Efficient Pairings on Twisted Elliptic Curve

This paper proposes an efficient implementation of Ate pairing on twisted elliptic curve. Suppose that a pairing-friendly elliptic curve E has a twisted elliptic curve E' of degree d, and let psi(d) be an isomorphic map from E'(F-p(e)) to the corresponding subgroup of E(F-p(k)). Then, consider G' = psi(-1)(d) (G1) and G(2)' = psi(-1)(d) (G(2)) for G(1), G(2) at Ate pairing alpha. Let P is an element of G(1), Q is an element of G(2), P' is an element of G'(1) and Q' is an element of G'(2), the authors have shown alpha(Q,P) = F-t-q,F-Q (P)((pk-1)/r) = f(t-1,Q')(P')((pk-1)/r). This paper shows that this new Ate pairing, namely cross twisted (Xt) Ate pairing, provides an quite efficient implementation.