International Association
for Cryptologic Research

The BGV scheme is one of the most popular FHE schemes for computing homomorphic integer arithmetic. The bootstrapping technique of BGV is necessary to evaluate arbitrarily deep circuits homomorphically. However, the BGV bootstrapping performs poorly for large plaintext prime $p$ due to its digit removal procedure exhibiting a computational complexity of at least $O(\sqrt{p})$. In this paper, we propose optimizations for the digit removal procedure with large $p$ by leveraging the properties of null polynomials over the ring $\mathbb{Z}_{p^e}$. Specifically, we demonstrate that it is possible to construct low-degree null polynomials based on two observations of the input to the digit removal procedure: 1) the support size of the input can be upper-bounded by $(2B+1)^2$; 2) the size of the lower digits to be removed can be upper-bounded by $B$. Here $B$ can be controlled within a narrow interval $[22,23]$ in our parameter selection, making the degree of these null polynomials much smaller than $p$ for large values of $p$. These low-degree null polynomials can significantly reduce the polynomial degrees during homomorphic digit removal, thereby decreasing both running time and capacity consumption. Theoretically, our optimizations reduce the computational cost of extracting a single digit from $O(\sqrt{pe})$ (by Chen and Han) or $O(\sqrt{p}\sqrt[4]{e})$ (by Geelen et al.) to $\min(2B+1,\sqrt{\lceil e/t\rceil(2B+1)})$ for some $t\ge 1$. We implement and benchmark our method on HElib with $p=17,127,257,8191$ and $65537$. With our optimized digit removal, we achieve a bootstrapping throughput $1.38\sim151$ times that in HElib, with the speedup increasing with the value of $p$. For $p=65537$, we accelerate the digit removal step by 80 times and reduce the bootstrapping time from more than 12 hours to less than 14 minutes.

The functional bootstrap in FHEW/TFHE allows for fast table lookups on ciphertexts and is a powerful tool for privacy-preserving computations. However, the functional bootstrap suffers from two limitations: the negacyclic constraint of the lookup table (LUT) and the limited ability to evaluate large-precision LUTs. To overcome the first limitation, several full-domain functional bootstraps (FDFB) have been developed, enabling the evaluation of arbitrary LUTs. Meanwhile, algorithms based on homomorphic digit decomposition have been proposed to address the second limitation. Although these algorithms provide effective solutions, they are yet to be optimized. This paper presents four new FDFB algorithms and two new homomorphic decomposition algorithms that improve the state-of-the-art. Our FDFB algorithms reduce the output noise, thus allowing for more efficient and compact parameter selection. Across all parameter settings, our algorithms reduce the runtime by up to 39.2%. Our homomorphic decomposition algorithms also run at 2.0x and 1.5x the speed of prior algorithms. We have implemented and benchmarked all previous FDFB and homomorphic decomposition algorithms and our methods in OpenFHE.