crnjak.diplom.ps
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	end
	%%EndProlog
	%%BeginSetup
	%%Feature: *Resolution 600dpi
	TeXDict begin
	%%BeginPaperSize: a4
	a4
	%%EndPaperSize

	%%EndSetup
	%%Page: 0 1
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	%%BeginDocument: Pics/athene.eps
	%!PS-Adobe-3.0 EPSF-3.0
	%%Title: c:\tmp\athene.eps
	%%Creator: Micrografx Graphics Engine
	%%CreationDate: Tue Nov 25 14:11:00 1997
	%%BoundingBox: 125 0 470 413
	%%Pages: (atend)
	%%LanguageLevel: 1
	%%CMYKCustomColor: (atend)
	%%DocumentCustomColors: (atend)
	%%DocumentProcessColors: (atend)
	%%DocumentSuppliedResources: (atend)
	%%EndComments
	%%BeginProlog
	%%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
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	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
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	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
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	-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
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	(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
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	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
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	2323 y(Kryptosystemen)g(spielen)f(sie)h(in)f(der)h(letzten)i(Zeit)e
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	(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
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	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)
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	2166 y(Metho)s(de)d(der)g(V)-8 b(er-)23 b(und)f(En)m(tsc)m(hl)1453
	2170 y(\177)1450 2166 y(usselung)f(mit)h(einem)g(v)m(orher)h(v)m(erein)
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	2279 y(Im)j(Gegensatz)j(dazu)d(ist)g(dies)g(b)s(ei)f(den)h(asymmetrisc)
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	2392 y(Dort)34 b(b)s(esteh)m(t)f(der)g(Sc)m(hl)1138 2396
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	2392 y(o\013en)m(tlic)m(hen)h(und)d(einem)i(geheimen)f(An)m(teil.)270
	2504 y(Der)449 2508 y(\177)449 2504 y(o\013en)m(tlic)m(he)h(T)-8
	b(eil)32 b(dien)m(t)h(zum)f(V)-8 b(ersc)m(hl)1786 2508
	y(\177)1783 2504 y(usseln.)32 b(Un)m(ter)i(V)-8 b(erw)m(endung)32
	b(des)h(geheimen)270 2617 y(An)m(teils)d(erh)706 2621
	y(\177)706 2617 y(alt)f(man)h(wieder)g(die)f(urspr)1691
	2621 y(\177)1688 2617 y(unglic)m(he)e(Nac)m(hric)m(h)m(t.)411
	2744 y(Der)g(V)-8 b(orteil)27 b(der)f(asymmetrisc)m(hen)h(V)-8
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	2857 y(Kenn)m(tnis)h(des)809 2861 y(\177)809 2857 y(o\013en)m(tlic)m
	(hen)h(Sc)m(hl)1457 2861 y(\177)1454 2857 y(usselteils)d(nic)m(h)m(t)j
	(bzw.)g(n)m(ur)f(sehr)g(sc)m(h)m(w)m(er)i(den)f(anderen)270
	2970 y(T)-8 b(eil)20 b(b)s(erec)m(hnen)h(k)-5 b(ann.)21
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	(ersonen)h(Daten)i(un)m(ter)e(V)-8 b(erw)m(endung)270
	3083 y(desselb)s(en)30 b(Sc)m(hl)842 3087 y(\177)839
	3083 y(ussels)f(an)i(einen)g(b)s(estimm)m(ten)g(T)-8
	b(eilnehmer)29 b(v)m(ersc)m(hl)2703 3087 y(\177)2700
	3083 y(usselt)i(w)m(erden.)g(Es)270 3196 y(ist)26 b(also)i(m)649
	3200 y(\177)649 3196 y(oglic)m(h,)e(die)1074 3200 y(\177)1074
	3196 y(o\013en)m(tlic)m(hen)h(Sc)m(hl)1721 3200 y(\177)1718
	3196 y(ussel)e(allen)h(zug)2282 3200 y(\177)2282 3196
	y(anglic)m(h)g(zu)h(mac)m(hen,)h(ohne)f(die)270 3309
	y(Sic)m(herheit)j(des)g(V)-8 b(erfahrens)31 b(zu)f(b)s(eein)m(tr)1704
	3313 y(\177)1704 3309 y(ac)m(h)m(tigen.)i(Daher)f(b)s(ezeic)m(hnet)g
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	b(auc)m(h)g(als)g Fu(Public)h(Key)42 b Fw(V)-8 b(erfahren.)35
	b(Es)g(soll)f(jedo)s(c)m(h)h(nic)m(h)m(t)f(der)270 3535
	y(Eindruc)m(k)25 b(erw)m(ec)m(kt)j(w)m(erden,)f(diese)f(seien)g(den)g
	(asymmetrisc)m(hen)g(in)f(jeder)i(Hinsic)m(h)m(t)f(un-)270
	3648 y(terlegen.)g(In)e(b)s(ezug)g(auf)h(die)f(E\016zienz)h(sind)e(sie)
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	3877 y(\177)1993 3873 y(ur)h(die)f(eigen)m(tlic)m(he)2706
	3854 y(\177)2695 3873 y(Ub)s(ermittlung)e(der)270 3986
	y(Nac)m(hric)m(h)m(t)31 b(mit)f(einem)g(symmetrisc)m(hen)f(V)-8
	b(erfahren)30 b(b)s(en)m(utzt.)411 4113 y(Es)f(gibt)f(im)g(w)m(esen)m
	(tlic)m(hen)h(zw)m(ei)g(Ans)1728 4117 y(\177)1728 4113
	y(atze)h(b)s(ei)d(der)i(Konstruktion)e(v)m(on)i(Public)e(Key)270
	4226 y(V)-8 b(erfahren.)27 b(Die)h(einen)e(b)s(eruhen)g(auf)h(dem)g(F)
	-8 b(aktorisierungsproblem,)25 b(die)i(anderen)f(auf)270
	4339 y(dem)43 b(Problem)f(des)i(Diskreten)f(Logarithm)m(us.)g(Bei)h
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	4339 y(oglic)m(h,)270 4452 y(elliptisc)m(he)27 b(Kurv)m(en)h(zu)h(v)m
	(erw)m(enden.)g(Diese)g(bieten)f(in)g(der)g(Regel)h(V)-8
	b(orteile,)30 b(w)m(enn)e(man)270 4565 y(gewisse)i(Dinge)h(b)s(ei)e
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	4692 y(Wir)e(w)m(erden)g(in)f(dieser)h(Arb)s(eit)f(eine)h(An)m(w)m
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	4692 y(usseln)27 b(v)m(on)i(Daten)270 4805 y(en)m(t)m(wic)m(k)m(eln,)40
	b(die)e(im)g(folgenden)g(als)h Fu(Pr)-5 b(ovider)50 b
	Fw(b)s(ezeic)m(hnet)39 b(w)m(erden)g(soll,)f(da)h(sie)f(eine)270
	4918 y(F)-8 b(unktionalit)760 4922 y(\177)760 4918 y(at)28
	b(b)s(ereitstellt.)f(Dab)s(ei)h(w)m(erden)g(elliptisc)m(he)e(Kurv)m(en)
	h(als)h(mathematisc)m(he)270 5030 y(Grundlage)37 b(b)s(en)m(utzt.)i
	(Zur)e(T)-8 b(ec)m(hnik)38 b(z)1659 5034 y(\177)1659
	5030 y(ahlt)g(die)f(JA)-10 b(V)g(A)39 b(Cryptograph)m(y)e(Arc)m
	(hitecture,)273 5147 y(\177)270 5143 y(ub)s(er)29 b(die)g(am)i(Ende)e
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	5270 y(Damit)37 b(hab)s(en)e(wir)g(sc)m(hon)i(die)e(wic)m(h)m(tigsten)i
	(Asp)s(ekte)f(angespro)s(c)m(hen,)h(die)e(in)h(die-)270
	5383 y(ser)30 b(Diplomarb)s(eit)e(V)-8 b(erw)m(endung)30
	b(\014nden.)e(Im)i(folgenden)f(soll)g(jedo)s(c)m(h)h(no)s(c)m(h)g(ein)f
	(kurzer)281 5477 y(\177)270 5496 y(Ub)s(erblic)m(k)700
	5500 y(\177)698 5496 y(ub)s(er)42 b(die)h(in)g(den)g(einzelnen)g
	(Kapiteln)f(b)s(ehandelten)g(Themen)h(gegeb)s(en)270
	5609 y(w)m(erden.)g(Die)g(erforderlic)m(he)f(Theorie)h(der)f(elliptisc)
	m(hen)f(Kurv)m(en)h(wird)f(in)h(Kapitel)h(2)1807 5858
	y(5)p eop
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	528 y(n)m(ur)27 b(die)h(Grundlagen)e(b)s(ereitgestellt)i(w)m(erden,)g
