CryptoComputing with Rationals

Abstract

In this paper we describe a method to compute with encrypted rational numbers. It is well-known that homomorphic schemes allow calculations with hidden integers, i.e. given integers x and y encrypted in ɛ(x) and ɛ(y), one can compute the encrypted sum ɛ(x + y) or the encrypted product ɛ(kx) of the encrypted integer x and a known integer k without having to decrypt the terms ɛ(x) or ɛ(y). Such cryptosystems have a lot of applications in electronic voting schemes, lottery or in multiparty computation since they allow to keep the privacy of the terms and return the result in encrypted form. However, from a practical point of view, it might be interesting to compute with rationals. For instance, a lot of financial applications require algorithms to compute with rational values instead of integers such as bank accounts, electronic purses in order to make payments or micropayments, or secure spreadsheets. We present here a way to solve this problem using the Paillier cryptosystem which offers the largest bandwidth among all homomorphic schemes. The method uses two-dimensional lattices to recover the numerator and denominator of the rationals. Finally we implement this technique and our results in order to build an encrypted spreadsheet showing the practical possibilities of the homomorphic properties applied on rationals.