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Tate and Ate pairings for \(y^2=x^5-\alpha x\) in characteristic five. (English) Zbl 1222.14044

Summary: We consider the Tate and Ate pairings for the genus-2 supersingular hyperelliptic curves \(y^2 =x^5-\alpha x\) (\(\alpha = \pm 2\)) 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.

MSC:

14G15 Finite ground fields in algebraic geometry
14G50 Applications to coding theory and cryptography of arithmetic geometry
14Q20 Effectivity, complexity and computational aspects of algebraic geometry
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