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	b(die)g(F)-8 b(unktionsw)m(eise)30 b(des)h(Pro)m(viders)g(erkl)2065
	758 y(\177)2065 754 y(art)f(Kapitel)g(4.)i(Dort)g(w)m(erden)f(eb)s
	(enfalls)390 867 y(v)m(ersc)m(hiedene)e(Asp)s(ekte)g(der)g(Implemen)m
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	980 y(v)m(on)39 b(Kapitel)f(5,)i(w)m(o)f(auc)m(h)g(v)m(ersuc)m(h)m(t)h
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	1093 y(Ausblic)m(k)45 b(auf)i(w)m(eitere)g(En)m(t)m(wic)m(klungen)f
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	1093 y(angen)f(wird)390 1205 y(sc)m(hlie\031lic)m(h)40
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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
	1435 y(\177)2271 1431 y(uhrungen)2682 1435 y(\177)2680
	1431 y(ub)s(er)f(elliptisc)m(he)g(Kur-)390 1544 y(v)m(en)k(k)m(einen)g
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	1544 y(andigk)m(eit.)37 b(Daher)i(soll)d(n)m(un)h(eine)h(Au\015istung)e
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	n 500 1977 m 488 1990 l gs col0 s gr
	% Polyline
	n 500 1977 m 511 1969 l gs col0 s gr
	% Polyline
	n 517 1964 m 511 1969 l gs col0 s gr
	% Polyline
	n 517 3804 m 539 3796 l gs col0 s gr
	% Polyline
	n 556 3791 m 539 3796 l gs col0 s gr
	% Polyline
	n 517 1964 m 533 1951 l gs col0 s gr
	% Polyline
	n 556 1934 m 533 1951 l gs col0 s gr
	% Polyline
	n 556 3791 m 568 3785 l gs col0 s gr
	% Polyline
	n 594 3775 m 568 3785 l gs col0 s gr
	% Polyline
	n 556 1934 m 558 1934 l gs col0 s gr
	% Polyline
	n 559 1931 m 558 1934 l gs col0 s gr
	% Polyline
	n 559 1931 m 582 1917 l gs col0 s gr
	% Polyline
	n 594 1909 m 582 1917 l gs col0 s gr
	% Polyline
	n 594 3775 m 597 3772 l gs col0 s gr
	% Polyline
	n 600 3770 m 597 3772 l gs col0 s gr
	% Polyline
	n 600 3770 m 622 3759 l gs col0 s gr
	% Polyline
	n 633 3753 m 622 3759 l gs col0 s gr
	% Polyline
	n 594 1909 m 606 1902 l gs col0 s gr
	% Polyline
	n 633 1887 m 606 1902 l gs col0 s gr
	% Polyline
	n 633 3753 m 648 3743 l gs col0 s gr
	% Polyline
	n 671 3729 m 648 3743 l gs col0 s gr
	% Polyline
	n 632 1887 m 634 1887 l gs col0 s gr
	% Polyline
	n 634 1887 m 633 1887 l gs col0 s gr
	% Polyline
	n 634 1887 m 661 1873 l gs col0 s gr
	% Polyline
	n 671 1869 m 661 1873 l gs col0 s gr
	% Polyline
	n 671 3729 m 673 3727 l gs col0 s gr
	% Polyline
	n 676 3724 m 673 3727 l gs col0 s gr
	% Polyline
	n 676 3724 m 697 3710 l gs col0 s gr
	% Polyline
	n 708 3703 m 697 3710 l gs col0 s gr
	% Polyline
	n 671 1869 m 687 1861 l gs col0 s gr
	% Polyline
	n 708 1852 m 687 1861 l gs col0 s gr
	% Polyline
	n 708 3703 m 720 3693 l gs col0 s gr
	% Polyline
	n 740 3679 m 720 3693 l gs col0 s gr
	% Polyline
	n 740 3679 m 745 3674 l gs col0 s gr
	% Polyline
	n 747 3672 m 745 3674 l gs col0 s gr
	% Polyline
	n 708 1852 m 717 1849 l gs col0 s gr
	% Polyline
	n 744 1840 m 717 1849 l gs col0 s gr
	% Polyline
	n 744 1840 m 746 1839 l gs col0 s gr
	% Polyline
	n 747 1838 m 746 1839 l gs col0 s gr
	% Polyline
	n 747 3672 m 767 3656 l gs col0 s gr
	% Polyline
	n 785 3639 m 767 3656 l gs col0 s gr
	% Polyline
	n 747 1838 m 778 1831 l gs col0 s gr
	% Polyline
	n 785 1828 m 778 1831 l gs col0 s gr
	% Polyline
	n 785 3639 m 788 3636 l gs col0 s gr
	% Polyline
	n 793 3632 m 788 3636 l gs col0 s gr
	% Polyline
	n 793 3632 m 810 3617 l gs col0 s gr
	% Polyline
	n 824 3605 m 810 3617 l gs col0 s gr
	% Polyline
	n 785 1828 m 810 1824 l gs col0 s gr
	% Polyline
	n 824 1821 m 810 1824 l gs col0 s gr
	% Polyline
	n 824 3605 m 832 3597 l gs col0 s gr
	% Polyline
	n 842 3587 m 832 3597 l gs col0 s gr
	% Polyline
	n 842 3587 m 853 3576 l gs col0 s gr
	% Polyline
	n 861 3566 m 853 3576 l gs col0 s gr
	% Polyline
	n 824 1821 m 843 1818 l gs col0 s gr
	% Polyline
	n 861 1816 m 843 1818 l gs col0 s gr
	% Polyline
	n 861 3566 m 873 3554 l gs col0 s gr
	% Polyline
	n 888 3541 m 873 3554 l gs col0 s gr
	% Polyline
	n 888 3541 m 894 3534 l gs col0 s gr
	% Polyline
	n 901 3526 m 894 3534 l gs col0 s gr
	% Polyline
	n 861 1816 m 880 1814 l gs col0 s gr
	% Polyline
	n 901 1813 m 880 1814 l gs col0 s gr
	% Polyline
	n 901 3526 m 914 3511 l gs col0 s gr
	% Polyline
	n 929 3496 m 914 3511 l gs col0 s gr
	% Polyline
	n 929 3496 m 934 3489 l gs col0 s gr
	% Polyline
	n 938 3484 m 934 3489 l gs col0 s gr
	% Polyline
	n 901 1813 m 915 1813 l gs col0 s gr
	% Polyline
	n 938 1812 m 915 1813 l gs col0 s gr
	% Polyline
	n 938 3484 m 954 3467 l gs col0 s gr
	% Polyline
	n 969 3449 m 954 3467 l gs col0 s gr
	% Polyline
	n 969 3449 m 973 3445 l gs col0 s gr
	% Polyline
	n 977 3441 m 973 3445 l gs col0 s gr
	% Polyline
	n 938 1812 m 954 1813 l gs col0 s gr
	% Polyline
	n 977 1814 m 954 1813 l gs col0 s gr
	% Polyline
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	% Polyline
	n 1007 3403 m 993 3422 l gs col0 s gr
	% Polyline
	n 1007 3403 m 1011 3398 l gs col0 s gr
	% Polyline
	n 1015 3393 m 1011 3398 l gs col0 s gr
	% Polyline
	n 977 1814 m 994 1816 l gs col0 s gr
	% Polyline
	n 1015 1817 m 994 1816 l gs col0 s gr
	% Polyline
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	% Polyline
	n 1043 3356 m 1029 3375 l gs col0 s gr
	% Polyline
	n 1043 3356 m 1049 3351 l gs col0 s gr
	% Polyline
	n 1053 3344 m 1049 3351 l gs col0 s gr
	% Polyline
	n 1015 1817 m 1037 1821 l gs col0 s gr
	% Polyline
	n 1053 1821 m 1037 1821 l gs col0 s gr
	% Polyline
	n 1053 3344 m 1067 3326 l gs col0 s gr
	% Polyline
	n 1079 3311 m 1067 3326 l gs col0 s gr
	% Polyline
	n 1079 3311 m 1085 3303 l gs col0 s gr
	% Polyline
	n 1091 3294 m 1085 3303 l gs col0 s gr
	% Polyline
	n 1053 1821 m 1083 1828 l gs col0 s gr
	% Polyline
	n 1091 1831 m 1083 1828 l gs col0 s gr
	% Polyline
	n 1091 3294 m 1103 3278 l gs col0 s gr
	% Polyline
	n 1113 3265 m 1103 3278 l gs col0 s gr
	% Polyline
	n 1113 3265 m 1120 3253 l gs col0 s gr
	% Polyline
	n 1130 3240 m 1120 3253 l gs col0 s gr
	% Polyline
	n 1130 1842 m 1126 1840 l 1091 1831 l gs col0 s gr
	% Polyline
	n 1130 3240 m 1138 3229 l gs col0 s gr
	% Polyline
	n 1145 3218 m 1138 3229 l gs col0 s gr
	% Polyline
	n 1145 3218 m 1156 3204 l gs col0 s gr
	% Polyline
	n 1168 3187 m 1156 3204 l gs col0 s gr
	% Polyline
	n 1130 1842 m 1131 1842 l gs col0 s gr
	% Polyline
	n 1168 1853 m 1131 1842 l gs col0 s gr
	% Polyline
	n 1168 3187 m 1173 3179 l gs col0 s gr
	% Polyline
	n 1176 3173 m 1173 3179 l gs col0 s gr
