The Establishment Of ElGamal Public-Key Cryptosystem Based On Algebraic Method

With the development of the network techniques, the transmission of wireless information has become the main mode. The following information security is getting up more and more important position. Cryptography is the core of information security. Public-Key Cryptosystem has become more and more popular. Public-Key Cryptosystem has gained the research focus.It has opened up the new epoch of cryptography since Whirefield Diffie and Martin Hellman put forword the thought of Public-Key. The cryptosystem has a new direction. Encryption key is different from decryption key for the cryptosystem, therefore, in order to be different from symmetric encryption, which is called Public-Key Cryptosystem. It is based one-way trapdoor function a cryptosystem and a mathematical difficult problem, it could such say, it studies on the core Utilization for mathematical function to structure encryption. During the late 20th Public-Key Cryptosystem has made a great progress. It has gotton a greater advance in encryption, digital signature, authentication, etc. Now the popular cryptography scheme is based two mathematical problems, the first is based on the integer factorization problem and the second is based on discrete logarithm problem. Preset the Public-Key Cryptosystem such as RSA, ElGamal, Diffie-Hellman, Elliptic Curve applied widely.This paper is mainly based on algebraic method, studies on public-key algorithm especially for ElGamal public-key algorithm. First it introduces the history and the development of Public-Key Cryptosystem, meanwhile learning basic knowledge of algebra and number theory. It analysizes the feature of semigroup,group,rings,field and so on. And it researches the mainly cryptographic algorithms. It discusses the relationship of the algebraic structure and cryptography. Following, it summarizes based on algebraic method of ElGamal Public-Key Cryptosystem and conceives a generalized ElGamal Public-Key Cryptosystem. At the same time, it studies on integral quaternions group and proposes based on integral quaternions of ElGamal Public-Key Cryptosystem. Its security is based on the difficulty of large integer factorization and discrete logarithms problem, and simulation and realization in computer in using Visual C++ for little Parameters example, furthermore discusses and analyzes the security of it.