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October 2007 Tate and Ate Pairings for $y^{2}=x^{5}-\alpha x$ in Characteristic Five
Ryuichi Harasawa, Yutaka Sueyoshi, Aichi Kudo
Japan J. Indust. Appl. Math. 24(3): 251-274 (October 2007).


In this paper, we consider the Tate and Ate pairings for the genus-$2$ supersingular hyperelliptic curves $y^{2}=x^{5} -\alpha x$ ($\alpha = \pm2$) defined over finite fields of characteristic five. More precisely, we construct a distortion map explicitly, and show that the map indeed gives an input for which the value of the Tate pairing is not trivial. We next describe a computation of the Tate pairing by using the proposed distortion map. We also see that this type of curve is equipped with a simple quintuple operation on the Jacobian group, which leads to an improvement for computing the Tate pairing. We further show the Ate pairing, a variant of the Tate pairing for elliptic curves, can be applied to this curve. The Ate pairing yields an algorithm which is about $50\,\%$ more efficient than the Tate pairing in this case.


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Ryuichi Harasawa. Yutaka Sueyoshi. Aichi Kudo. "Tate and Ate Pairings for $y^{2}=x^{5}-\alpha x$ in Characteristic Five." Japan J. Indust. Appl. Math. 24 (3) 251 - 274, October 2007.


Published: October 2007
First available in Project Euclid: 17 December 2007

zbMATH: 1222.14044
MathSciNet: MR2374990

Keywords: distortion map , hyperelliptic curves , Tate and Ate pairings

Rights: Copyright © 2007 The Japan Society for Industrial and Applied Mathematics

Vol.24 • No. 3 • October 2007
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