Paper 2013/740

An efficient FHE proposal based on the hardness of solving systems of nonlinear multivariate equations (II)

Gérald Gavin

Abstract

We propose a general framework to develop fully homomorphic encryption schemes (FHE) without using Gentry's technique. Initially, a private-key cryptosystem is built over $\mathbb{Z}_n$ ($n$ being an RSA modulus). An encryption of $x\in \mathbb{Z}_n$ is a randomly chosen vector $e$ such that $\Phi(e)=x$ where $\Phi$ is a secret multivariate polynomial. This private-key cryptosystem is not homomorphic in the sense that the vector sum is not a homomorphic operator. Non-linear homomorphic operators are then developed. The security relies on the difficulty of solving systems of nonlinear equations (which is a $\mathcal{NP}$-complete problem). While the security of our scheme has not been reduced to a provably hard instance of this problem, its security is globally investigated.