Algorithms and Arithmetic Operators for Computing the $\eta_T$ Pairing in Characteristic Three

Jean-Luc Beuchat; Nicolas Brisebarre; Jérémie Detrey; Eiji Okamoto; Masaaki Shirase; Tsuyoshi Takagi

Paper 2007/417

Algorithms and Arithmetic Operators for Computing the $\eta_T$ Pairing in Characteristic Three

Jean-Luc Beuchat, Nicolas Brisebarre, Jérémie Detrey, Eiji Okamoto, Masaaki Shirase, and Tsuyoshi Takagi

Abstract

Since their introduction in constructive cryptographic 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 architectures became an active research area. In this paper, we discuss several algorithms to compute the $\eta_T$ pairing in characteristic three and suggest further improvements. These algorithms involve addition, multiplication, cubing, inversion, and sometimes cube root extraction over $\mathbb{F}_{3^m}$. 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 $\mathbb{F}_{3^{97}}$ given by $\mathbb{F}_3[x]/(x^{97}+x^{12}+2)$, which compares favorably with other solutions described in the open literature.

Metadata

Available format(s): PDF
Category: Implementation
Publication info: Published elsewhere. Unknown where it was published
Keywords: $\eta_T$ pairing finite field arithmetic elliptic curve hardware accelerator FPGA
Contact author(s): beuchat @ risk tsukuba ac jp
History: 2008-09-10: last of 3 revisions; 2007-11-06: received; See all versions
Short URL: https://ia.cr/2007/417
License: CC BY

BibTeX

@misc{cryptoeprint:2007/417,
      author = {Jean-Luc Beuchat and Nicolas Brisebarre and Jérémie Detrey and Eiji Okamoto and Masaaki Shirase and Tsuyoshi Takagi},
      title = {Algorithms and Arithmetic Operators for Computing the $\e{ta_T}$ Pairing in Characteristic Three},
      howpublished = {Cryptology {ePrint} Archive, Paper 2007/417},
      year = {2007},
      url = {https://eprint.iacr.org/2007/417}
}