Paper 2026/1026

Sparse Hermite Interpolation Method for Discrete-CKKS Functional Bootstrapping

Abstract

Discrete CKKS is a promising approach for performing high-throughput homomorphic computations over encrypted discrete data. Although it relies on CKKS, an approximate FHE scheme, as the computation engine, discrete CKKS can achieve exact correctness. The core operation of discrete CKKS is functional bootstrapping, a mechanism which enables evaluating an arbitrary function over a bounded discrete domain by representing it as a lookup table and computing it as part of bootstrapping. Simultaneously, the same procedure enables reducing the input ciphertext noise using Hermite interpolation methods. This noise reduction feature is critical for both supporting arbitrary computations and improving the efficiency of their evaluation, by providing more noise budget between bootstrapping invocations. In this paper, we first show that both state-of-the-art Hermite interpolation noise reduction methods by Bae et al. (ASIACRYPT'24) and Alexandru et al. (CRYPTO'25) have a limited noise reduction ability for distinct structural reasons. We then propose a new method that can efficiently overcome these limitations by using a CKKS-friendly *arbitrary-order* Hermite interpolation. We call this method "sparse" trigonometric Hermite interpolation because both constraints and coefficients have convenient sparsity properties, which allow us to achieve efficiency comparable to the fastest prior method by Alexandru et al., while attaining superior noise reduction. In the process, we develop a metric that measures the noise budget between consecutive functional bootstrapping invocations, and use it to compare all methods on equal footing. We implement our new method in OpenFHE and experimentally demonstrate its noise reduction advantage over prior methods.