An Optimized Hardware Implementation of a Non-Adjacent Form Algorithm Using Radix-4 Multiplier for Binary Edwards Curves

Abstract

Binary Edwards Curves (BEC) play a pivotal role in modern cryptographic processes and applications, offering a combination of robust security as well as computational efficiency. For robust security, this article harnesses the inherent strengths of BEC for the cryptographic point multiplication process by utilizing the Non-Adjacent Form (NAF) algorithm. For computational efficiency, a hardware architecture for the NAF algorithm is proposed. Central to this architecture is an Arithmetic Logic Unit (ALU) designed for streamlined execution of essential operations, including addition, squaring, and multiplication. One notable innovation in our ALU design is the integration of multiplexers, which maximize ALU efficiency with minimal additional hardware requirements. Complementing the optimized ALU, the proposed architecture incorporates a radix-4 multiplier, renowned for its efficiency in both multiplication and reduction. It eliminates resource-intensive divisions, resulting in a substantial boost to overall computational speed. The architecture is implemented on Xilinx Virtex series Field-Programmable Gate Arrays (FPGAs). It achieves throughput-to-area ratios of 14.819 (Virtex-4), 25.5 (Virtex-5), 34.58 (Virtex-6), and 37.07 (Virtex-7). These outcomes underscore the efficacy of our optimizations, emphasizing an equilibrium between computational performance and area utilization.

Keywords

• cryptography
• security
• computational efficiency
• resource utilization
• NAF algorithm
• Montgomery radix-4 multiplier