8 b(ac)m(h-)646
2717 y(b)s(ereic)m(h)30 b(Mathematik,)h(Oktob)s(er)f(1999.)270
2923 y([Hus87])81 b Ft(Husem)941 2919 y(\177)937 2923
y(oller,)44 b(D)m(ale)p Fw(:)e Fu(El)5 b(liptic)43 b(Curves)p
Fw(.)75 b(Springer)39 b(V)-8 b(erlag,)43 b(New)f(Y)-8
b(ork,)646 3036 y(1987.)270 3242 y([ISD98])83 b Ft(IEEE)37
b(St)-6 b(and)n(ards)33 b(Dep)-6 b(ar)g(tment,)33 b(Pisca)-6
b(t)g(a)e(w)g(a)i(y,)34 b(USA)p Fw(:)d Fu(IEEE)h(P1363)646
3355 y(/)40 b(D8)f(Standar)-5 b(d)42 b(Sp)-5 b(e)g(ci\014c)g(ations)41
b(for)f(Public)e(Key)h(Crypto)-5 b(gr)g(aphy)p Fw(,)41
b(Oktob)s(er)646 3468 y(1998.)270 3674 y([Kn)m(u98])66
b Ft(Knudsen,)39 b(Jona)-6 b(than)p Fw(:)34 b Fu(JA)-9
b(V)g(A)36 b(Crypto)-5 b(gr)g(aphy)p Fw(.)62 b(O'Reilly)34
b(&)i(Asso)s(ciates,)646 3787 y(Sebastop)s(ol,)30 b(USA,)h(1998.)270
3994 y([Kob94])69 b Ft(K)m(oblitz,)53 b(Neal)p Fw(:)48
b Fu(A)h(Course)i(in)e(Numb)-5 b(er)50 b(The)-5 b(ory)51
b(and)g(Crypto)-5 b(gr)g(aphy)p Fw(.)646 4106 y(Springer)28
b(V)-8 b(erlag,)32 b(New)e(Y)-8 b(ork,)31 b(2.)g(Au\015age,)h(1994.)270
4313 y([LiD99])84 b Ft(LiDIA-Gr)n(oup)p Fw(,)21 b(T)-8
b(ec)m(hnisc)m(he)21 b(Univ)m(ersit)2107 4317 y(\177)2107
4313 y(at)f(Darmstadt,)h(F)-8 b(ac)m(h)m(b)s(ereic)m(h)22
b(Infor-)646 4426 y(matik,)34 b(Institut)f(f)1301 4430
y(\177)1298 4426 y(ur)f(Theoretisc)m(he)i(Informatik:)f
Fu(LiDIA-Manual)p Fw(,)g(F)-8 b(ebruar)646 4539 y(1999.)270
4745 y([Men93])62 b Ft(Menezes,)23 b(Alfred)h(J.)p Fw(:)d
Fu(El)5 b(liptic)24 b(Curve)g(Public)g(Key)g(Cryptosystems)p
Fw(.)k(Klu-)646 4858 y(w)m(er)j(Academic)f(Publishers,)e(Boston,)j
(1993.)270 5064 y([M)381 5068 y(\177)378 5064 y(u95])102
b Ft(M)739 5060 y(\177)736 5064 y(uller,)25 b(V)m(olker)p
Fw(:)e Fu(Die)i(Ber)-5 b(e)g(chnung)26 b(der)h(Punktanzahl)g(el)5
b(liptischer)26 b(Kur-)646 5177 y(ven)805 5181 y(\177)805
5177 y(ub)-5 b(er)23 b(end)5 b(lichen)24 b(K)1463 5181
y(\177)1464 5177 y(orp)-5 b(ern)26 b(der)e(Char)-5 b(akteristik)26
b(gr)2554 5181 y(\177)2555 5177 y(o\031er)e(3)p Fw(.)h(Doktorarb)s
(eit,)646 5290 y(Univ)m(ersit)1009 5294 y(\177)1009 5290
y(at)30 b(des)g(Saarlandes,)g(Saarbr)2024 5294 y(\177)2021
5290 y(uc)m(k)m(en,)g(1995.)270 5496 y([Oak98])72 b Ft(O)m(aks,)45
b(Scott)p Fw(:)40 b Fu(JA)-9 b(V)g(A)40 b(Se)-5 b(curity)p
