High-performance montgomery modular multiplier with NTT and negative wrapped convolution

Abstract

Modular multiplication plays a crucial role in modern cryptography. Montgomery modular multiplication(MMM), one of the most classic and practical modular multiplication algorithms, has been widely used in cryptographic algorithms such as RSA, Diffie–Hellman algorithm, and Elliptic Curve Cryptography. In this paper, we incorporate negative wrapped convolution (NWC) into the FFT-based Montgomery modular multiplication to avoid the issue of zero-padding and use carry-save arithmetics for parallel computation. By utilizing coefficient pairs (pos_part and neg_part), we reconstruct the final result and eliminate the restrictions imposed by nega-cyclic parts. Moreover, Karatsuba-like algorithm is introduced for building fine-grained large integer multipliers. We have modified the parameter specifications for our design to meet requirements from diverse application scenarios. We implement the design on Xilinx Virtex-7 FPGA under different conditions and compare the results with the state-of-the-art MMM designs. The comparisons confirm that our design has the following characteristics: low latency for process, competitive area-latency-product(ALP), efficient DSP usage, and constant delay, which enhances security against timing attacks.