From 1d9ab1d601b3dab9d874463140429da96226186a Mon Sep 17 00:00:00 2001 From: jude Date: Tue, 11 Apr 2023 15:39:49 +0100 Subject: [PATCH] jacobi based paillier --- static/js/modules/crypto/paillier.js | 81 +++++++++++++++++++-------- whitepaper/Dissertation.bib | 42 ++++++++++++++ whitepaper/Dissertation.pdf | Bin 383383 -> 385443 bytes whitepaper/Dissertation.tex | 14 +++-- 4 files changed, 111 insertions(+), 26 deletions(-) diff --git a/static/js/modules/crypto/paillier.js b/static/js/modules/crypto/paillier.js index 04fb92d..5be998d 100644 --- a/static/js/modules/crypto/paillier.js +++ b/static/js/modules/crypto/paillier.js @@ -1,22 +1,17 @@ import { cryptoRandom, generate_prime } from "./random_primes.js"; -import { mod_exp } from "./math.js"; +import { gcd, mod_exp } from "./math.js"; class Ciphertext { constructor(key, plainText, r) { if (r === undefined) { - r = cryptoRandom(4096); - - // Resample to avoid modulo bias. - while (r >= key.n) { - r = cryptoRandom(4096); - } + r = cryptoRandom(2048); } // Compute g^m by binomial theorem. let gm = (1n + key.n * plainText) % key.n2; - // Compute g^m r^n from crt. - this.cipherText = (gm * mod_exp(r, key.n, key.n2)) % 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) { @@ -206,10 +201,25 @@ class ValueProofSessionVerifier { window.ReadOnlyCiphertext = ReadOnlyCiphertext; export class PaillierPubKey { - constructor(n) { + constructor(n, h) { this.n = n; - this.n2 = this.n ** 2n; + + if (h === undefined) { + let x = cryptoRandom(4096); + + while (x >= this.n) { + x = cryptoRandom(4096); + } + + this.h = ((-1n * x ** 2n) % this.n) + this.n; + } else { + this.h = h; + } + this.g = this.n + 1n; + + this.n2 = this.n ** 2n; + this.hn = mod_exp(this.h, this.n, this.n2); } encrypt(m, r) { @@ -219,11 +229,12 @@ export class PaillierPubKey { toJSON() { return { n: "0x" + this.n.toString(16), + h: "0x" + this.h.toString(16), }; } static fromJSON(data) { - return new PaillierPubKey(BigInt(data.n)); + return new PaillierPubKey(BigInt(data.n), BigInt(data.h)); } } @@ -241,22 +252,48 @@ class PaillierPrivKey { } } +function check_gcd(primes, new_prime) { + for (let prime of primes) { + if (gcd(prime - 1n, new_prime - 1n) === 2n) { + return prime; + } + } + return null; +} + export function generate_keypair() { let p, q, pubKey, privKey; - if (window.sessionStorage.getItem("p") === null) { - p = generate_prime(); - window.sessionStorage.setItem("p", p); - } else { + if ( + window.sessionStorage.getItem("p") !== null && + window.sessionStorage.getItem("q") !== null + ) { p = BigInt(window.sessionStorage.getItem("p")); + q = BigInt(window.sessionStorage.getItem("q")); + } else { + let p1 = generate_prime(); + while (p1 % 4n !== 3n) { + p1 = generate_prime(); + } + + let primes = [p1]; + + while ( + check_gcd(primes.slice(0, primes.length - 1), primes[primes.length - 1]) === + null + ) { + q = generate_prime(); + while (q % 4n !== 3n) { + q = generate_prime(); + } + primes.push(q); + } + + p = check_gcd(primes.slice(0, primes.length - 1), primes[primes.length - 1]); } - if (window.sessionStorage.getItem("q") === null) { - q = generate_prime(); - window.sessionStorage.setItem("q", q); - } else { - q = BigInt(window.sessionStorage.getItem("q")); - } + window.sessionStorage.setItem("p", p); + window.sessionStorage.setItem("q", q); pubKey = new PaillierPubKey(p * q); privKey = new PaillierPrivKey(p, q); diff --git a/whitepaper/Dissertation.bib b/whitepaper/Dissertation.bib index 5ccb5dd..fb6bd60 100644 --- a/whitepaper/Dissertation.bib +++ b/whitepaper/Dissertation.bib @@ -338,4 +338,46 @@ doi={10.1109/SP.2014.36}} address= {Washington, D.C.}, DOI = {10.6028/NIST.FIPS.202}, year = {2015}, +} + +@inproceedings{Jurik2003ExtensionsTT, + title={Extensions to the Paillier Cryptosystem with Applications to Cryptological Protocols}, + author={Mads Jurik}, + year={2003} +} + +@inproceedings{10.1145/3485832.3485842, + author = {Ma, Huanyu and Han, Shuai and Lei, Hao}, + title = {Optimized Paillier’s Cryptosystem with Fast Encryption and Decryption}, + year = {2021}, + isbn = {9781450385794}, + publisher = {Association for Computing Machinery}, + address = {New York, NY, USA}, + url = {https://doi.org/10.1145/3485832.3485842}, + doi = {10.1145/3485832.3485842}, + abstract = {In this paper, we propose a new optimization for the Paillier’s additively homomorphic encryption scheme (Eurocrypt’99). At the heart of our optimization is a well-chosen subgroup of the underlying , which is used as the randomness space for masking messages during encryption. The size of the subgroup is significantly smaller than that of , leading to faster encryption and decryption algorithms of our optimization.     