Fw(.)72 b(O'Reilly)39 b(&)i(Asso)s(ciates,)g(Sebastop)s(ol,)646
5609 y(USA,)31 b(1998.)1785 5858 y(87)p eop
%%Page: 88 89
88 88 bop 390 82 a Fw(88)1862 b(LITERA)-8 b(TUR)e(VERZEICHNIS)390
415 y([PM98])91 b Ft(P)-8 b(a)n(ulus,)39 b(Sa)n(char)34
b Fw(und)g Ft(V)m(olker)39 b(M)2172 411 y(\177)2169 415
y(uller)p Fw(:)34 b Fu(Zur)j(Erzeugung)g(krypto)-5 b(gr)g(a-)766
528 y(phisch)30 b(starker)g(El)5 b(liptischer)29 b(Kurven)p
Fw(.)34 b(Datensc)m(h)m(utz)28 b(und)d(Datensic)m(herheit,)766
641 y(Septem)m(b)s(er)30 b(1998.)390 829 y([Ros99])88
b Ft(R)m(osing,)30 b(Michael)p Fw(:)c Fu(Implementing)k(El)5
b(liptic)29 b(Curve)f(Crypto)-5 b(gr)g(aphy)p Fw(.)38
b(Man-)766 941 y(ning)29 b(Publications,)g(Green)m(wic)m(h,)h(1999.)390
1129 y([Sc)m(h95])97 b Ft(Schoof,)40 b(Ren)1346 1121
y(\023)1346 1129 y(e)p Fw(:)35 b Fu(Counting)k(p)-5 b(oints)40
b(of)f(el)5 b(liptic)39 b(curves)f(over)g(\014nite)h(\014elds)p
Fw(.)766 1242 y(Journal)27 b(de)i(th)m(\023)-43 b(eorie)30
b(des)e(nom)m(bres)h(de)f(Bordeaux)h(7,)h(Seiten)e(219{254,)k(1995.)390
1430 y([Sil86])133 b Ft(Sil)-8 b(verman,)30 b(Joseph)f(H.)p
Fw(:)f Fu(The)i(A)n(rithmetic)g(of)g(El)5 b(liptic)30
b(Curves)p Fw(.)36 b(Springer)766 1543 y(V)-8 b(erlag,)32
b(New)e(Y)-8 b(ork,)31 b(1986.)390 1730 y([Sti95])124
b Ft(Stinson,)38 b(Douglas)g(R.)p Fw(:)d Fu(Crypto)-5
b(gr)g(aphy)41 b({)c(The)-5 b(ory)39 b(and)e(Pr)-5 b(actic)g(e)p
Fw(.)55 b(CR)m(C)766 1843 y(Press,)30 b(Bo)s(ca)i(Raton,)f(USA,)g
(1995.)p eop
%%Page: 89 90
89 89 bop 270 415 a Fe(Erkl)480 420 y(\177)481 415 y(arung)270
567 y Fw(Hiermit)38 b(v)m(ersic)m(here)i(ic)m(h,)f(da\031)h(ic)m(h)e
(die)h(v)m(orliegende)g(Diplomarb)s(eit)e(selbst)2983
571 y(\177)2983 567 y(andig)g(und)270 680 y(n)m(ur)29
b(mit)h(den)g(angegeb)s(enen)h(Hilfsmitteln)c(angefertigt)k(hab)s(e.)
270 793 y(Ic)m(h)38 b(bin)f(damit)h(ein)m(v)m(erstanden,)g(da\031)g
(der)g(Betreuer)h(meiner)e(Diplomarb)s(eit)f(diese)i(zu)270
906 y(wissensc)m(haftlic)m(hen)33 b(Zw)m(ec)m(k)m(en)k(w)m(eitergibt)e
(und)f(ein)g(Exemplar)g(zum)h(Zw)m(ec)m(k)h(der)f(Ein-)270
1019 y(sic)m(h)m(tnahme)30 b(in)f(die)h(Bibliothek)e(eingestellt)i
(wird.)2219 1623 y(Darmstadt,)h(29.)g(Oktob)s(er)f(1999)p
eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF
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