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\chapter{Implementation of EC-ElGamal Library}
\section{Limitations of Current EC-ElGamal Schemes}
After long research, no existing cryptographic library provides a scheme for ElGamal on Elliptic Curve with homomorphic operations, as none performs an actual mapping of message onto the elliptic curve and back. This motivates our need to implement a new modules providing Elliptic Curve ElGamal with homomorphic operations.
\section{Cryptographic Library Selection}
In order to build that module, we want to rely on cryptographic primitives provided by a standard cryptographic library. The desired library should have the following features. It should
\begin{itemize}
\item Provide a low level API for handling elliptic curve points,
\item Have a wide catalog of elliptic curves, including NIST recommended curves,
\item Be an open source project, as this project is academic and we want to have full control over the algorithms used,
\item Have a good support in case we run into any kind of bug.
\end{itemize}
The first library inspected was the default Java Cryptography Extension (JCE) \citep{jca-ref-guide} delivered with all Java Runtime Environments (JRE). Its main limitation is that it does not provide an API to handle elliptic curve (EC) points.
The second library inspected was the Java BouncyCastle library \citep{bc-ref}. This second one offers good EC point handling with compression capabilities (\lstinline{org.bouncycastle.math.ec.ECPoint.getEncoded(boolean compression)}). It is also open source and is well supported. There are weekly improvements, and it offers a broad range of support interfaces through Github \citep{git-bc-activity} and JIRA \citep{bc-jira} issue trackers. As BouncyCastle also allows to fine-tune the usage of asymmetric and symmetric encryption schemes, we finally chose it as our cryptographic primitive library and Java Security Provider (providing the implementations of the encryption schemes available in Java Cryptography Architecture).
\section{Design Decisions}
\subsection{Reverse Mapping from EC Space onto Message Space}
As described in \refsec{ecceg-mapping-decision}, we generate and store a lookup table mapping every element of the curve to the message space. The Java implementation uses the \verb#java.util.HashMap<K,V># class to map \verb#org.bouncycastle.math.ec.ECPoint# to \verb#java.math.BigInteger#. We iterated over the whole message space which is of limited size. For every message $m$, we computed $mG$ and stored the $(mG, m)$ entry into the \verb#HashMap<ECPoint, BigInteger>#.
After having evaluated the strengths and weaknesses of elliptic curves in \refsec{sec:perf-analysis}, the elliptic curve chosen for the application is unlikely to vary. Hence it is reasonable to generate the lookup table only once and distribute it as part of the application package. Please also note that the performance of this implementation will strongly depend on the implementation of the \lstinline{hashCode()} method in \lstinline{org.bouncycastle.math.ec.ECPoint}.
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