Universal Construction of a 12th Degree Extension Field for Asymmetric Pairing
It is necessary to perform arithmetic in F(p)12 to use an Ate pairing on a Barreto-Naehrig (BN) curve, where p is a prime given by p(z) = 36z(4) + 36z(3) + 24z(2) + 6z + I for some integer z. In many implementations of Ate pairings, F(p12) has been regarded as a 6th degree extension of F(p)2, and it has been constructed by F(p)12 = F(p2)[v]/(v(6) - xi) for an element xi epsilon F(p2) such that v(6) - xi is irreducible in F(p)(2) [v]. Such a xi depends on the value of p, and we may use a mathematical software package to find xi. In this paper it is shown that when z 7, 11 (mod 12), we can universally construct F(P)(12)r as F(p)(12) = F(p2)[v]/(v(6) - u - 1), where F(P2) = F(P) [U]/(U(2) + 1).
extension field pairing barreto-naehrig curve quadratic residue cubic residue euler's conjecture