Faster Bootstrapping of FHE over the Integers

Abstract

In FHE over the integers, decryption function is simplified by sparse subset subset sum problem (SSSP) assumption, which is introduced by Dijk et al. (Eurocrypt 2010), so that bootstrapping can be achieved successfully. Later, Nuida and Kurowasa (Eurocrypt 2015) proposed an advanced method of which the degree is very low and the message space is non-binary. These previous methods require low degree but more than O(λ4) homomorphic multiplications which make them very slow. For a general bootstrapping method in FHE over the integers, the number of homomorphic multiplications and the degree of decryption function are important factors for the efficiency of bootstrapping procedure. In this paper, we propose a new bootstrapping method for FHE over the integers requiring only O(log2λ) homomorphic multiplications which is significantly lower than previous methods. Implementing our bootstrapping method on the scale-invariant FHE over the integers called CLT scheme, it takes 6 s for 500-bit message space and 80-bit security on a desktop. We also apply our bootstrapping method to the homomorphic evaluation of AES-128 circuit: It takes about 8 s per 128-bit block and is faster than the previous results of homomorphic AES evaluation using FHEs over the integers without bootstrapping.