Paper 2023/1304

Homomorphic polynomial evaluation using Galois structure and applications to BFV bootstrapping

Abstract

BGV and BFV are among the most widely used fully homomorphic encryption (FHE) schemes. Both schemes have a common plaintext space, with a rich algebraic structure. Our main contribution is to show how this structure can be exploited to more efficiently homomorphically evaluate polynomials. Namely, using Galois automorphisms, we present an algorithm to homomorphically evaluate a polynomial of degree $d$ in only $3\log(d)$ (in some cases only $2\log(d)$) many ciphertext-ciphertext multiplications and automorphism evaluations, where $d$ is bounded by the ring degree. In other words, as long as the degree of the polynomial is bounded, we achieve an exponential speedup compared to the state of the art. In particular, the approach also improves on the theoretical lower bound of $2\sqrt{d}$ many ciphertext-ciphertext multiplications, which would apply if automorphisms were not available. We investigate how to apply our improved polynomial evaluation to the bootstrapping procedure for BFV, and show that we are able to significantly improve its performance. We demonstrate this by providing an implementation of our improved BFV bootstrapping using the Microsoft SEAL library. More concretely, we obtain a $1.6\times$ speed up compared to the prior implementation given by Chen and Han (Eurocrypt 2018). The techniques are independent of, and can be combined with, the more recent optimisations presented by Geelen et al. (Eurocrypt 2023). As an additional contribution, we show how the bootstrapping approach used in schemes such as FHEW and TFHE can be applied in the BFV context. In particular, we demonstrate that programmable bootstrapping can be achieved for BFV. Moreover, we show how this bootstrapping approach can be improved in the BFV context to make better use of the Galois structure. However, we estimate that its complexity is around three orders of magnitude slower than the classical approach to BFV bootstrapping.