Optimized FPGA Architecture for Modular Reduction in NTT

Abstract

In this paper, we present a comprehensive analysis of various modular multiplication methods for Number Theoretic Transform (NTT) on FPGA. NTT is a critical and time-intensive component of Fully Homomorphic Encryption (FHE) applications while modular multiplication consumes a significant portion of the design resources in an NTT implementation. We study the existing modular reduction approaches from the literature, and implement particular methods on FPGA. Specifically Word-Level Montgomery (WLM) for NTT friendly primes [20] and K2RED [4]. For improvements, we explore the trade-offs between the number of available primes in special forms and hardware cost of the reduction methods. We develop a DSP multiplication-optimized version of WLM, which we call WLM-Mixed. We also introduce a subclass of Proth primes, referred to as Proth-l primes, characterized by a low and fixed signed Hamming Weight. This special class of primes allows us to design multiplication-free shift-add versions of K2RED and naive Montgomery reduction [21], referred to as K2RED-Shift and Montgomery-Shift. We provide in-depth evaluations of these five reduction methods in an NTT architecture on FPGA. Our results indicate that WLM-Mixed is highly resource-efficient, utilizing only 3 DSP multiplications for 64-bit coefficient moduli. On the other hand, K2RED-Shift and Montgomery-Shift offer DSP-free alternatives, which can be beneficial in specific scenarios. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2026.