Universal Construction of a 12th Degree Extension Field for Asymmetric Pairing

Masaaki SHIRASE  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E94-A   No.1   pp.156-164
Publication Date: 2011/01/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E94.A.156
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Cryptography and Information Security)
Category: Mathematics
pairing,  Barreto-Naehrig curve,  extension field,  quadratic residue,  cubic residue,  Euler's conjecture,  

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It is necessary to perform arithmetic in Fp12 to use an Ate pairing on a Barreto-Naehrig (BN) curve, where p is a prime given by p(z)=36z4+36z3+24z2+6z+1 for some integer z. In many implementations of Ate pairings, Fp12 has been regarded as a 6th degree extension of Fp2, and it has been constructed by Fp12=Fp2[v]/(v6-ξ) for an element ξ ∈ Fp2 such that v6-ξ is irreducible in Fp2[v]. Such a ξ depends on the value of p, and we may use a mathematical software package to find ξ. In this paper it is shown that when z ≡ 7,11 (mod 12), we can universally construct Fp12 as Fp12=Fp2[v]/(v6-u-1), where Fp2=Fp[u]/(u2+1).