Hardware Acceleration of the Prime-Factor and Rader NTT for BGV Fully Homomorphic Encryption

Fully Homomorphic Encryption (FHE) enables computation on encrypted data, holding immense potential for enhancing data privacy and security in various applications. Presently, FHE adoption is hindered by slow computation times, caused by data being encrypted into large polynomials. Optimized FHE libraries and hardware acceleration are emerging to tackle this performance bottleneck. Often, these libraries implement the Number Theoretic Transform (NTT) algorithm for efficient polynomial multiplication. Existing implementations mostly focus on the case where the polynomials are defined over a power-of-two cyclotomic ring, allowing to make use of the simpler Cooley-Tukey NTT. However, generalized cyclotomics have several benefits in the BGV FHE scheme, including more SIMD plaintext slots and a simpler bootstrapping algorithm.We present a hardware architecture for the NTT targeting generalized cyclotomics within the context of the BGV FHE scheme. We explore different non-power-of-two NTT algorithms, including the Prime-Factor, Rader, and Bluestein NTTs. Our most efficient architecture targets the 21845-th cyclotomic polynomial — a practical parameter for BGV — with ideal properties for use with a combination of the Prime-Factor and Rader algorithms. The design achieves high throughput with optimized resource utilization, by leveraging parallel processing, pipelining, and reusing processing elements. Compared to Wu et al.’s VLSI architecture of the Bluestein NTT, our approach showcases 2× to 5× improved throughput and area efficiency. Simulation and implementation results on an AMD Alveo U250 FPGA demonstrate the feasibility of the proposed hardware design for FHE.