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_pc) to the corresponding subgroup of E(F_pk). Then,consider G´₁= psi_d ^-1(G₁) and G´₂ = psi_d ^-1(G₂) for G₁,G₂ at Ate pairing alpha. Let P isin G₁, Q isin ,G₂, P´ isin G´₁ and Q´ isin G´₂, the authors have shown alpha(Q, P) = f_t-1,Q(P) ⁽p^{K -1)/r} = f_t-1,Q´(P´) ⁽p^{K -1)/r} .This paper shows that this new Ate pairing, namely cross twisted (Xt) Ate pairing, provides an quite efficient implementation.