Paillier cryptosystem is a probabilistic asymmetric algorithm for public key cryptography and it is particularly interesting as it offers additive homomorphism.
And what is additive homomorphism?
It simply means that you can add together encrypted output and when you decrypt the result it will be the addition of the decrypted inputs.
In this article we are not going to talk about the mathematics behind the Paillier cryptosystem, which you can find also in the Wikipedia page, but the main point is that it is based on the difficulty to solve the problem of computing n-th residue classes.
What are the applications of an additive homomorphic system?
There are basically two main applications of Paillier that is particularly useful:
- Electronic Voting
- Electronic Cash
For example in electronic voting we can use it so people can vote anonymously, make sure that they have voted only once and that the vote is among the valid ones. If you would like to learn more about how Paillier works and why it works the way it does you can find a lot of information simply by googling about it, so I will not expand any further here.
The following implementation is in Python and with a key of 2048 bits using gmpy2. The use of gmpy2 is drastically improving the performance versus a pure python implementation.
import math import random from random import shuffle import sys import gmpy2 from time import time from Crypto.Util.number import getPrime # In: def gcd(a,b): while b > 0: a, b = b, a % b return a def lcm(a, b): return a * b / gcd(a, b) def int_time(): return int(round(time() * 1000)) class PrivateKey(object): def __init__(self, p, q, n): #self.l = lcm(p-1,q-1)----This is added as requested by the setup BUT not used, shortcut is used! self.l = (p-1) * (q-1) #self.m = gmpy2.invert(gmpy2.f_div(gmpy2.sub(gmpy2.powmod(n+1,self.l,n*n),gmpy2.mpz(1)),pub.n),n) --- Shortcut used instead of it self.m = gmpy2.invert(self.l, n) #1/fi(n) def __repr__(self): return '<PrivateKey: %s %s>' % (self.l, self.m) class PublicKey(object): @classmethod def from_n(cls, n): return cls(n) def __init__(self, n): self.n = n self.n_sq = n * n self.g = n + 1 def __repr__(self): return '<PublicKey: %s>' % self.n def generate_keypair(bits): p_equal_q = True while p_equal_q: p = getPrime(bits / 2) q = getPrime(bits / 2) if (p!=q): p_equal_q = False n = p * q return PrivateKey(p, q, n), PublicKey(n) def encrypt(pub, plain): one = gmpy2.mpz(1) state = gmpy2.random_state(int_time()) r = gmpy2.mpz_random(state,pub.n) while gmpy2.gcd(r,pub.n) != one: state = gmpy2.random_state(int_time()) r = gmpy2.mpz_random(state,pub.n) x = gmpy2.powmod(r,pub.n,pub.n_sq) cipher = gmpy2.f_mod(gmpy2.mul(gmpy2.powmod(pub.g,plain,pub.n_sq),x),pub.n_sq) return cipher def decrypt(priv, pub, cipher): one = gmpy2.mpz(1) x = gmpy2.sub(gmpy2.powmod(cipher,priv.l,pub.n_sq),one) plain = gmpy2.f_mod(gmpy2.mul(gmpy2.f_div(x,pub.n),priv.m),pub.n) if plain >= gmpy2.f_div(pub.n,2): plain = plain - pub.n return plain def addemup(pub, a, b): return gmpy2.mul(a,b) def multime(pub, a, n): return gmpy2.powmod(a, n, pub.n_sq)
A thing to note is that the implementation above does support negative numbers, which is something missing from many implementations you will find upon searching for Paillier in Python.\
Finally for those interested in the topic you can check about zero-knowledge-proofs using Paillier and the honest-but-curious protocol. For the later one you can find a python implementation for union and intersection here.