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

Masataka AKANE  Yasuyuki NOGAMI  Yoshitaka MORIKAWA  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E92-A    No.2    pp.508-516
Publication Date: 2009/02/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E92.A.508
Print ISSN: 0916-8508
Type of Manuscript: PAPER
Category: Cryptography and Information Security
Ate pairing,  twist,  subfield arithmetic operation,  fast computing,  

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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, aFp. For the curve, an isomorphic substitution from G2E(Fp12 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.