Tate and Ate Pairings for $y^{2}=x^{5}-\alpha x$ in Characteristic Five
Harasawa, Ryuichi ; Sueyoshi, Yutaka ; Kudo, Aichi
Japan J. Indust. Appl. Math., Tome 24 (2007) no. 1, p. 251-274 / Harvested from Project Euclid
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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.
Publié le : 2007-10-14
Classification:  hyperelliptic curves,  distortion map,  Tate and Ate pairings 
@article{1197909112,
     author = {Harasawa, Ryuichi and Sueyoshi, Yutaka and Kudo, Aichi},
     title = {Tate and Ate Pairings for $y^{2}=x^{5}-\alpha x$ in Characteristic Five},
     journal = {Japan J. Indust. Appl. Math.},
     volume = {24},
     number = {1},
     year = {2007},
     pages = { 251-274},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1197909112}
}

Harasawa, Ryuichi; Sueyoshi, Yutaka; Kudo, Aichi. Tate and Ate Pairings for $y^{2}=x^{5}-\alpha x$ in Characteristic Five. Japan J. Indust. Appl. Math., Tome 24 (2007) no. 1, pp.  251-274. http://gdmltest.u-ga.fr/item/1197909112/