Efficient Homomorphic Integer Polynomial Evaluation Based on GSW FHE

Abstract

In this paper, we introduce new methods to evaluate integer polynomials with the Gentry-Sahai-Waters fully homomorphic encryption (GSW FHE) scheme. Our solution has much slower noise growth and per homomorphic integer multiplication cost, with a ratio O((logk)w+1kw·n) in comparison to the original GSW scheme where k is the input plaintext width, n is the LWE dimension and ω=2.373. Technically, we reduce the integer multiplication noise by restricting the evaluation to be between two kinds of ciphertexts: one is for the message space Zq and the other is for message space F2 [logq ]. To achieve generality, we propose an integer bootstrapping scheme which converts these two kinds of ciphertexts into each other. To solve the ciphertext expansion problem due to ciphertexts in F2 [logq], we propose a solution based on symmetric-key encryption with stream ciphers.