Accelerating beta weil pairing with precomputation and multi-pairing techniques

Abstract

In this paper, we present the efficient computation methods of $$\beta $$ Weil pairing with the precomputation and multi-pairing techniques. We use the following two ideas to facilitate applying those techniques to $$\beta $$ Weil pairing. 1. Elimination of denominators of $$\beta $$ Weil pairing: The original $$\beta $$ Weil pairing proposed by Aranha et al. is the products of quotients of extended Miller functions. Then, we propose the method to eliminate denominators of $$\beta $$ Weil pairing. 2. Elimination of exponents of $$\beta $$ Weil pairing: The original $$\beta $$ Weil pairing has distinct exponent in each term. Then, we propose the method to eliminate exponents of the $$\beta $$ Weil pairing. Thereby, we can apply the precomputation and multi-pairing technique to $$\beta $$ Weil pairing in order to accelerate the computation. Moreover, we estimate the computational costs of the proposed methods for the BLS-48 curve with 256 bit level of security against the Kim-Barbulescu attack, proposed by Kiyomura et al. Then, we compare the results with the normal $$\beta $$ Weil pairing and the optimal ate pairing. Furthermore, we also provide implementation results.