	% Polyline
	n 1176 3173 m 1189 3153 l gs col0 s gr
	% Polyline
	n 1207 3130 m 1189 3153 l gs col0 s gr
	% Polyline
	n 1168 1853 m 1185 1860 l gs col0 s gr
	% Polyline
	n 1207 1869 m 1185 1860 l gs col0 s gr
	% Polyline
	n 1207 3130 m 1207 3129 l gs col0 s gr
	% Polyline
	n 1207 3127 m 1207 3129 l gs col0 s gr
	% Polyline
	n 1207 3127 m 1224 3102 l gs col0 s gr
	% Polyline
	n 1237 3082 m 1224 3102 l gs col0 s gr
	% Polyline
	n 1237 3082 m 1239 3075 l gs col0 s gr
	% Polyline
	n 1244 3070 m 1239 3075 l gs col0 s gr
	% Polyline
	n 1207 1869 m 1243 1883 l gs col0 s gr
	% Polyline
	n 1244 1884 m 1243 1883 l gs col0 s gr
	% Polyline
	n 1244 3070 m 1257 3049 l gs col0 s gr
	% Polyline
	n 1265 3034 m 1257 3049 l gs col0 s gr
	% Polyline
	n 1265 3034 m 1272 3023 l gs col0 s gr
	% Polyline
	n 1283 3007 m 1272 3023 l gs col0 s gr
	% Polyline
	n 1283 1904 m 1249 1887 l 1244 1884 l gs col0 s gr
	% Polyline
	n 1283 3007 m 1290 2997 l gs col0 s gr
	% Polyline
	n 1294 2989 m 1290 2997 l gs col0 s gr
	% Polyline
	n 1294 2989 m 1305 2969 l gs col0 s gr
	% Polyline
	n 1321 2943 m 1305 2969 l gs col0 s gr
	% Polyline
	n 1320 2943 m 1322 2943 l gs col0 s gr
	% Polyline
	n 1283 1904 m 1310 1919 l gs col0 s gr
	% Polyline
	n 1321 1925 m 1310 1919 l gs col0 s gr
	% Polyline
	n 1321 2943 m 1335 2916 l gs col0 s gr
	% Polyline
	n 1347 2896 m 1335 2916 l gs col0 s gr
	% Polyline
	n 1347 2896 m 1353 2888 l gs col0 s gr
	% Polyline
	n 1360 2876 m 1353 2888 l gs col0 s gr
	% Polyline
	n 1360 1949 m 1333 1931 l 1321 1925 l gs col0 s gr
	% Polyline
	n 1360 2876 m 1367 2861 l gs col0 s gr
	% Polyline
	n 1373 2852 m 1367 2861 l gs col0 s gr
	% Polyline
	n 1373 2852 m 1382 2832 l gs col0 s gr
	% Polyline
	n 1397 2805 m 1382 2832 l gs col0 s gr
	% Polyline
	n 1360 1949 m 1394 1975 l gs col0 s gr
	% Polyline
	n 1397 1977 m 1394 1975 l gs col0 s gr
	% Polyline
	n 1397 2805 m 1412 2777 l gs col0 s gr
	% Polyline
	n 1422 2759 m 1412 2777 l gs col0 s gr
	% Polyline
	n 1422 2759 m 1427 2748 l gs col0 s gr
	% Polyline
	n 1437 2731 m 1427 2748 l gs col0 s gr
	% Polyline
	n 1437 2008 m 1400 1977 l 1397 1977 l gs col0 s gr
	% Polyline
	n 1437 2731 m 1441 2719 l gs col0 s gr
	% Polyline
	n 1445 2712 m 1441 2719 l gs col0 s gr
	% Polyline
	n 1445 2712 m 1455 2691 l gs col0 s gr
	% Polyline
	n 1467 2667 m 1455 2691 l gs col0 s gr
	% Polyline
	n 1467 2667 m 1469 2662 l gs col0 s gr
	% Polyline
	n 1474 2650 m 1469 2662 l gs col0 s gr
	% Polyline
	n 1474 2046 m 1453 2024 l 1437 2008 l gs col0 s gr
	% Polyline
	n 1474 2650 m 1482 2632 l gs col0 s gr
	% Polyline
	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
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	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
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	n 2404 875 m 2400 886 l gs col0 s gr
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	n 2404 875 m 2413 853 l gs col0 s gr
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	n 2422 830 m 2413 853 l gs col0 s gr
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	n 2422 830 m 2425 822 l gs col0 s gr
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	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
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	n 2431 807 m 2437 791 l gs col0 s gr
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	n 2439 782 m 2449 759 l gs col0 s gr
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	n 2459 736 m 2449 759 l gs col0 s gr
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	n 2459 736 m 2461 729 l gs col0 s gr
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	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
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	n 2468 711 m 2473 696 l gs col0 s gr
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	n 2476 690 m 2473 696 l gs col0 s gr
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	n 2476 690 m 2486 667 l gs col0 s gr
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	n 2494 646 m 2486 667 l gs col0 s gr
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	n 2494 646 m 2498 634 l gs col0 s gr
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	n 2507 612 m 2498 634 l gs col0 s gr
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	n 2545 2982 m 2518 2943 l 2507 2925 l gs col0 s gr
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	n 2507 612 m 2510 603 l gs col0 s gr
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	n 2512 599 m 2510 603 l gs col0 s gr
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	n 2512 599 m 2522 571 l gs col0 s gr
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	n 2530 553 m 2522 571 l gs col0 s gr
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	n 2530 553 m 2534 540 l gs col0 s gr
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	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
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	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
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	n 2565 461 m 2560 478 l gs col0 s gr
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	n 2565 461 m 2572 447 l gs col0 s gr
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	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
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	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
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
	0 slj
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	% Polyline
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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 969 1371 m 937 1350 l 906 1333 l gs col0 s gr
	% Polyline
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	% Polyline
	1 slj
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	% 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
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	% Polyline
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	% Polyline
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	% Polyline
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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 1038 1420 m 1030 1413 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
	n 1158 1522 m 1162 1528 l gs col0 s gr
	% Polyline
	n 1158 1522 m 1153 1519 l gs col0 s gr
