Innovative Dual-Binary-Field Architecture for Point Multiplication of Elliptic Curve Cryptography

Abstract

High-performance Elliptic Curve Cryptography (ECC) implementation in encryption authentication severs has become a challenge due to the explosive growth of e-commerce's demand for speed and security. Point multiplication (PM) is the most common and complex operation in ECC which directly determines the performance of the whole system. This article proposes a 6CC-6CC (clock cycle) dual-field PM architecture and a 6CC-4CC dual-field PM architecture based on maximizing utilization of Karatsuba multipliers and re-ordering schedule strategy in PM. The Montgomery Ladder algorithm used in PM is modified to a 4CC algorithm for better resource utilization and parallel computation. To solve the frequency drop problem while working on large finite field, the PM architectures for high and low field are carefully studied to have universal critical path length and balanced performance. Both of the architectures are implemented over GF(2 571 ) and GF(2 283 ) on Xilinx Virtex-5 and Virtex-7 FPGAs (Field-Programmable Gate Array) for comparison. The 6CC-6CC architecture is shown to have the best performance on GF(2 571 ), which achieves one PM operation in 17.44 µs using 81549 LUTs (Look-Up-Table ) with the frequency of 197.2 MHz on Virtex-5, and 12.55 µs using 80970 LUTs with the frequency of 274.1 MHz on Virtex-7. The 6CC-4CC architecture performs better on GF(2 283 ) with the shortest computation time. It takes only 3.21 µs to finish one PM operation on Virtex-5 and 2.22 µs on Virtex-7, which are faster than all the previous designs. The implementation results prove that the proposed architectures have state-of-the-art performance as well as higher versatility for ECC designs.INDEX TERMS Elliptic curve cryptography (ECC), dual-field point multiplication, montgomery ladder, field-programmable gate array (FPGA) implementation.

An Optimized Architecture for Binary Huff Curves With Improved Security

“…The SPA attacks for Weierstrass model are counter measured using Montgomery ladder PM algorithm [5]. In addition to it, the PA and PD computations are exactly similar in Weierstrass model.…”

“…On the other hand, asymmetric algorithms employ some mathematical problems which are difficult to resolve [3]. Rivest Shamir Adleman (RSA) [4] and Elliptic Curve Cryptography (ECC) [5] are the frequently used asymmetric algorithms. The security strength of RSA and ECC is based on solving the prime numbers using exponential and logarithms respectively.…”

“…Due to the involvement of complex mathematical operations, software implementations of ECC lead to slower designs. As a result, the hardware implementations of ECC, using various implementation models, are proposed [5], [8]- [10]. The commonly used implementation models of ECC are Weierstrass [5], Hessian [8], Edward [9] and Huff [10].…”

“…As a result, the hardware implementations of ECC, using various implementation models, are proposed [5], [8]- [10]. The commonly used implementation models of ECC are Weierstrass [5], Hessian [8], Edward [9] and Huff [10]. All of these ECC models (Weierstrass, Hessian, Edward and Huff) consist of four layers: The first layer is an arithmetic layer which includes finite field additions, multiplications, squaring and inversions.…”

“…The projective coordinate system, on the other hand, requires only a single FF inversion. It implies that the projective coordinate system is more useful for reducing the number of clock cycles (CCs) and increasing the throughputs [5]. Consequently, this work has targeted the binary GF (2 m ) field with the projective coordinate system representation.…”