High-Radix/Mixed-Radix NTT Multiplication Algorithm/Architecture Co-Design Over Fermat Modulus

Yile Xing, Guangyan Li, Zewen Ye, Ryan W. L. Luk, Donglong Chen, Hong Yan, Ray C. C. Cheung

Abstract: Polynomial multiplication using Number Theoretic Transform (NTT) is crucial in lattice-based post-quantum cryptography (PQC) and fully homomorphic encryption (FHE), with modulus $q$ significantly affecting performance. Fermat moduli of the form $2^{2^{n}}+1$, such as 65537, offer efficiency gains due to simplified modular reduction and powers-of-2 twiddle factors in NTT. While Fermat moduli have been directly applied or explored for incorporation into existing schemes, Fermat NTT-based polynomial multiplication designs remain underexplored in fully exploiting the benefits of Fermat moduli. This work presents a high-radix/mixed-radix NTT architecture tailored for Fermat moduli, which improves the utilization of the powers-of-2 twiddle factors in large transform sizes. In most cases, our design achieves a 30%–85% reduction in DSP area-time product (ATP) and a 70%–100% reduction in BRAM ATP compared to state-of-the-art designs with smaller or equivalent modulus, while maintaining competitive LUT and FF ATP, underscoring the potential of Fermat NTT-based polynomial multipliers in lattice-based cryptography.

External IDs:dblp:journals/tc/XingLYLCYC25