	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
	n 1249 2074 m 1239 2086 l gs col0 s gr
	% Polyline
	n 1249 2074 m 1257 2061 l gs col0 s gr
	% Polyline
	n 1273 2039 m 1257 2061 l gs col0 s gr
	% Polyline
	n 1273 2039 m 1274 2037 l gs col0 s gr
	% Polyline
	n 1277 2032 m 1274 2037 l gs col0 s gr
	% Polyline
	n 1277 1668 m 1273 1660 l 1271 1656 l gs col0 s gr
	% Polyline
	n 1249 1626 m 1271 1656 l gs col0 s gr
	% Polyline
	n 1249 1626 m 1226 1597 l gs col0 s gr
	% Polyline
	n 1222 1591 m 1226 1597 l gs col0 s gr
	% Polyline
	n 1222 1591 m 1215 1585 l gs col0 s gr
	% Polyline
	n 1277 2032 m 1290 2012 l gs col0 s gr
	% Polyline
	n 1295 2004 m 1290 2012 l gs col0 s gr
	% Polyline
	n 1295 2004 m 1305 1985 l gs col0 s gr
	% Polyline
	n 1313 1970 m 1305 1985 l gs col0 s gr
	% Polyline
	n 1313 1970 m 1317 1958 l gs col0 s gr
	% Polyline
	n 1325 1935 m 1317 1958 l gs col0 s gr
	% Polyline
	n 1325 1935 m 1327 1929 l gs col0 s gr
	% Polyline
	n 1334 1901 m 1327 1929 l gs col0 s gr
	% Polyline
	n 1334 1901 m 1335 1898 l gs col0 s gr
	% Polyline
	n 1339 1866 m 1335 1898 l gs col0 s gr
	% Polyline
	n 1338 1866 m 1340 1866 l gs col0 s gr
	% Polyline
	n 1339 1832 m 1339 1866 l gs col0 s gr
	% Polyline
	n 1338 1832 m 1340 1832 l gs col0 s gr
	% Polyline
	n 1334 1798 m 1339 1832 l gs col0 s gr
	% Polyline
	n 1334 1798 m 1333 1794 l gs col0 s gr
	% Polyline
	n 1325 1763 m 1333 1794 l gs col0 s gr
	% Polyline
	n 1325 1763 m 1321 1753 l gs col0 s gr
	% Polyline
	n 1313 1729 m 1321 1753 l gs col0 s gr
	% Polyline
	n 1313 1729 m 1302 1708 l gs col0 s gr
	% Polyline
	n 1295 1695 m 1302 1708 l gs col0 s gr
	% Polyline
	n 1295 1695 m 1277 1668 l gs col0 s gr
	% Polyline
	n 1895 2041 m 1894 2039 l 1894 2038 l gs col0 s gr
	% Polyline
	n 1878 2004 m 1894 2038 l gs col0 s gr
	% Polyline
	n 1878 2004 m 1874 1993 l gs col0 s gr
	% Polyline
	n 1866 1970 m 1874 1993 l gs col0 s gr
	% Polyline
	n 1866 1970 m 1861 1951 l gs col0 s gr
	% Polyline
	n 1857 1935 m 1861 1951 l gs col0 s gr
	% Polyline
	n 1857 1935 m 1852 1912 l gs col0 s gr
	% Polyline
	n 1851 1901 m 1852 1912 l gs col0 s gr
	% Polyline
	n 1851 1901 m 1848 1875 l gs col0 s gr
	% Polyline
	n 1847 1866 m 1848 1875 l gs col0 s gr
	% Polyline
	n 1847 1866 m 1847 1840 l gs col0 s gr
	% Polyline
	n 1847 1832 m 1847 1840 l gs col0 s gr
	% Polyline
	n 1847 1832 m 1850 1806 l gs col0 s gr
	% Polyline
	n 1851 1798 m 1850 1806 l gs col0 s gr
	% Polyline
	n 1851 1798 m 1855 1775 l gs col0 s gr
	% Polyline
	n 1857 1763 m 1855 1775 l gs col0 s gr
	% Polyline
	n 1857 1763 m 1862 1744 l gs col0 s gr
	% Polyline
	n 1866 1729 m 1862 1744 l gs col0 s gr
	% Polyline
	n 1866 1729 m 1871 1715 l gs col0 s gr
	% Polyline
	n 1878 1695 m 1871 1715 l gs col0 s gr
	% Polyline
	n 1878 1695 m 1882 1687 l gs col0 s gr
	% Polyline
	n 1894 1660 m 1882 1687 l gs col0 s gr
	% Polyline
	n 1894 1660 m 1895 1659 l gs col0 s gr
	% Polyline
	n 1895 1658 m 1895 1659 l gs col0 s gr
	% Polyline
	n 1957 2162 m 1945 2141 l 1938 2131 l gs col0 s gr
	% Polyline
	n 1925 2108 m 1938 2131 l gs col0 s gr
	% Polyline
	n 1925 2108 m 1914 2083 l gs col0 s gr
	% Polyline
	n 1909 2074 m 1914 2083 l gs col0 s gr
	% Polyline
	n 1909 2074 m 1895 2041 l gs col0 s gr
	% Polyline
	n 1895 1658 m 1906 1632 l gs col0 s gr
	% Polyline
	n 1909 1626 m 1906 1632 l gs col0 s gr
	% Polyline
	n 1909 1626 m 1919 1604 l gs col0 s gr
	% Polyline
	n 1925 1591 m 1919 1604 l gs col0 s gr
	% Polyline
	n 1925 1591 m 1932 1578 l gs col0 s gr
	% Polyline
	n 1945 1557 m 1932 1578 l gs col0 s gr
	% Polyline
	n 1945 1557 m 1948 1552 l gs col0 s gr
	% Polyline
	n 1957 1537 m 1948 1552 l gs col0 s gr
	% Polyline
	n 2019 2268 m 2003 2245 l 1996 2232 l gs col0 s gr
	% Polyline
	n 1982 2211 m 1996 2232 l gs col0 s gr
	% Polyline
	n 1982 2211 m 1967 2181 l gs col0 s gr
	% Polyline
	n 1964 2176 m 1967 2181 l gs col0 s gr
	% Polyline
	n 1964 2176 m 1957 2162 l gs col0 s gr
	% Polyline
	n 1957 1537 m 1962 1526 l gs col0 s gr
	% Polyline
	n 1964 1522 m 1962 1526 l gs col0 s gr
	% Polyline
	n 1964 1522 m 1977 1499 l gs col0 s gr
	% Polyline
	n 1982 1489 m 1977 1499 l gs col0 s gr
	% Polyline
	n 1982 1489 m 1991 1473 l gs col0 s gr
	% Polyline
	n 2003 1454 m 1991 1473 l gs col0 s gr
	% Polyline
	n 2003 1454 m 2008 1448 l gs col0 s gr
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	n 2019 1431 m 2008 1448 l gs col0 s gr
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	n 2081 2368 m 2067 2348 l 2060 2336 l gs col0 s gr
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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
	2952 y(o\031e)h Fq(t)f Fu(Spur)44 b Fw(v)m(on)34 b Fq(E)39
	b Fw(genann)m(t.)34 b(Mit)f(Hilfe)f(v)m(on)i(Diskriminan)m(te)270
	3065 y(und)23 b(Spur)g(lassen)i(sic)m(h)f(Eigensc)m(haften)h(wie)f
	Fu(Singularit)2204 3069 y(\177)2205 3065 y(at)35 b Fw(und)24
	b Fu(Sup)-5 b(ersingularit)3085 3069 y(\177)3086 3065
	y(at)36 b Fw(einer)270 3178 y(elliptisc)m(hen)28 b(Kurv)m(e)i
	(de\014nieren.)e(Dazu)j(sp)1778 3182 y(\177)1778 3178
	y(ater)g(jedo)s(c)m(h)f(mehr.)411 3293 y(Es)f(sollte)g(auc)m(h)g(no)s
	(c)m(h)g(erw)1346 3297 y(\177)1346 3293 y(ahn)m(t)g(w)m(erden,)g
	(da\031)h(die)e(Bestimm)m(ung)h(der)f(Punktanzahl)270
	3406 y(v)m(on)36 b(elliptisc)m(hen)c(Kurv)m(en)i(k)m(eine)i(leic)m(h)m
	(te)f(Aufgab)s(e)g(ist.)g(Es)g(sind)e(zw)m(ar)j(in)e(der)g(letzten)270
	3519 y(Zeit)25 b(b)s(ei)f(den)h(p)s(olynomiellen)d(Algorithmen)i(v)m
	(on)i(Sc)m(ho)s(of,)g(A)m(tkin)f(und)f(Elkies)g(b)s(etr)3194
	3523 y(\177)3194 3519 y(ac)m(h)m(t-)270 3632 y(lic)m(he)39
	b(F)-8 b(ortsc)m(hritte)42 b(erzielt)e(w)m(orden.)g(Man)g(darf)g(jedo)s
	(c)m(h)g(nic)m(h)m(t)g(die)2735 3636 y(\177)2735 3632
	y(au\031erst)h(umfang-)270 3745 y(reic)m(he)h(Theorie)e(v)m(ergessen,)j
	(die)e(hin)m(ter)g(diesen)f(Algorithmen)g(steh)m(t.)j(Da)g(hier)d(nic)m
	(h)m(t)270 3858 y(ausf)433 3862 y(\177)430 3858 y(uhrlic)m(her)e
	(darauf)j(eingegangen)h(w)m(erden)f(k)-5 b(ann,)40 b(wird)g(auf)h([Sc)m
	(h95)q(])g(und)f([M)3226 3862 y(\177)3223 3858 y(u95)q(])270
	3971 y(v)m(erwiesen.)270 4270 y Fz(2.2)135 b(Elliptisc)l(he)47
	b(Kurv)l(en)e(als)h(ab)t(elsc)l(he)f(Grupp)t(en)270 4478
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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	% Polyline
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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 10987 2411 m 10989 2411 l gs col0 s gr
