Paper 2006/114

Tate pairing for $y^{2}=x^{5}-\alpha x$ in Characteristic Five

Ryuichi Harasawa, Yutaka Sueyoshi, and Aichi Kudo


In this paper, for the genus-$2$ hyperelliptic curve $y^{2}=x^{5} -\alpha x$ ($\alpha = \pm2$) defined over finite fields of characteristic five, we construct a distortion map explicitly, and show the map indeed gives an input for which the value of the Tate pairing is not trivial. Next we describe a computation of the Tate pairing by using the proposed distortion map. Furthermore, we also see that this type of curve is equipped with a simple quintuple operation on the Jacobian group, which leads to giving an improvement for computing the Tate pairing. We indeed show that, for the computation of the Tate pairing for genus-$2$ hyperelliptic curves, our method is about twice as efficient as a previous work.

Note: We simplified the representation of the distortion map and revalued the cost.

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Publication info
Published elsewhere. The full version, entitled "Tate and Ate Pairings for $y^{2} = x^{5} - \alpha x$ in Characteristic Five", is published in Japan Journal of Industrial and Applied Mathematics (JJIAM), Vol. 24, No. 3, pp. 251 - 274, 2007.
Distortion mapTate pairingHyperelliptic curves
Contact author(s)
harasawa @ cis nagasaki-u ac jp
2008-01-09: last of 2 revisions
2006-03-26: received
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      author = {Ryuichi Harasawa and Yutaka Sueyoshi and Aichi Kudo},
      title = {Tate pairing for $y^{2}=x^{5}-\alpha x$  in Characteristic Five},
      howpublished = {Cryptology ePrint Archive, Paper 2006/114},
      year = {2006},
      note = {\url{}},
      url = {}
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