An Efficient and Scalable Hardware Architecture for Number Theoretic Transform on FPGA with Design Automation

Fully Homomorphic Encryption (FHE) has become a promising approach to protecting data privacy in emerging application scenarios. Unfortunately, FHE suffers from significant processing speed degradation compared to plaintext computation, with one of the primary bottlenecks being the time-consuming Number Theoretic Transform (NTT). Therefore, accelerating NTT to accommodate various FHE parameters is crucial to advancing FHE towards practical use. With highly reconfigurable and performant logical fabrics, Field Programmable Gate Arrays (FPGAs) have exhibited great potential in NTT acceleration. By decomposing large-point NTT with strong data dependency into independent and simple small-point NTTs, the emerging Ten-step NTT (TNTT) algorithms intuitively enable higher parallelism and thereby have the potential to explore better performance compared to traditional algorithms. However, our quantitative analysis reveals that TNTT exhibits significant performance degradation as parallelism increases due to additional varying-size transpositions and Hadamard products. This paper proposes AutoNest, an efficient and scalable hardware architecture, along with an accelerator auto-generation framework for TNTT. The proposed hardware architecture maximizes performance by 1) adopting a 2D block decomposition dataflow to address critical path delays in transpose logic, thereby improving clock frequency. 2) integrating algorithm-level costfree twiddle factor fusion to reduce the number of modular multiplications in Hadamard products, thereby allowing higher parallelism on chip. Moreover, we also deliver an accelerator generation framework conducting automated design space exploration to elaborate a performant TNTT architecture under the target FPGAs' resource budget for user-defined FHE parameters. Experimental results on the AMD-Xilinx U280 FPGA demonstrate that NTT accelerators generated by AutoNest achieve an average speedup of$2.31 \times$compared to prior designs.