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	n 10992 2407 m 10988 2411 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11062 5721 m 11037 5689 l 10992 5620 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 10992 2407 m 11019 2369 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11037 2342 m 11019 2369 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11062 2306 m 11047 2326 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11132 5826 m 11086 5761 l 11062 5721 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11086 2268 m 11077 2285 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11086 2268 m 11106 2242 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11132 2203 m 11106 2242 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11205 5930 m 11184 5901 l 11143 5841 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11137 5832 m 11143 5841 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11137 5832 m 11132 5826 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11132 2203 m 11135 2200 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11137 2198 m 11135 2200 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11137 2198 m 11163 2158 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11184 2127 m 11191 2115 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11205 2097 m 11191 2115 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11275 6038 m 11230 5973 l 11205 5930 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11205 2097 m 11220 2071 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11230 2055 m 11220 2071 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11230 2055 m 11249 2029 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11275 1988 m 11249 2029 l gs col7 0.00 shd ef gr gs col0 s gr
	% Polyline
	n 11345 6150 m 11324 6117 l 11283 6055 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11278 6045 m 11283 6055 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11278 6045 m 11275 6038 l gs col7 0.00 shd ef gr gs col0 s gr
	% Polyline
	n 11275 1988 m 11276 1986 l gs col7 0.00 shd ef gr gs col0 s gr
	% Polyline
	n 11278 1984 m 11276 1986 l gs col7 0.00 shd ef gr gs col0 s gr
	% Polyline
	n 11278 1984 m 11304 1942 l gs col7 0.00 shd ef gr gs col0 s gr
	% Polyline
	n 11324 1912 m 11304 1942 l gs col7 0.00 shd ef gr gs col0 s gr
	% Polyline
	n 11324 1912 m 11333 1899 l gs col7 0.00 shd ef gr gs col0 s gr
	% Polyline
	n 11345 1880 m 11333 1899 l gs col7 0.00 shd ef gr gs col0 s gr
	% Polyline
	n 11417 6261 m 11416 6259 l 11412 6255 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11370 6188 m 11412 6255 l gs col7 0.00 shd ef gr gs col0 s gr
	% Polyline
	n 11370 6188 m 11345 6150 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11345 1880 m 11360 1856 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11370 1841 m 11360 1856 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11370 1841 m 11388 1813 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11416 1771 m 11388 1813 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11416 1771 m 11417 1769 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11487 6373 m 11460 6329 l 11417 6261 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11417 1769 m 11443 1727 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11460 1699 m 11443 1727 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11460 1699 m 11470 1681 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11487 1655 m 11470 1681 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11559 6489 m 11548 6472 l 11530 6445 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11504 6401 m 11530 6445 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11504 6401 m 11487 6373 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11487 1655 m 11498 1638 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11504 1628 m 11498 1638 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11504 1628 m 11525 1594 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11548 1555 m 11525 1594 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11548 1555 m 11552 1550 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11559 1539 m 11552 1550 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11630 6606 m 11591 6544 l 11559 6489 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11559 1539 m 11580 1507 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11591 1486 m 11580 1507 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11591 1486 m 11606 1463 l gs col7 0.00 shd ef gr gs col0 s gr
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	n 11630 1424 m 11606 1463 l gs col7 0.00 shd ef gr 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
	n 1078 3479 m 1072 3495 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
	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
	n 2005 5386 m 2053 5384 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 2346 5334 m 2345 5334 l gs col0 s gr
	% Polyline
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	% Polyline
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	% 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
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	% 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
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	% 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
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	% Polyline
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	% Polyline
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	@endspecial 2316 3172 a Fq(P)2670 3474 y(P)33 b Fo(\003)21
	b Fq(P)2878 2940 y Fo(O)2469 3765 y Fw(2)p Fq(P)957 3162
	y(P)1267 3247 y(Q)1413 3495 y(P)33 b Fo(\003)21 b Fq(Q)1661
	2940 y Fo(O)1175 3802 y Fq(P)33 b Fw(+)20 b Fq(Q)1169
	4488 y Fw(Abbildung)27 b(2.4:)32 b(Addition)c(zw)m(eier)i(Punkte)531
	4704 y(Abgesehen)c(v)m(om)g(Nac)m(h)m(w)m(eis)h(der)e(Assoziativit)2179
	4708 y(\177)2179 4704 y(at)g(ist)g(es)h(n)m(un)f(nic)m(h)m(t)g(mehr)g
	(sc)m(h)m(w)m(er)h(zu)390 4817 y(zeigen,)f(da\031)g(die)f(Menge)i(der)e
	(Punkte)g(einer)g(elliptisc)m(hen)e(Kurv)m(e)i(mit)g(der)h(V)-8
	b(erkn)3238 4821 y(\177)3235 4817 y(upfung)390 4930 y(der)29
	b(Addition)e(zu)i(einer)g Fu(ab)-5 b(elschen)32 b(Grupp)-5
	b(e)38 b Fw(wird.)28 b(Dab)s(ei)g(wirkt)h Fo(O)j Fw(als)d(neutrales)f
