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 Ψ_d be an isomorphic map from E′ (double struck F sign_p^e) to the corresponding subgroup of E(double struck F sign_p^k). Then, consider double struck G sign′₁ = Ψ_d ^-1(double struck G sign₁) and double struck G sign′₂ = Ψ_d^-1(double struck G sign₂) for double struck G sign₁, double struck G sign₂ at Ate pairing α. Let P ∈ double struck G sign ₁, Q ∈ double struck G sign₂, P′ ∈ double struck G sign′₁ and Q′ ∈ double struck G sign′₂, the authors have shown α(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.