Speed-Oriented Architecture for Binary Field Point Multiplication on Elliptic Curves

Abstract

This paper introduces a novel speed-oriented architecture of point multiplication in elliptic curve cryptography. A balanced full-precision multiplier is proposed to shorten latency, and a new modular inversion architecture is integrated to reduce the total number of clock cycles in point multiplication. A modified Montgomery Ladder algorithm that takes three clock cycles to calculate one input bit is proposed to best utilize hardware resources. A mixed-pipeline technique is used to balance the delay of different paths and increase frequency. The proposed architecture is implemented on GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">163</sup> ) and GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">571</sup> ), based on Xilinx Virtex-5 and Virtex-7 FPGA. For GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">163</sup> ), the design reaches 211 MHz, with 29309 LUTs, and 547 clock cycles or <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2.6~\mu \text{s}$ </tex-math></inline-formula> latency on Virtex-5; 320.5 MHz, with 28911 LUTs and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1.7~\mu \text{s}$ </tex-math></inline-formula> latency on Virtex-7. For GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">571</sup> ), the design reaches 186 MHz, with 286400 LUTs, and 1813 clock cycles or <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$9.6~\mu \text{s}$ </tex-math></inline-formula> latency on Virtex-5; 267 MHz, 290001 LUTs and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$6.79~\mu \text{s}$ </tex-math></inline-formula> latency on Virtex-7. The proposed design achieves the lowest latency among all existing works, and its performance is also among the top. Furthermore, it is demonstrated that the proposed architecture maintains a high speed for larger binary fields, making it more suitable to be implemented in large-bit-length platforms with a higher security level. Since the multiplier and its segments work in different bit-length and refer to different fields, the proposed architecture can also be upgraded to a reconfigurable design to support multiple-field point multiplication in the future.