A Full RNS Variant of Approximate Homomorphic Encryption

Abstract

The technology of homomorphic encryption has improved rapidly in a few years. The cutting edge implementations are efficient enough to use in practical applications. Recently, Cheon et al. (ASI-ACRYPT’17) proposed a homomorphic encryption scheme which supports an arithmetic of approximate numbers over encryption. This scheme shows the current best performance in computation over the real numbers, but its implementation could not employ core optimization techniques based on the Residue Number System (RNS) decomposition and the Number Theoretic Transformation (NTT). In this paper, we present a variant of approximate homomorphic encryption which is optimal for implementation on standard computer system. We first introduce a new structure of ciphertext modulus which allows us to use both the RNS decomposition of cyclotomic polynomials and the NTT conversion on each of the RNS components. We also suggest new approximate modulus switching procedures without any RNS composition. Compared to previous exact algorithms requiring multi-precision arithmetic, our algorithms can be performed by using only word size (64-bit) operations. Our scheme achieves a significant performance gain from its full RNS implementation. For example, compared to the earlier implementation, our implementation showed speed-ups 17.3, 6.4, and 8.3 times for decryption, constant multiplication, and homomorphic multiplication, respectively, when the dimension of a cyclotomic ring is 32768. We also give experimental result for evaluations of some advanced circuits used in machine learning or statistical analysis. Finally, we demonstrate the practicability of our library by applying to machine learning algorithm. For example, our single core implementation takes 1.8 minutes to build a logistic regression model from encrypted data when the dataset consists of 575 samples, compared to the previous best result 3.5 minutes using four cores.