Fast Ate Pairing Computation of Embedding Degree 12 Using Subfield-Twisted Elliptic Curve
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 is an element of F(p). For the curve, an isomorphic substitution from G(2) is an element of 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.
ate pairing subfield arithmetic operation twist fast computing