The Verheul homomorphism is a group homomorphism from a finite subgroup of the multiplicative group of a field to an elliptic curve. The hardness of computation of the Verheul homomorphism was shown by Verheul to be closely related to the hardness of the computational Diffie-Hellman problem. Let p ≥ 5 be a prime, and let N be a prime satisfying √(12p) < N < 2p / √3, where N ≠ p. Let E be an ordinary elliptic curve over Fp, and let C ⊂ E be a cyclic subgroup of order N. Let H be the group of all Nth roots of unity (contained in the algebraic closure of Fp), and let phi be the Verheul isomorphism from H to C.
We consider a polynomial P such that P(z) is the X-coordinate of phi(z) for all z ∈ H – {1}. We show that, for at least approximately 58% of pairs (E, C), none of the coefficients of the non-constant terms of P vanishes.