Fully homomorphic encryption over the integers for non-binary plaintexts without the sparse subset sum problem

Abstract

Abstract In this work, we solve the open problem of designing a fully homomorphic encryption scheme over the integers for non-binary plaintexts in Z Q for prime Q ( Q -FHE-OI) without the hardness of the sparse subset sum problem (SSSP). Furthermore, we show that our Q -FHE-OI scheme is a useful optimization for evaluating arithmetic circuits on encrypted data for some primes. To that end, we provide a natural extension of the somewhat homomorphic encryption (SHE) scheme over the integers proposed by Cheon and Stehle (Eurocrypt 2015) to support non-binary plaintexts. Then, a novel bootstrapping algorithm is proposed for this extended SHE scheme by introducing generalizations of several functions in binary arithmetic. As a result, we obtain a Q -FHE-OI scheme for any constant-sized prime Q ≥ 3 without the hardness of the SSSP, whose bootstrapping algorithm is asymptotically as efficient as previous best results. Beyond that, we compare the efficiency of our scheme against a Q -FHE-OI scheme obtained by emulating mod- Q gates with boolean circuits as proposed by Kim and Tibouchi (CANS 2016). Our analysis indicates our proposed scheme performs better for prime Q up to 11287, which improves on the result of Kim and Tibouchi, who showed there is at most one prime, Q = 3 where the Q -FHE-OI scheme by Nuida and Kurosawa (Eurocrypt 2015) is a better approach. This overturns our previous understanding that Q -FHE-OI schemes do not provide significant benefits.