Algorithms and arithmetic operators for computing the ηT pairing in characteristic three

Abstract — Since their introduction in constructive crypto-graphic applications, pairings over (hyper)elliptic curves are at the heart of an ever increasing number of protocols. Software implementations being rather slow, the study of hardware archi-tectures became an active research area. In this paper, we discuss several algorithms to compute the ηT pairing in characteristic three and suggest further improvements. These algorithms involve addition, multiplication, cubing, inver-sion, and sometimes cube root extraction over F3m. We propose a hardware accelerator based on a unified arithmetic operator able to perform the operations required by a given algorithm. We describe the implementation of a compact coprocessor for the field F397 given by F3[x]/(x97+x12+2), which compares favorably with other solutions described in the open literature. Index Terms—ηT pairing, finite field arithmetic, elliptic curve, hardware accelerator, FPGA. I