International Association for Cryptologic Research

International Association
for Cryptologic Research


Paper: A Coprocessor for the Final Exponentiation of the $\eta_T$ Pairing in Characteristic Three

Jean-Luc Beuchat
Nicolas Brisebarre
Masaaki Shirase
Tsuyoshi Takagi
Eiji Okamoto
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Abstract: Since the introduction of pairings over (hyper)elliptic curves in constructive cryptographic applications, an ever increasing number of protocols based on pairings have appeared in the literature. Software implementations being rather slow, the study of hardware architectures became an active research area. Beuchat et al. proposed for instance a coprocessor which computes the characteristic three $\eta_T$ pairing, from which the Tate pairing can easily be derived, in $33$\,$\mu$s on a Cyclone II FPGA. However, a final exponentiation is required to ensure a unique output value and the authors proposed to supplement their $\eta_T$ pairing accelerator with a coprocessor for exponentiation. Thus, the challenge consists in designing the smallest possible piece of hardware able to perform this task in less than $33$\,$\mu$s on a Cyclone~II device. In this paper, we propose a novel arithmetic operator implementing addition, cubing, and multiplication over $\mathbb{F}_{3^{97}}$ and show that a coprocessor based on a single such operator meets this timing constraint.
  title={A Coprocessor for the Final Exponentiation of the $\eta_T$ Pairing in Characteristic Three},
  booktitle={IACR Eprint archive},
  keywords={implementation / $\eta_T$ pairing, characteristic three, hardware accelerator, FPGA},
  note={ 13557 received 13 Feb 2007},
  author={Jean-Luc Beuchat and Nicolas Brisebarre and Masaaki Shirase and Tsuyoshi Takagi and Eiji Okamoto},