International Association
for Cryptologic Research

In this paper, we perform a systematic study of functions $f: \mathbb{Z}_{p^e} \to \mathbb{Z}_{p^e}$ and categorize those functions that can be represented by a polynomial with integer coefficients. More specifically, we cover the following properties: necessary and sufficient conditions for the existence of an integer polynomial representation; computation of such a representation; and the complete set of equivalent polynomials that represent a given function. As an application, we use the newly developed theory to speed up bootstrapping for the BGV and BFV homomorphic encryption schemes. The crucial ingredient underlying our improvements is the existence of null polynomials, i.e. non-zero polynomials that evaluate to zero in every point. We exploit the rich algebraic structure of these null polynomials to find better representations of the digit extraction function, which is the main bottleneck in bootstrapping. As such, we obtain sparse polynomials that have 50% fewer coefficients than the original ones. In addition, we propose a new method to decompose digit extraction as a series of polynomial evaluations. This lowers the time complexity from $\mathcal{O}(\sqrt{pe})$ to $\mathcal{O}(\sqrt{p}\sqrt[^4]{e})$ for digit extraction modulo $p^e$, at the cost of a slight increase in multiplicative depth. Overall, our implementation in HElib shows a significant speedup of a factor up to 2.6 over the state-of-the-art.

We unify the state-of-the-art bootstrapping algorithms for BGV and BFV in a single framework and show that both schemes can be bootstrapped with identical complexity. This result corrects a claim by Chen and Han (Eurocrypt 2018) that BFV is more efficient to bootstrap than BGV. We also fix an error in their optimized procedure for power-of-two cyclotomics, which occurs for some parameter sets. Our analysis is simpler, yet more general than earlier work, in that it simultaneously covers both BGV and BFV. Furthermore, we also design and implement a high-level open source software library for bootstrapping in the Magma Computer Algebra System. It is the first library to support both BGV and BFV bootstrapping in full generality, with all recent techniques (including the above fixes) and trade-offs.