Based On The Discrete Logarithm Problem, A Number Of Cryptography Research

The security of morden cryptology is mainly based on the two mathematical puzzles, factoring the large number and discrete logarithm in Number Theory. The cryptography based on the problem of discrete logarithm was generally applied in the security agency and widely used to protect information, because of better security. Many cryptosystem, such as Diffie-Hellman Key Exchange Protocol, ElGamal Public-key Cryptosystem and Digital Signature Scheme and the official U.S. Digital Signature Algorithm, are even world-famous.The cryptography based on the problem of discrete logarithm can be devided into two forms, one is the cryptography in finite field, and another in the curves of elliptic. Moreover, because of its advantages of little integrated circuit, higher efficiency, stronger security, the cryptography based on the problem of elliptic curves discrete logarithm was extensively applied in the devices where computational power and integrated circuit was limited, such as Smart Card, PDA and Wireless.The application of discrete logarithm in the cryptography was researched and discussed in this paper, and main results are obtained as follows:1. Some basic properties of the general linear groups and the solution of the order of their elements were utilized in order to propose the Discrete Logarithm Public-Key Cryptosystem based on general linear groups. Furthermore, a general linear groups digital signature was presented, which is of stronger security and higher efficiency.2. The structure of group ring was studied and the structure and properties of the group ring of general linear group was obtained. By used the one good property that the order of the group ring increases in the form of exponent, a public-key cryptography based on group ring was proposed. At last, the security level of this cryptosystem, the cost of computation and the method of implementation is discussed, and an example is given for support to illustrate the feasibility of this cryptography.3. The theory of self-bilinear pairing was applied to elliptic curves cryptography over the ring, and the security and complexity of this system was discussed. Meanwhile, it is pointed out that the problems of factoring the integer and elliptic curve discrete logarithm are considered simultaneously, for this system, which enhance the security.