Public-Key Cryptosystem Based on the Discrete Logarithm Problem

In mathematics,spciticially in abstract algebra and its applications, discrete loga-rithms are group -theoretic analogues of ordinary logarithms. The problem of computing both problems are difficult, alogrithms from one problem are often adapted to the other, and the difficulty of both problems has been exploited to construct various cryptographic system. This work describes an application of the discrete logarithm that is cryp-tosystem based on the discrete logarith problem. The paper defined at first the classical diffie-helman key exchange, then extend this cryptosystem to the domain of gaussian integer that make it more difficult to break. Then we mention the elgamal public key cryptosystem at the domain of integer number, to extend it later to the domain of gaussian interger also to be more difficult to break that is the main objective in this work.