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

摘要

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 ∈ F_ρ For the curve, an isomorphic substitution from G_2 is contained in E(F_P 12) into G'_2 in subfield-twisted elliptic curve E'(F_P 2) speeds up scalar multiplications over G_2 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 (3.6 GHz), it is shown that the proposed algorithm computes Ate pairing in 13.3 milli-seconds including final exponentiation.