A Public-Key Cryptosystem Based on Lucas Sequences

Based on Lucas functions, an improved version of Diffie-hellman key distribution, El Gamal public key crypto-system scheme and El Gamal signature scheme are proposed, together with an implementation and computational cost. The security relies on the difficulty of factoring an RSA integer and on the difficulty of computing the discrete logarithm. Introduction In [1], Diffie and Hellman introduced a practical solution to the key distribution problem, allowing two parties, Alice and Bob never met, to share a secret key by exchanging information over an open channel. In [2], El Gamal used Diffie-Hellman ideas to design a crypto-system whose security is based on the difficulty of solving the discrete logarithm problem. In [3], It was suggested that linear sequences could be used instead of the standard RSA. In this paper, based on Lucas sequences, an improved of Diffie-hellman key distribution,El Gamal public key crypto-system and El Gamal digital signature were proposed. This considerably reduces the computation cost of these methods. The security relies on the difficulty of factoring an RSA integer. In section 1, an investigation of cryptographic properties of Lucas sequences, and a computational method to evaluate the k term of a Lucas sequence are given. In section 2, two cryptographic applications are given, their security and computational cost were analyzed.