Universally Constructing 12-th Degree Extension Field for Ate Pairing
Masaaki Shirase · 2009
Abstract. 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 4 + 36z 3 + 24z 2 + 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 6 −ξ) for an element ξ ∈ Fp(z) 2 such that v 6 − 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 6 −u−1), where Fp(z) 2 = Fp(z)[u]/(u 2 +1). Key words: pairing, Barreto-Naehrig curve, extension field, quadratic residue, cubic residue, Euler’s conjecture. 1