Modulo Reduction for Paillier Encryptions and Application to Secure Statistical Analysis

For the homomorphic Paillier cryptosystem we construct a protocol for secure modulo reduction, that on input of an encryption JxK with x of bit length `x and a public ‘modulus’ a of bit length `a outputs an encryption Jx mod aK. As a result, a protocol for computing an encrypted integer division Jxdiv aK is obtained. Surprisingly, efficiency of the protocol is independent of `x: the broadcast complexity of the protocol varies between O(nk`a) and O(n k`a), for n parties and security parameter k, and it is very efficient in case of small `a (in practical cases `a often is much smaller than `x). Our protocol allows for efficient multiparty computation of statistics such as the mean, the variance and the median, and it is therefore very applicable to surveys for the benefit of statistical analysis.