An efficient FHE based on the hardness of solving systems of non-linear multivariate equations

We propose a general framework to develop fully homomorphic encryption schemes (FHE) without using the Gentry’s technique. Initially, a private-key cryptosystem is built over Zn (n being an RSA modulus). An encryption of x ∈ Zn is a randomly chosen vector e such that Φ(e) = x where Φ 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 non-linear equations (which is a NP-complete problem). While the security of our scheme has not been reduced to a provably hard instance of this problem, security is globally investigated.