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\subsubsection{Time complexity} -It is generally remarked that Paillier performs considerably slower than RSA on all key sizes. %todo cite +It is remarked that Paillier encryption performs considerably slower than RSA on all key sizes. \cite{paillier1999public} provides a table of theoretic results, suggesting that Paillier encryption can be over 1,000 times slower than RSA for the same key size. + +\cite{paillier1999public} also remarks that the choice of the public parameter $g$ can improve the time complexity. The selection of $g = n + 1$ is optimal in this regard, as binomial theorem allows the modular exponentiation $g^m \mod n^2$ to be reduced to the computation $1 + gm \mod n^2$. + +These results are backed experimentally by my implementation. The following benchmarking code was executed. + +\begin{minted}{javascript} + console.log("Warming up") + + for (let i = 0n; i < 100n; i++) { + keyPair.pubKey.encrypt(i); + } + + console.log("Benching") + + performance.mark("start") + for (let i = 0n; i < 250n; i++) { + keyPair.pubKey.encrypt(i); + } + performance.mark("end") + + console.log(performance.measure("duration", "start", "end").duration) +\end{minted} + +Performing 250 Paillier encrypts required 49,100ms. On the other hand, performing 250 RSA encrypts required 60ms. This is a difference of over 1,000 times. + +Decryption is remarked as being optimisable to constant time through application of Chinese Remainder Theorem. \subsection{Quantum resistance}