A Low-Cost ECC Coprocessor for Smartcards

Abstract

Abstract. In this article we present a low-cost coprocessor for smartcards which supports all necessary mathematical operations for a fast calculation of the Elliptic Curve Digital Signature Algorithm (ECDSA) based on the finite field GF(2 m ). These ECDSA operations are GF(2 m ) addition, 4-bit digit-serial multiplication in GF(2 m ), inversion in GF(2 m ), and inversion in GF(p). An efficient implementation of the multiplicative inversion which breaks the 11:1 limit regarding multiplications makes it possible to use affine instead of projective coordinates for point operations on elliptic curves. A bitslice architecture allows an easy adaptation for different bit lengths. A small chip area is achieved by reusing the hardware registers for different operations.

Hardware/Software Co-design of Elliptic Curve Cryptography on an 8051 Microcontroller

“…AVR or 8051) have been published [1,2,6,7,14,15,17]. However, the co-design approach for ECC presented in this paper differs from previous work in two important aspects.…”

“…We emphasize that attaining scalability in hardware/software co-design affects all abstraction levels and layers between hardware and software (including the operand transfers), and is not a "pure" hardware design issue as in [19]. For instance, when 1 Batina et al presented a hyperelliptic curve cryptosystem of genus 2 over the field GF(2 83 ). The security level of this HECC system is approximately 166 bits, and thus it is comparable to the ECC system over the field GF(2 163 ) that we have used.…”

“…On the other hand, this approach may entail a significant communication overhead, especially when the accelerator hardware does not provide local storage for auxiliary variables. The fastest implementations following this approach have been reported by Ernst et al [7] and Aigner et al [1]. The latter implements all field arithmetic operations in hardware (including squaring and inversion) and uses affine coordinates.…”

“…One could, for instance, implement the point addition/doubling in hardware and the rest in software [15]. An alternative approach is to implement the field arithmetic in hardware and the curve/point arithmetic in software [1,2,7,14]. Furthermore, hardware acceleration at the granularity of instruction set extensions for the finite field multiplication has also been investigated [6,11,17].…”

An Elliptic Curve Cryptography Coprocessor over GF(2m) on a Low-Cost Embedded System

“…There are numerous papers dealing with the hardware/software co-design of ECC on 8-bit CPU platforms [2,3,4,6,7,8]. The performance of ECC coprocessors in previous work highly depended on the efficiency of the platforms, in [3,6,7,8] the performance built around an AVR microcontroller is faster than that of those using 8051 [2,4].…”