	(Ele-)390 5043 y(men)m(t)j(in)e(der)h(Grupp)s(e.)542
	5138 y(\177)531 5157 y(Ub)s(errasc)m(henderw)m(eise)i(liefert)g(diese)g
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	b(gleic)m(he)i(Grupp)s(enstruktur.)d(Im)i(allgemeinen)g(en)m(tsc)m
	(heidet)h(man)g(sic)m(h)f(ab)s(er)g(f)3427 5274 y(\177)3424
	5270 y(ur)390 5383 y(den)44 b(F)-8 b(ernpunkt)43 b(\()p
	Fo(1)p Fw(\).)i(Das)g(hat)f(den)g(V)-8 b(orteil,)44 b(da\031)h(die)e
	(In)m(v)m(ertierung)g(eines)h(Punk-)390 5496 y(tes)e(sehr)e(sc)m(hnell)
	g(geh)m(t,)i(da)f(hierzu)f(die)g(V)-8 b(erbindungsgerade)40
	b(eines)g(Punktes)h(mit)f Fo(O)390 5609 y Fw(der)j(Senkrec)m(h)m(ten)g
	(im)f(a\016nen)h(Ko)s(ordinatensystem)f(en)m(tspric)m(h)m(t.)h(Das)h
	(In)m(v)m(erse)g(eines)p eop
	%%Page: 11 12
	11 11 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)1210 b(11)1248 1741 y @beginspecial
	0 @llx 0 @lly 341 @urx 426 @ury 1700 @rhi @clip @setspecial
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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
	1431 y(ac)m(hlic)m(h)f(an)h(der)g(zeitauf-)390 1544 y(w)m(endigen)k
	(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
	1661 y(\177)1549 1657 y(ub)s(erstellung)f(der)i(b)s(eiden)e(V)-8
	b(erfahren)38 b(\014ndet)e(man)h(in)390 1770 y([PM98)q(].)g(Dort)g(w)m
	(erden)f(so)m(w)m(ohl)h(die)e(Theorie)h(kurz)f(erl)2381
	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
	15 15 bop 270 1040 a Fr(Kapitel)66 b(3)270 1457 y Fv(Der)78
	b(Algorithm)-6 b(us)270 1904 y Fw(Es)43 b(gibt)g(v)m(ersc)m(hiedene)g
	(mathematisc)m(he)h(Konstruktionen,)e(die)g(die)g(Grundlage)h(v)m(on)
	270 2017 y(V)-8 b(ersc)m(hl)559 2021 y(\177)556 2017
	y(usselungsv)m(erfahren)30 b(bilden.)h(Bei)h(den)g(Public)e(Key)j
	(Kryptosystemen)f(gibt)g(es)270 2130 y(im)37 b(w)m(esen)m(tlic)m(hen)g
	(zw)m(ei)h(Ans)1301 2134 y(\177)1301 2130 y(atze.)h(Die)f(einen)f(V)-8
	b(erfahren)37 b(v)m(erdank)m(en)h(ihre)e(Sic)m(herheit)270
	2243 y(dem)26 b(Problem,)g(da\031)h(es)f(sc)m(h)m(w)m(er)i(ist,)e(sehr)
	f(gro\031e)j(Zahlen)d(zu)i(faktorisieren)e(\(z.)16 b(B.)27
	b Fu(RSA)p Fw(\).)270 2356 y(Andere)k(V)-8 b(erfahren)32
	b(hingegen)f(v)m(ertrauen)h(auf)f(die)g(Sc)m(h)m(wierigk)m(eit,)g
	(diskrete)g(Logarith-)270 2469 y(men)f(in)f(endlic)m(hen)g(ab)s(elsc)m
	(hen)g(Grupp)s(en)f(zu)j(b)s(erec)m(hnen)e(\(z.)16 b(B.)31
	b Fu(ElGamal)9 b Fw(\).)411 2584 y(Dieses)29 b(Kapitel)f(hat)i(den)e
	(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
	2588 y(\177)3197 2584 y(usse-)270 2697 y(lungsv)m(erfahren)34
	b(b)s(ereitzustellen.)g(Dab)s(ei)h(gehen)h(wir)e(zuerst)i(auf)f(das)h
	Fu(Diskr)-5 b(ete)37 b(L)-5 b(o)g(ga-)270 2810 y(rithmus)39
	b(Pr)-5 b(oblem)38 b($DLP$)h Fw(ein.)c(Ansc)m(hlie\031end)f(k)m
	(ommen)j(wir)d(zur)h(allgemeinen)f(De\014-)270 2923 y(nition)g(des)i
	Fu(ElGamal)9 b Fw(-V)-8 b(erfahrens.)39 b(Sc)m(hlie\031lic)m(h)34
	b(wird)g(die)i(sp)s(ezielle)e(V)-8 b(arian)m(te)38 b(mit)d(el-)270
	3036 y(liptisc)m(hen)k(Kurv)m(en)h(f)1053 3040 y(\177)1050
	3036 y(ur)h(den)f(hier)g(en)m(t)m(wic)m(k)m(elten)j(Pro)m(vider)d(v)m
	(orgestellt.)i(Auf)f(andere)270 3149 y(Alternativ)m(en)30
	b(soll)f(am)i(Sc)m(hlu\031)e(dieses)h(Kapitels)f(kurz)h(eingegegangen)i
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	%%Page: 16 17
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	%%Page: 17 18
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	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
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	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)
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	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
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	m(h)g(der)f(zugrundeliegenden)390 5609 y(mathematisc)m(hen)c
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	%%Page: 21 22
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	(liegt)i(vielmehr)e(in)h(der)270 641 y(standardisierten)38
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	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)
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	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
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	(Au\031erdem)f(ist)g(die)f(Institution)f Ft(IEEE)j Fw(eine)f(f)3075
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	b(k)m(ommen)g(n)m(un)f(zu)h(den)g(f)1522 2356 y(\177)1519
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	y(v)m(en)60 b(b)s(efassen,)f(w)m(ar)h(das)f(Kapitel)f(7.1)j(in)d
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	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
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	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
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	(enpunk-)497 4493 y(ten\))406 4704 y Fo(\017)46 b Fw(Die)31
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	(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
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	%%Page: 22 23
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	1283 y(ur)f(elliptisc)m(he)f(Kurv)m(en)i(w)m(eiterhin)e(sc)m(h)m(w)m
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	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
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	(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)
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	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)
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	(spielt,)f(sind)g(andere)h(masc)m(hinenn)3055 2451 y(\177)3055
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	(gibt)f(damit)f(ab)s(er)h(die)g(Plattform)m(unabh)3153
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	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
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	(zw)m(ei)i(Stellen,)e(an)h(denen)f(LiDIA)h(b)s(en)m(utzt)g(wird.)e(Bei)
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	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
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	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
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	2609 4141 V 32 w(P1363)f Fw(\(eine)g(Besc)m(hrei-)390
	4254 y(bung)63 b(\014ndet)h(man)g(in)f(dem)h(Man)m(ual)g([LiD99)q(]\).)