We establish the one-wayness and semantic security of our optimized Paillier scheme upon those of an optimization (i.e., “Scheme 3”) made by Paillier in Eurocrypt’99. Thus, our optimized scheme is one-way under the partial discrete logarithm (PDL) assumption, and is semantically secure under the decisional PDL (DPDL) assumption. On the other hand, we present a detailed analysis on the concrete security of our optimized scheme under several known methods. To provide 112-bit security, our analysis suggests that a 2048-bit modulus N and a well-chosen subgroup of size 448-bit would suffice. We compare our optimization with existing optimized Paillier schemes, including the Jurik’s optimization proposed by Jurik in his Ph.D. thesis and the Paillier’s optimization in Eurocrypt’99. Our experiments show that, – the encryption of our optimization is about 2.7 times faster than that of the Jurik’s optimization and is about 7.5 times faster than that of the Paillier’s optimization; – the decryption of our optimization is about 4.1 times faster than that of the Jurik’s optimization and has a similar performance with that of the Paillier’s optimization.}, + booktitle = {Annual Computer Security Applications Conference}, + pages = {106–118}, + numpages = {13}, + keywords = {public-key encryption, homomorphic encryption, optimization, provable security}, + location = {Virtual Event, USA}, + series = {ACSAC '21} +} + +@inproceedings{10.1145/2809695.2809723, + author = {Shafagh, Hossein and Hithnawi, Anwar and Droescher, Andreas and Duquennoy, Simon and Hu, Wen}, + title = {Talos: Encrypted Query Processing for the Internet of Things}, + year = {2015}, + isbn = {9781450336314}, + publisher = {Association for Computing Machinery}, + address = {New York, NY, USA}, + url = {https://doi.org/10.1145/2809695.2809723}, + doi = {10.1145/2809695.2809723}, + abstract = {The Internet of Things, by digitizing the physical world, is envisioned to enable novel interaction paradigms with our surroundings. This creates new threats and leads to unprecedented security and privacy concerns. To tackle these concerns, we introduce Talos, a system that stores IoT data securely in a Cloud database while still allowing query processing over the encrypted data. We enable this by encrypting IoT data with a set of cryptographic schemes such as order-preserving and partially homomorphic encryption. In order to achieve this in constrained IoT devices, Talos relies on optimized algorithms that accelerate order-preserving and partially homomorphic encryption by 1 to 2 orders of magnitude. We assess the feasibility of Talos on low-power devices with and without cryptographic accelerators and quantify its overhead in terms of energy, computation, and latency. With a thorough evaluation of our prototype implementation, we show that Talos is a practical system that can provide a high level of security with a reasonable overhead. We envision Talos as an enabler of secure IoT applications.}, + booktitle = {Proceedings of the 13th ACM Conference on Embedded Networked Sensor Systems}, + pages = {197–210}, + numpages = {14}, + keywords = {homomorphic encryption, cloud computing, data security, internet of things, computing on encrypted data}, + location = {Seoul, South Korea}, + series = {SenSys '15} } \ No newline at end of file diff --git a/whitepaper/Dissertation.pdf b/whitepaper/Dissertation.pdf index e0bb78c6bd4ddb6b41445aa0be43eddf8c2e94d0..301a1695191630c860466df3859d889b9dcf09e8 100644 GIT binary patch delta 76930 zcmV)SK(fD=vlpYo7qF8Pf8w^?ugkcun=RKDyeH%ol(h>=OBA_GOmy9V%LS3BIdMam z+c*d!zmAwjcM$Gbo%OOtFshknDvtah@#vaHWHz4@ArSHc8Sb5e`SmuWv#gHGgkL+> z%tMP8UT$Z67ewVIh#Zd^z%LLES9f8f4i5M?5d&;TNeu!_5la#Ef2b`WEmYl+DAKJ^ zlWy@g(InKl_=lP*kQ(qSEvh?GXE{|{Gd^2-J2@_am_shEZ!dwFmzGYDI@gJd1LzIc zUe|EKm>ahoQa@x8+^Nu&1E)2)B*5+5T@prDKm1YdO)hg@ze&fmY2qs7&d(gJZ%cz> zyR?DZ(y@RW% 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