High-Performance Pipelined Architecture of Point Multiplication on Koblitz Curves

Abstract

Point multiplication is the key operation that dominates the speed and area of each elliptic curve cryptosystem. In this brief, we propose a highly efficient architecture for point multiplication on Koblitz curves. The right-to-left point multiplication algorithm is adopted to allow parallel computation of Frobenius maps and point additions. In order to achieve short latency as well as high frequency, we explore operation rescheduling in consecutive point additions with a pipelined bit-parallel finite field multiplier accumulator. Furthermore, the pipelined mixed-form double-digit scalar converter with compact recoding is employed to reduce the wait time and consumed logic resources by scalar conversion. Fast inversion based on optimal addition chains is used to further reduce the latency. Implementation results on Xilinx Virtex V show that the proposed architecture can perform a point multiplication in as few as 2.50, 4.09, 5.81, 9.50, $18.51~ {\mu }\text{s}$ over the five NIST Koblitz curves K-163, K-233, K-283, K-409, and K-571, respectively. To the best of our knowledge, the proposed architecture of point multiplication has the fastest speed in comparison to the reported results.