In this work, we propose a secure version of ElGamal public key cryptosystem, and prove that it is semantically secure assuming the hardness of what we call the two-dimensional decisional Diffie-Hellman (2DDDH) problem, this cryptosystem is distinguished by the speed of encryption and decryption processes and by its resistance to active adversaries. Since the 2DDDH problem is harder than the decisional Diffie-Hellman (DDH) problem (as it will be seen), one may conclude that our model reinforces the exchange security compared to the existed cryptosystems falling within the same context, also we discuss the difficult problems that guarantee its security.

In this paper, we propose a new and practical variant of ElGamal encryption which is secure against every passive and active adversary. Under the hardiness of the decisional Diffie-Hellman assumption, we can prove that the proposed scheme is secure against an adaptive chosen ciphertext attacks in the standard model. Such security verifies not only the confidentiality but also verifies the integrity and the authentication of communications. We display that the modified scheme furthermore achieves anonymity as well as strong robustness.

In this paper we consider the problem of the asymptotic expansion of double Laplace-type integrals, in the case when the set γ of points where the phase achieves its absolute minimum is a simple curve. It will be shown that the asymptotic behaviour of such integrals is governed by the order of degeneracy of normal derivatives of the phase with respect to the curve γ. Complete asymptotic expansions will be constructed if that order is constant along γ, and the first two coefficients will be explicitly computed. If not, a uniform asymptotic expansion method, involving special functions, is suggested.