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Paillier Threshold Encryption WebService

This document details the implementation of a Threshold Paillier Encryption Web Service, designed for secure voting and lottery applications. The service utilizes a public key encryption method where secret key shares are distributed among participants using Shamir's Secret Sharing scheme. This enables decryption only when a predefined threshold of participants cooperate, ensuring maximum privacy. Key algorithms for generating keys, encryption, and share decryption are discussed, alongside public verifiability of encrypted votes to support transparent electoral processes. Implementation is realized via Visual Studio 2005 with VB.NET.

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Paillier Threshold Encryption WebService

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  1. Paillier Threshold Encryption WebService by Brett Wilson

  2. Paillier Encryption • Trapdoor Discrete Logarithm Scheme • c = gMrn mod n2 • n is an RSA modulus • g is an integer of order nα mod n2 • r is a random number in Zn* • M = L(cλ(n) mod n2)/L(gλ(n) mod n2) mod n • L(u) = (u-1)/n, λ(n)=lcm((p-1)(q-1)) • Important Properties • Homomorphic • E(M1 + M2) = E(M1) x E(M2), E(k x M) = E(M)k • Self-blinding • Re-encryption with a different r doesn’t change M

  3. Threshold Encryption • Public key encryption as usual • Distribute secret key “shares” among i participants • Decryption can only be accomplished if a threshold number t of the i participants cooperate • No information about m can be obtained with less than t participants cooperating

  4. Threshold Paillier Encryption • Different public key and secret key generation algorithm • Distribute secret key shares using Shamir Secret Sharing scheme • “Sharing Decryption in the Context of Voting or Lotteries” Fouque, Poupard, and Stern 2000

  5. Threshold Paillier Encryption WebService • Key generation algorithm • Input • k – size of key • l – number of shares to generate • One RSA public key (of the designated participant) for each share • t – threshold parameter • Output • Public Key PK • List SK1, …, SKl of private key shares • Encrypted with supplied RSA keys so only designated participant can recover the key share • List of Verifier Keys VK, VK1, …,VKl

  6. Threshold Paillier Encryption WebService • Encryption Algorithm • Input • Public Key PK • Random string r • Cleartext M • Output • Ciphertext c

  7. Threshold Paillier Encryption WebService • Share Decryption Algorithm • Input • Ciphertext c • Private Key Share Ski • Encrypted with public key of webservice • Output • Decryption share ci • Validity proof pi

  8. Threshold Paillier Encryption WebService • Combining Algorithm • Input • Ciphertext c • List of decryption shares c1,…,cl • List of verification keys VK, VK1…VKl • List of validity proofs P1,…Pl • Output • M

  9. Use of WebService in Secure Voting • Ballot format: pick 1 out of c candidates • Vote = 2c*log2v where c is the desired candidate number (0…c) and v is the next power of 2 greater than the maximum number of voters • All Paillier-encrypted votes could be publicly posted • At end of election, all encrypted votes could be multiplied together (publicly verifiable) • With cooperation of the required threshold number of “authorities”, the final product could be decrypted to reveal the vote total (sum of individual votes). • A threshold number of authorities would not agree to decrypt a single particular vote, and thus the individual votes would remain private • All computations are publicly verifiable given the validity proofs

  10. Implementation Tools • Visual Studio 2005 • VB.NET • Gnu Multiprecision Library (Gmp) • Open source arbitrary precision numeric library • Compiled under Visual Studio 2005 • NGmp • Open source VB.NET binding of gmp.dll • Enables calling of gmp library functions through VB.NET • Compiled under Visual Studio 2005

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