## Cryptology ePrint Archive: Report 2013/119

Speeding up Ate Pairing Computation in Affine Coordinates

Duc-Phong Le and Chik How Tan

Abstract: At Pairing 2010, Lauter et al's analysis showed that Ate pairing computation in affine coordinates may be much faster than projective coordinates at high security levels. In this paper, we further investigate techniques to speed up Ate pairing computation in affine coordinates. We first analyze Ate pairing computation using $4$-ary Miller algorithm in affine coordinates. This technique allows us to trade one multiplication in the full extension field and one field inversion for several multiplications in a smaller field. Then, we focus on pairing computations over elliptic curves admitting a twist of degree $3$. We propose new fast explicit formulas for Miller function that are comparable to formulas over even twisted curves. We further analyze pairing computation on cubic twisted curves by proposing efficient subfamilies of pairing-friendly elliptic curves with embedding degrees $k = 9$, and $15$. These subfamilies allow us not only to obtain a very simple form of curve, but also lead to an efficient arithmetic and final exponentiation.

Category / Keywords: implementation / Ate pairing, Pairing computation, final exponentiation, affine coordinates, cubic twisted curves, pairing-friendly elliptic curves

Publication Info: This paper was presented at ICISC 2012