PolyFHEmus: Rethinking Multiplication in Fully Homomorphic Encryption

Homomorphic encryption (HE) is a powerful technology that solves key privacy concerns in cloud computing by enabling computation on encrypted data. However, it has not seen widespread adoption due to high latencies resulting from extensive operations over high-degree polynomials with large coefficients. In this paper, we identify polynomial multiplication as a bottleneck and investigate alternative algorithms to accelerate encrypted computing. Most popular open-source HE implementations utilize the NTT or FFT algorithms to perform pointwise multiplication in the Fourier domain; on the other hand, we investigate applying alternative strategies that decompose the problem of multiplying polynomials to that of multiplying large integers. We combine the Schönhage-Strassen (SS) algorithm for integer multiplication and the discrete Galois transform (DGT) to achieve faster ciphertext multiplication speeds for large polynomial degrees. For a degree-5 polynomial approximation of the nonlinear Sigmoid function utilized in logistic regression, we observe that this alternative approach can result in a speedup of approximately $1.3 \times$ relative to the Microsoft SEAL library for the CKKS scheme. Notably, the proposed approach can also apply to other HE schemes that utilize large polynomial degrees, such as the BFV and BGV cryptosystems.