An efficient residue group multiplication for the η T pairing over

Abstract

When we implement the η T pairing, which is one of the fastest pairings, we need multiplications in a base field and in a group G. We have previously regarded elements in G as those in to implement the η T pairing. Gorla et al. proposed a multiplication algorithm in that takes 5 multiplications in , namely 15 multiplications in . This algorithm then reaches the theoretical lower bound of the number of multiplications. On the other hand, we may also regard elements in G as those in the residue group in which βa is equivalent to a for and . This paper proposes an algorithm for computing a multiplication in the residue group. Its cost is asymptotically 12 multiplications in as m → ∞, which reaches beyond the lower bound the algorithm of Gorla et al. reaches. The proposed algorithm is especially effective when multiplication in the finite field is implemented using a basic method such as shift-and-add. © 2009 Springer-Verlag Berlin Heidelberg.