Separating Decision Diffie–Hellman from Computational Diffie–Hellman in Cryptographic Groups

In many cases the security of a cryptographic scheme based on computational Diffie–Hellman does in fact rely on the hardness of the decision Diffie–Hellman problem. In this paper we construct concrete examples of groups where the stronger hypothesis, hardness of the decision Diffie–Hellman problem, no longer holds, while the weaker hypothesis, hardness of computational Diffie–Hellman, is equivalent to the hardness of the discrete logarithm problem and still seems to be a reasonable hypothesis.