Abstract
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.
Citation
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.
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