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	(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
	y(andige)g(Erkl)1988 2188 y(\177)1988 2184 y(arung)e(des)i(Pro)m
	(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
	b(hier)f(v)m(orgestellte)i(Pro)m(vider)e(b)s(esteh)m(t)h(aus)g
	(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
	(wurden)f(die)270 2858 y(Klassen)d(auf)h(mehrere)h(T)-8
	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
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	%%Page: 26 27
	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
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	%%Page: 27 28
	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
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	(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
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	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
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	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
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	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
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	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
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	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
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	1883 y(Klassen)38 b Ff(ECElGamalGFPPublicKey)32 b Fw(und)37
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	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,)
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	(Klassen)g(hab)s(en.)p eop
	%%Page: 28 29
	28 28 bop eop
	%%Page: 29 30
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	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
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	%%Page: 49 50
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	%%Page: 50 51
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	5643 V eop
	%%Page: 51 52
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	%%Page: 52 53
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	%%Page: 53 54
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	%%Page: 54 55
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	%%Page: 55 56
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	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
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	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\(betaZer)o(oXBy)o(tes,)o(0,o)o(ut,o)o(utOf)
	o(f+)p 3507 4062 V 274 4141 a Fa(361)p 390 4175 V 531
	w Ff(mCipherBlockSize/4*3-betaZ)o(eroX)o(Byt)o(es.l)o(engt)o(h,)p
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	(betaZeroXBytes.length\);)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\(byte[])c(in,int)j(inOff,int)f(inLen,)p
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	(outOff\))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
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	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
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	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
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	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
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	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
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	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
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	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
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	390 4288 V 3507 4288 V 274 4367 a(457)p 390 4401 V 436
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	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
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	390 5304 V 245 w Ff(protected)f(byte[])h(engineGetIV)p
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	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
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	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
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	3387 901 V 154 980 a Fa(474)p 270 1014 V 245 w Ff(protected)e(void)i
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	154 1544 a Fa(479)p 270 1578 V 245 w Ff(})p 3387 1578
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	270 2256 V 245 w Ff(protected)e(void)i(engineSetPadding\(String)41
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	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
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	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
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	(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
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	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
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	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)
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	5078 V 293 w Ff(*)g(@param)f(percentage)f(the)i(lBigInt)f(specifies)f
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	154 5270 a Fa(310)p 270 5304 V 293 w Ff(*)g(@param)f(info)h(the)g
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	3387 5304 V 154 5383 a Fa(311)p 270 5417 V 293 w Ff(*)g(messages)f(\(1)
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	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
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	1014 V 293 w Ff(*)g(integer)p 3507 1014 V 274 1093 a
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	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
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	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
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	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
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	V 3507 2481 V 274 2560 a(333)p 390 2594 V 245 w Ff(/**)p
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	245 w Ff(public)h(native)g(LBigInt)g(LgetK;)p 3507
	3498 V 274 3577 a Fa(342)p 390 3610 V 3507 3610 V 274
	3689 a(343)p 390 3723 V 245 w Ff(/**)p 3507 3723 V 274
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	Fa(345)p 390 3949 V 293 w Ff(*/)p 3507 3949 V 274 4028
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	V 274 4254 a(348)p 390 4288 V 245 w Ff(/**)p 3507 4288
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	V 274 4593 a Fa(351)p 390 4627 V 245 w Ff(public)h(native)g(LBigInt)g
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	390 4767 3120 4 v eop
	%%Page: 79 80
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	(arametersP1363)49 b($C++-Co)t(de$)p 270 650 3120 4
	v 270 675 4 25 v 3387 675 V 215 754 a Fa(1)p 270 788
	4 113 v 53 w Ff(/**********************)o()o()o(*)o(**)o
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	V 215 867 a Fa(2)p 270 901 V 53 w Ff(**)p 3387 901 V
	215 980 a Fa(3)p 270 1014 V 53 w Ff(**)e(Implementierung)d(of)j(the)g
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	1014 V 215 1093 a Fa(4)p 270 1126 V 53 w Ff(**)f(LiDIA-class)e
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	1352 V 215 1431 a Fa(7)p 270 1465 V 53 w Ff(**)g(parameters)e(of)j
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	1578 V 215 1657 a Fa(9)p 270 1691 V 53 w Ff(**)p 3387
	1691 V 184 1770 a Fa(10)p 270 1804 V 54 w Ff(**)g(The)g(class)g(and)g
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	V 184 2673 a Fa(18)p 270 2707 V 3387 2707 V 184 2786
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	2933 V 54 w Ff(#include)g("LJNI.h")p 3387 2933 V 184
