Integer Variable chi-Based Cross Twisted Ate Pairing and Its Optimization for Barreto-Naehrig Curve
It is said that the lower bound of the number of iterations of Miller's algorithm for pairing calculation is log(2) r/phi(k), where phi(.) is the Euler's function, r is the group order, and k is the embedding degree. Ate pairing reduced the number of the loops of Miller's algorithm of Tate pairing from left perpendicularlog(2) right perpendicular to left perpendicularlog(2)(t-1)right perpendicular, where t is the Frobenius trace. Recently, it is known to systematically prepare a pairing-friendly elliptic curve whose parameters are given by a polynomial of integer variable "chi." For such a curve, this paper gives integer variable chi-based Ate (Xate) pairing that achieves the lower bound. In the case of the well-known Barreto-Naehrig pairing-friendly curve. it reduces the number of loops to left perpendicularlog(2)chi right perpendicular. Then, this paper optimizes Xate pairing for Barreto-Naehrig curve and shows its efficiency based on some simulation results.
ate pairing miller's algorithm