## Cryptology ePrint Archive: Report 2007/417

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

*Jean-Luc Beuchat and Nicolas Brisebarre and J\'er\'emie Detrey and Eiji Okamoto and 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.

**Category / Keywords: **implementation / $\eta_T$ pairing, finite field arithmetic, elliptic curve, hardware accelerator, FPGA

**Date: **received 1 Nov 2007, last revised 9 Sep 2008

**Contact author: **beuchat at risk tsukuba ac jp

**Available format(s): **PDF | BibTeX Citation

**Version: **20080910:013639 (All versions of this report)

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