Paper 2026/1533

Correcting the modulus switch error in TFHE bootstrapping for real-valued computation

Abstract

Torus Fully Homomorphic Encryption (TFHE) enables the homomorphic evaluation of arbitrary functions via Programmable Bootstrapping (PBS). However, the modulus switching step inherent to bootstrapping introduces a rounding error that forces the discretization of the input space, limiting the achievable precision on real-valued inputs. We propose a correction algorithm based on a first-order Taylor expansion, applied after bootstrapping, that directly mitigates this rounding error. Our method leverages the many-LUT technique to simultaneously recover encryptions of the function and its derivative within a single PBS, making the correction essentially free in terms of bootstrapping latency. We support our construction with a heuristic average-case noise analysis, validated by empirical measurements, and demonstrate a tenfold reduction in bootstrapping noise standard deviation. As a proof of concept, we apply our method to the numerical integration of ordinary differential equations under encryption.