Information has been one of the leading role naw. With the fast development of Internet applications, information security has beome more and more important. The problem of the trustworthiness and security has also been more and more important. So we need best Digital Signature Algorithm and cryptographic algorithm. The core of information security is cryptography including traditional block ciphers and public key ciphers.Another significant factor in information security is cryptographic protocols in which electronic commerce protocols play an wide spread important role.Much of the security research and development over the last thirty years has been in the area of cryptography.This is worth it,but far from enough.Information security still needs research and development on cryptography protocols.It is well know that designing a correct and secure cryptographic protocol is an error-prone task.Furthermore, the burst of computer hardware in recent years has a great impact on cryptography. So we should be devoted to the researching and improving of implementation of digital signature algorithm.In this paper,we brought forward the concept of ergodic matrix over finite field Fq , simply discussed theory of finite field including property theory of finite field, calculation in finite field and construction from low-degree finite field to high-degree finite field. There are important uses in encryption because operations in the finite field has many characteristics such as without carrying bits,equal bits and without rounding error. In this paper, We brought out the concept of"ergodic matrix", and gave out the recursion formula(II) which can construct the ergodic matrix Q with n×n dimensions using a vector with n dimension. That is, we can delegate n2 bits ergodic matrix Q using n bits integer vector ,at the same time save the complexity of space and time. Then we brought out arithmetic to construct ergodic matrix according to the recursion formula(II) and expatiate characteristic theorems about"ergodic matrix"by the numbers,the difficult problem and one-way (trapdoor) function based on the ergodic matrix over finite field.Then we put forward the concept of strong matrix based on ergodic matrix according to a problem and the algorithm of finding and constructing the strong matrix of ergodic... |