Fast Ate Pairing Computation of Embedding Degree 12 Using Subfield-Twisted Elliptic Curve

Abstract

This paper presents implementation techniques of fast Ate pairing of embedding degree 12. In this case, we have no trouble in finding a prime order pairing friendly curve E such as the Barreto-Naehrig curve $y^2=x^3+a, a\\in\\Fp{}$. For the curve, an isomorphic substitution from $\\Gii\\subset \\EFpxii$ into $\\Gii'$ in subfield-twisted elliptic curve $\\EdFpii$ speeds up scalar multiplications over $\\Gii$ and wipes out denominator calculations in Miller's algorithm. This paper mainly provides about 30% improvement of the Miller's algorithm calculation using proper subfield arithmetic operations. Moreover, we also provide the efficient parameter settings of the BN curves. When p is a 254-bit prime, the embedding degree is 12, and the processor is Pentium4 (3.6GHz), it is shown that the proposed algorithm computes Ate pairing in 13.3 milli-seconds including final exponentiation.