Universally Constructing 12-th Degree Extension Field for Ate Pairing

We need to perform arithmetic in Fp(z)12 to use Ate pairing on a Barreto-Naehrig (BN) curve, where p(z) is a prime given by p(z) = 36z + 36z + 24z + 6z+ 1 with an integer z. In many implementations of Ate pairing, Fp(z)12 has been regarded as the 6-th extension of Fp(z)2 , and it has been constructed as Fp(z)12 = Fp(z)2 [v]/(v−ξ) for an element ξ ∈ Fp(z)2 such that v − ξ is irreducible in Fp(z)2 [v]. Such ξ depends on the value of p(z), and we may use mathematic software to find ξ. This paper shows that when z ≡ 7, 11 (mod 12) we can universally construct Fp(z)2 as Fp(z)12 = Fp(z)2 [v]/(v−u−1), where Fp(z)2 = Fp(z)[u]/(u+1).