From b7150d6547a77fe4f4ded845ac680d51de3d4db3 Mon Sep 17 00:00:00 2001 From: jude Date: Thu, 13 Apr 2023 13:32:29 +0100 Subject: [PATCH] Fixed stuff --- static/js/modules/crypto/paillier.js | 64 ++++++++++++++-------- static/js/modules/crypto/random_primes.js | 6 +- static/js/modules/interface/proofs.js | 6 +- whitepaper/Dissertation.pdf | Bin 385443 -> 387139 bytes whitepaper/Dissertation.tex | 28 ++++++---- 5 files changed, 69 insertions(+), 35 deletions(-) diff --git a/static/js/modules/crypto/paillier.js b/static/js/modules/crypto/paillier.js index 5be998d..256112a 100644 --- a/static/js/modules/crypto/paillier.js +++ b/static/js/modules/crypto/paillier.js @@ -1,24 +1,45 @@ -import { cryptoRandom, generate_prime } from "./random_primes.js"; +import { cryptoRandom, generate_prime, KEY_SIZE } from "./random_primes.js"; import { gcd, mod_exp } from "./math.js"; +const PAILLIER = 0; +const JURIK = 1; + class Ciphertext { constructor(key, plainText, r) { if (r === undefined) { - r = cryptoRandom(2048); + // Use the optimised form using Jacobi classes + r = cryptoRandom(); + + // Compute g^m by binomial theorem. + let gm = (1n + key.n * plainText) % key.n2; + + // Compute g^m h^r. + this.cipherText = (gm * mod_exp(key.hn, r, key.n2)) % key.n2; + + // Force into range. + while (this.cipherText < 0n) { + this.cipherText += key.n2; + } + + this.mode = JURIK; + this.r = mod_exp(key.h, r, key.n); + } else { + // Use the standard form + // Compute g^m by binomial theorem. + let gm = (1n + key.n * plainText) % key.n2; + + // Compute g^m r^n. + this.cipherText = (gm * mod_exp(r, key.n, key.n2)) % key.n2; + + // Force into range. + while (this.cipherText < 0n) { + this.cipherText += key.n2; + } + + this.mode = PAILLIER; + this.r = r; } - // Compute g^m by binomial theorem. - let gm = (1n + key.n * plainText) % key.n2; - - // Compute g^m h^r. - this.cipherText = (gm * mod_exp(key.hn, r, key.n2)) % key.n2; - - // Force into range. - while (this.cipherText < 0n) { - this.cipherText += key.n2; - } - - this.r = r; this.pubKey = key; this.plainText = plainText; @@ -46,9 +67,9 @@ class Ciphertext { // Construct a non-interactive proof proveNI() { - let rp = cryptoRandom(4096); + let rp = cryptoRandom(KEY_SIZE * 2); while (rp >= this.pubKey.n) { - rp = cryptoRandom(4096); + rp = cryptoRandom(KEY_SIZE * 2); } let a = mod_exp(rp, this.pubKey.n, this.pubKey.n2); @@ -92,9 +113,9 @@ class ValueProofSessionProver { constructor(cipherText) { this.cipherText = cipherText; - this.rp = cryptoRandom(4096); + this.rp = cryptoRandom(KEY_SIZE * 2); while (this.rp >= this.cipherText.pubKey.n) { - this.rp = cryptoRandom(4096); + this.rp = cryptoRandom(KEY_SIZE * 2); } } @@ -168,7 +189,7 @@ class ValueProofSessionVerifier { if (challenge === undefined) { // Shift the challenge down by 1 to ensure it is smaller than either prime factor. - this.challenge = cryptoRandom(2048) << 1n; + this.challenge = cryptoRandom(KEY_SIZE) << 1n; } else { this.challenge = challenge; } @@ -205,10 +226,9 @@ export class PaillierPubKey { this.n = n; if (h === undefined) { - let x = cryptoRandom(4096); - + let x = cryptoRandom(KEY_SIZE * 2); while (x >= this.n) { - x = cryptoRandom(4096); + x = cryptoRandom(KEY_SIZE * 2); } this.h = ((-1n * x ** 2n) % this.n) + this.n; diff --git a/static/js/modules/crypto/random_primes.js b/static/js/modules/crypto/random_primes.js index 8729502..9f4c839 100644 --- a/static/js/modules/crypto/random_primes.js +++ b/static/js/modules/crypto/random_primes.js @@ -1,8 +1,10 @@ import { mod_exp } from "./math.js"; +export const KEY_SIZE = 512; + export function cryptoRandom(bits) { if (bits === undefined) { - bits = 2048; + bits = KEY_SIZE; } let length = bits / 64; @@ -29,7 +31,7 @@ function generate_bigint() { intRepr >>= 1n; // Add 2^2047 to force into range from below - intRepr += 2n ** 2047n; + intRepr += 2n ** BigInt(KEY_SIZE - 