Improving the Diffie-Hellman secure key exchange

Diffie-Hellman (DH) is a well-known cryptographic algorithm used for secure key exchange. The first appearance of DH was in 1976. The algorithm allows two users to exchange a symmetric secret key through an insecure wired or wireless channel and without any prior secrets. DH works under the domain of integers Z*_n where n = p. Here, p and α are the two parameters of DH where p is a large prime number and α is a generator selected from the cyclic group Z*_n. In this paper, we propose two modifications of DH. The first modification is to change the domain to integer with n=2p^t where Z*_n is still cyclic and the second modification is to change the domain to Gaussian arithmetic Z*_n. After implementing the three algorithms we found that the symmetric key size derived from the two modified algorithms is much greater than the classical one. Moreover, attacking the two modified algorithms using Pohlig-Hellman algorithm, using the same prime value p and private value a or b, needs much more time than the classical one.