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Paper: Some Efficient Algorithms for the Final Exponentiation of $\eta_T$ Pairing

Authors:
Masaaki Shirase
Tsuyoshi Takagi
Eiji Okamoto
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URL: http://eprint.iacr.org/2006/431
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Abstract: Recently Tate pairing and its variations are attracted in cryptography. Their operations consist of a main iteration loop and a final exponentiation. The final exponentiation is necessary for generating a unique value of the bilinear pairing in the extension fields. The speed of the main loop has become fast by the recent improvements, e.g., the Duursma-Lee algorithm and $\eta_T$ pairing. In this paper we discuss how to enhance the speed of the final exponentiation of the $\eta_T$ pairing in the extension field ${\mathbb F}_{3^{6n}}$. Indeed, we propose some efficient algorithms using the torus $T_2({\mathbb F}_{3^{3n}})$ that can efficiently compute an inversion and a powering by $3^{n}+1$. Consequently, the total processing cost of computing the $\eta_T$ pairing can be reduced by 17% for n=97.
BibTeX
@misc{eprint-2006-21922,
  title={Some Efficient Algorithms for the Final Exponentiation of $\eta_T$ Pairing},
  booktitle={IACR Eprint archive},
  keywords={public-key cryptography / Tate pairing, $\eta_T$ pairing, final exponentiation, torus},
  url={http://eprint.iacr.org/2006/431},
  note={ shirase@fun.ac.jp 13472 received 20 Nov 2006},
  author={Masaaki Shirase and Tsuyoshi Takagi and Eiji Okamoto},
  year=2006
}