Efficient Hardware Implementation of Modular Multiplier over GF (2m) on FPGA

TL;DR

This paper presents a hybrid hardware implementation of modular multiplication over GF(2^m) on FPGA, combining conventional and Karatsuba multiplication to improve efficiency for elliptic curve cryptography parameters.

Contribution

It introduces a hybrid multiplication method that optimally combines CM and KM for different operand sizes, reducing resource usage and enhancing performance in FPGA implementations.

Findings

Resource utilization reduced by up to 45.53%

Delay improved by 37.60% for m=163

Area-delay product outperforms existing designs

Abstract

Elliptic curve cryptography (ECC) has emerged as the dominant public-key protocol, with NIST standardizing parameters for binary field GF(2^m) ECC systems. This work presents a hardware implementation of a Hybrid Multiplication technique for modular multiplication over binary field GF(2m), targeting NIST B-163, 233, 283, and 571 parameters. The design optimizes the combination of conventional multiplication (CM) and Karatsuba multiplication (KM) to enhance elliptic curve point multiplication (ECPM). The key innovation uses CM for smaller operands (up to 41 bits for m=163) and KM for larger ones, reducing computational complexity and enhancing efficiency. The design is evaluated in three areas: Resource Utilization For m=163, the hybrid design uses 6,812 LUTs, a 39.82% reduction compared to conventional methods. For m=233, LUT usage reduces by 45.53% and 70.70% compared to overlap-free and bit-parallel implementations. Delay Performance For m=163, achieves 13.31ns delay, improving by 37.60% over bit-parallel implementations. For m=233, maintains 13.39ns delay. Area-Delay Product For m=163, achieves ADP of 90,860, outperforming bit-parallel (75,337) and digit-serial (43,179) implementations. For m=233, demonstrates 16.86% improvement over overlap-free and 96.10% over bit-parallel designs. Results show the hybrid technique significantly improves speed, hardware efficiency, and resource utilization for ECC cryptographic systems.