A Unified Point Multiplication Architecture of Weierstrass, Edward and Huff Elliptic Curves on FPGA

Abstract

This article presents an area-aware unified hardware accelerator of Weierstrass, Edward, and Huff curves over GF(2233) for the point multiplication step in elliptic curve cryptography (ECC). The target implementation platform is a field-programmable gate array (FPGA). In order to explore the design space between processing time and various protection levels, this work employs two different point multiplication algorithms. The first is the Montgomery point multiplication algorithm for the Weierstrass and Edward curves. The second is the Double and Add algorithm for the Binary Huff curve. The area complexity is reduced by efficiently replacing storage elements that result in a 1.93 times decrease in the size of the memory needed. An efficient Karatsuba modular multiplier hardware accelerator is implemented to compute polynomial multiplications. We utilized the square arithmetic unit after the Karatsuba multiplier to execute the quad-block variant of a modular inversion, which preserves lower hardware resources and also reduces clock cycles. Finally, to support three different curves, an efficient controller is implemented. Our unified architecture can operate at a maximum of 294 MHz and utilizes 7423 slices on Virtex-7 FPGA. It takes less computation time than most recent state-of-the-art implementations. Thus, combining different security curves (Weierstrass, Edward, and Huff) in a single design is practical for applications that demand different reliability/security levels.