1); return intRepr; } diff --git a/static/js/modules/interface/proofs.js b/static/js/modules/interface/proofs.js index ada55cd..93f24b7 100644 --- a/static/js/modules/interface/proofs.js +++ b/static/js/modules/interface/proofs.js @@ -1,4 +1,8 @@ -// Fisher-Yates shuffle +/** + * CSPRNG Fisher-Yates shuffle. + * + * Only works on lists up to 255 elements. + */ function cryptoShuffle(l) { for (let i = l.length - 1; i > 0; i--) { let value = new Uint8Array([0]); diff --git a/whitepaper/Dissertation.pdf b/whitepaper/Dissertation.pdf index 301a1695191630c860466df3859d889b9dcf09e8..6c214bddc8c5c25146571e77e06a5f50fa8017c0 100644 GIT binary patch delta 100871 zcmV(@K-Ryb!xzKQ7qCcK0W*`Z0V#i-T1$`PHWI$iuP}0n8W<~lN))m=Bta%SiwzL$ zHWFZyIjC)pEsS3y$eBJJzsM8$k)2Th4+NSP34ZXix%ItV2vzFHNTmG=|2jA9xb39Q0 z_?{+o=RbumR*gU0;q zQ$mT?cT1~8_S28QeE;uN+njp()X*92@oovf>+{O5NgPs}u{r-KZu24ip~pBrc;!ivZ8d>`e z*nbG@0;@M51zY+BhT0#cWa*8p8c?K_+1_M`*_%Jy{L2j zoPLNh*QFuKoBe6$_pE zsdtkptq;VDymBMjOP3p`iS~VsTx+ef$EM$6OJ%L!)a|;XVSu+DI z%&vEvV+$MYxf^KIiqr%KPIcQit23Yo@4n?7_o#CYY=8LV|2l;s1g%3j=76X64T06z zWRa=D(dr_c#13N_te%akh%{_2H6`r 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exponentiation. Selection of $g = n + 1$ is good in this regard, as binomial theorem allows the modular exponentiation $g^m \mod n^2$ to be reduced to the computation $1 + nm \mod n^2$. -\textbf{Smaller key size.} The complexity of Paillier encryption increases with key size. Using a smaller key could considerably reduce the time taken \cite{paillier1999public}. - \textbf{Caching.} As the main values being encrypted are 0 or 1, a peer could maintain a cache of encryptions of these values and transmit these instantly. Caching may be executed in a background "web worker". A consideration is whether a peer may be able to execute a timing-related attack by first exhausting a peer's cache of a known plaintext value, and then requesting an unknown value and using the time taken to determine if the value was sent from the exhausted cache or not. Taking this idea further, one may simply cache $r^n$ for a number of randomly generated $r$ (as this is the slowest part of encryption). This eliminates the timing attack concern, and grants full flexibility with the values being encrypted. \textbf{Alternative Paillier scheme.} \cite{Jurik2003ExtensionsTT} presents an optimised encryption scheme based on the subgroup of elements with Jacobi symbol $+1$. This forms a group as the Jacobi symbol is multiplicative, being a generalisation of the Legendre symbol. -I used this scheme to reduce the time to encrypt to half. Greater optimisations are possible through pre-computation of fixed-base exponentials, but this takes a considerable amount of time, and I found it unfeasible within my implementation, since keypairs are only used for a single session. +I used this scheme to reduce the time to encrypt to half. Greater optimisations are possible through pre-computation of fixed-base exponentials, but this takes a considerable amount of time, and I found it infeasible within my implementation, since keypairs are only used for a single session. + +\textbf{Smaller key size.} The complexity of Paillier encryption increases with key size. Using a smaller key could considerably reduce the time taken \cite{paillier1999public}. + +I tested this on top of the alternative Paillier scheme from above. This resulted in linear reductions in encryption time: encryption under a 1024-bit modulus took half the amount of time as under a 2048-bit modulus and so on. \textbf{Vectorised plaintexts.} The maximum size of a plaintext is $|n|$: in our case, this is 4096 bits. By considering this as a vector of 128 32-bit values, peers could use a single ciphertext to represent their entire state. \cite{10.1145/2809695.2809723} uses this process to allow embedded devices to make use of the homomorphic properties of Paillier.