International Association
for Cryptologic Research

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.

Recently, the authors proposed a method for computing the Tate pairing using a distortion map for $y^{2}=x^{5} -\alpha x$ ($\alpha = \pm2$) over finite fields of characteristic five. In this paper, we show the Ate pairing, an invariant of the Tate pairing, can be applied to this curve. This leads to about $50\%$ computational cost-saving over the Tate pairing.