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	3272 V 54 w Ff(//)p 3387 3272 V 184 3351 a Fa(24)p 270
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	270 3498 V 54 w Ff(//)p 3387 3498 V 184 3577 a Fa(26)p
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	w Ff(*)g(Class:)237 b(cdc_lidia_LECDomainParamet)o(ers)o(P136)o(3)p
	3387 3836 V 184 3915 a Fa(29)p 270 3949 V 102 w Ff(*)47
	b(Method:)189 b(public)46 b(static)h(void)f(staticDPInit)p
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	(V)p 3387 4062 V 184 4141 a Fa(31)p 270 4175 V 102
	w Ff(*/)p 3387 4175 V 184 4254 a Fa(32)p 270 4288 V 54
	w Ff(jclass)h(LECDomainParametersP1363)41 b(=)48 b(NULL;)p
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	(3)d(=)p 3387 4401 V 184 4480 a Fa(34)p 270 4514 V 150
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	V 3387 4627 V 184 4706 a(36)p 270 4740 V 54 w Ff(JNIEXPORT)j(void)i
	(JNICALL)p 3387 4740 V 184 4819 a Fa(37)p 270 4852 V
	150 w Ff(Java_cdc_lidia_LECDomai)o(nPar)o(ame)o(ters)o(P136)o(3_s)o
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	4965 V 150 w Ff($JNIEnv)e(*env,)i(jclass)f(jMyClass$)p
	3387 4965 V 184 5044 a Fa(39)p 270 5078 V 54 w Ff({)p
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	5270 a Fa(41)p 270 5304 V 150 w Ff(LECDomainParametersP136)o(3)42
	b(=)47 b($jclass$)f(env->NewGlobalRef)p 3387 5304 V
	184 5383 a Fa(42)p 270 5417 V 245 w Ff($jMyClass$;)p
	3387 5417 V 184 5496 a Fa(43)p 270 5530 V 150 w Ff(if)h
	($handleFatalJavaExceptio)o(n)42 b(\(env$\))p 3387
	5530 V 184 5609 a Fa(44)p 270 5643 V 245 w Ff(return;)p
	3387 5643 V eop
	%%Page: 80 81
	80 80 bop 390 82 a Fw(80)1950 b(ANHANG)31 b(C.)61 b(CDC.LIDIA)304
	415 y Fa(45)p 390 449 4 113 v 3507 449 V 304 528 a(46)p
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	(LECDomainParametersP1363.)p 3507 562 V 304 641 a Fa(47)p
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	(1363)41 b(=)p 3507 788 V 304 867 a Fa(49)p 390 901 V
	245 w Ff(env->GetMethodID)i(\(LECDomainParametersP1363,)e
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	1014 V 150 w Ff(if)47 b(\(handleFatalJavaExceptio)o(n)42
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	V 245 w Ff(return;)p 3507 1126 V 304 1205 a Fa(52)p 390
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	1352 V 3507 1352 V 304 1431 a(54)p 390 1465 V 54 w Ff(/*)p
	3507 1465 V 304 1544 a Fa(55)p 390 1578 V 102 w Ff(*)47
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	150 w Ff(Java_cdc_lidia_LECDomai)o(nPar)o(ame)o(ters)o(P136)o(3_i)o
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	2933 V 304 3012 a Fa(68)p 390 3046 V 150 w Ff(nThis.p)g(=)j(new)f
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	b(Method:)189 b(public)46 b(void)h(finalize)p 3507
	3949 V 304 4028 a Fa(77)p 390 4062 V 102 w Ff(*)g(Signature:)e(V)p
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	150 w Ff(Java_cdc_lidia_LECDomai)o(nPar)o(ame)o(ters)o(P136)o(3_f)o
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	%%Page: 81 82
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	1431 a Fa(101)p 270 1465 V 102 w Ff(*/)p 3387 1465 V
	154 1544 a Fa(102)p 270 1578 V 54 w Ff(JNIEXPORT)45 b(void)i(JNICALL)p
	3387 1578 V 154 1657 a Fa(103)p 270 1691 V 150 w Ff
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	3387 2820 V 154 2899 a Fa(302)p 270 2933 V 54 w Ff(})p
	3387 2933 V 154 3012 a Fa(303)p 270 3046 V 3387 3046
	V 154 3125 a(304)p 270 3159 V 54 w Ff(/*)p 3387 3159
	V 154 3238 a Fa(305)p 270 3272 V 102 w Ff(*)h(Class:)237
	b(cdc_lidia_LECDomainParamet)o(ers)o(P136)o(3)p 3387
	3272 V 154 3351 a Fa(306)p 270 3385 V 102 w Ff(*)47 b(Method:)189
	b(public)46 b(LBigInt)g(LgetX)p 3387 3385 V 154 3464
	a Fa(307)p 270 3498 V 102 w Ff(*)h(Signature:)e
	(Lcdc/lidia/LBigInt;)p 3387 3498 V 154 3577 a Fa(308)p
	270 3610 V 102 w Ff(*/)p 3387 3610 V 154 3689 a Fa(309)p
	270 3723 V 54 w Ff(JNIEXPORT)g(jobject)h(JNICALL)p 3387
	3723 V 154 3802 a Fa(310)p 270 3836 V 150 w Ff(Java_cdc_lidia_LECDomai)
	o(nPar)o(ame)o(ters)o(P136)o(3_L)o(getX)p 3387 3836 V
	154 3915 a Fa(311)p 270 3949 V 150 w Ff(\(JNIEnv)f(*env,)i(jobject)f
	(jThis\))p 3387 3949 V 154 4028 a Fa(312)p 270 4062 V
	54 w Ff({)p 3387 4062 V 154 4141 a Fa(313)p 270 4175
	V 150 w Ff(//)h(looks)f(for)h(the)g(EC_domain_parameters_P13)o(63)p
	3387 4175 V 154 4254 a Fa(314)p 270 4288 V 150 w Ff(jPointer)e(nThis;)p
	3387 4288 V 154 4367 a Fa(315)p 270 4401 V 150 w Ff(nThis.l)g(=)j
	(env->GetLongField)43 b($jThis,)j(LiDIAObject_nThis$;)p
	3387 4401 V 154 4480 a Fa(316)p 270 4514 V 150 w Ff(if)h
	($handleJavaException)42 b(\(env$\))p 3387 4514 V 154
	4593 a Fa(317)p 270 4627 V 245 w Ff(return)k(NULL;)p
	3387 4627 V 154 4706 a Fa(318)p 270 4740 V 3387 4740
	V 154 4819 a(319)p 270 4852 V 150 w Ff(//)h(calculates)p
	3387 4852 V 154 4932 a Fa(320)p 270 4965 V 150 w Ff(jobject)e(Result;)p
	3387 4965 V 154 5044 a Fa(321)p 270 5078 V 150 w Ff(jPointer)g
	(jpResult;)p 3387 5078 V 154 5157 a Fa(322)p 270 5191
	V 150 w Ff(bigint)h(*b_Result)f(=)i(new)g(bigint)f
	($\(\(EC_domain_parameters_P1)o(363)41 b(*$)p 3387
	5191 V 154 5270 a Fa(323)p 270 5304 V 245 w Ff
	(nThis.p\)->get_x.mantiss)o(a)o(\);)p 3387 5304
	V 154 5383 a Fa(324)p 270 5417 V 150 w Ff(jpResult.p)k(=)i(b_Result;)p
	3387 5417 V 154 5496 a Fa(325)p 270 5530 V 3387 5530
	V 154 5609 a(326)p 270 5643 V 150 w Ff(//)g(creates)f(the)h(return)f
	(object)p 3387 5643 V eop
	%%Page: 86 87
	86 86 bop 390 82 a Fw(86)1950 b(ANHANG)31 b(C.)61 b(CDC.LIDIA)274
	415 y Fa(327)p 390 449 4 113 v 150 w Ff(Result)46 b(=)h(env->NewObject)
	d(\(LBigInt,LBigInt_LBigInt,)o(jpR)o(esul)o(t.l,)p 3507
	449 V 274 528 a Fa(328)p 390 562 V 245 w Ff(JNI_TRUE\);)p
	3507 562 V 274 641 a Fa(329)p 390 675 V 150 w Ff(if)j
	($handleJavaException)42 b(\(env$\))p 3507 675 V 274
	754 a Fa(330)p 390 788 V 245 w Ff(return)k(NULL;)p 3507
	788 V 274 867 a Fa(331)p 390 901 V 3507 901 V 274 980
	a(332)p 390 1014 V 150 w Ff(//)h(and)g(back)p 3507 1014
	V 274 1093 a Fa(333)p 390 1126 V 150 w Ff(return)f(Result;)p
	3507 1126 V 274 1205 a Fa(334)p 390 1239 V 54 w Ff(})p
	3507 1239 V 274 1318 a Fa(335)p 390 1352 V 3507 1352
	V 274 1431 a(336)p 390 1465 V 54 w Ff(/*)p 3507 1465
	V 274 1544 a Fa(337)p 390 1578 V 102 w Ff(*)h(Class:)237
	b(cdc_lidia_LECDomainParamet)o(ers)o(P136)o(3)p 3507
	1578 V 274 1657 a Fa(338)p 390 1691 V 102 w Ff(*)47 b(Method:)189
	b(public)46 b(LBigInt)g(LgetY)p 3507 1691 V 274 1770
	a Fa(339)p 390 1804 V 102 w Ff(*)h(Signature:)e
	(Lcdc/lidia/LBigInt;)p 3507 1804 V 274 1883 a Fa(340)p
	390 1917 V 102 w Ff(*/)p 3507 1917 V 274 1996 a Fa(341)p
	390 2030 V 54 w Ff(JNIEXPORT)g(jobject)h(JNICALL)p 3507
	2030 V 274 2109 a Fa(342)p 390 2143 V 150 w Ff(Java_cdc_lidia_LECDomai)
	o(nPar)o(ame)o(ters)o(P136)o(3_L)o(getY)p 3507 2143 V
	274 2222 a Fa(343)p 390 2256 V 150 w Ff(\(JNIEnv)f(*env,)i(jobject)f
	(jThis\))p 3507 2256 V 274 2335 a Fa(344)p 390 2368 V
	54 w Ff({)p 3507 2368 V 274 2447 a Fa(345)p 390 2481
	V 150 w Ff(//)h(looks)f(for)h(the)g(EC_domain_parameters_P13)o(63)p
	3507 2481 V 274 2560 a Fa(346)p 390 2594 V 150 w Ff(jPointer)e(nThis;)p
	3507 2594 V 274 2673 a Fa(347)p 390 2707 V 150 w Ff(nThis.l)g(=)j
	(env->GetLongField)43 b($jThis,)j(LiDIAObject_nThis$;)p
	3507 2707 V 274 2786 a Fa(348)p 390 2820 V 150 w Ff(if)h
	($handleJavaException)42 b(\(env$\))p 3507 2820 V 274
	2899 a Fa(349)p 390 2933 V 245 w Ff(return)k(NULL;)p
	3507 2933 V 274 3012 a Fa(350)p 390 3046 V 3507 3046
	V 274 3125 a(351)p 390 3159 V 150 w Ff(//)h(calculates)p
	3507 3159 V 274 3238 a Fa(352)p 390 3272 V 150 w Ff(jobject)e(Result;)p
	3507 3272 V 274 3351 a Fa(353)p 390 3385 V 150 w Ff(jPointer)g
	(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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