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 y² = x³ + a, a ∈ F_p. For the curve, an isomorphic substitution from G₂ C E(F_p ¹²) into G ₂ in subfield-twisted elliptic curve E(Fp2 ) speeds up scalar multiplications over G₂ 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.