Key-Exchange Of Multiply Discrete Logarithm Based On The Ergodic Matrix

With the development of computer and population of network,people gradually enter the information age.The network influents people's lives increasingly.And providing abundant information,it brings a great of convenience to people.The most important feature of the network is open,that is,not only can people easily access infiormation,but also are faced with huge threat because of opening their privacies.Cryptography technique provides the security of information.They admit the owners access their resources freely,and refuse the intruders effectively.The security of the key in cryptography is essential.If the key is lost of security,it is nonsense even if the cryptography algorithm is strong.Therefore,it is critical whether the preservation and transmission of keys is safe or not.How to distribute and exchange keys securely is often a bottle-neck constraint in information security.This paper studies a method of exchanging keys,based on multi-DLP over Ergodic Matrices.It improves the old method of exchanging keys based on Diffie-Hellman Protocol, by the characteristic of Ergodic Matrices.That is,it changes Two-Side-Power-Multiply form to Three-Side-Power-Multiply form and proposes a new scheme of exchanging keys called "Multi-DLP Exchanging Keys Scheme over Ergodic Matrices".Its main content can be summarized as follows:There is an property of ergodic matrix as following:If E is an ergodic matrix,the period of E is(qn-1)in the finite with GF(qn)and the vector t∈Fq1×n{\0} come from E make into an set{tE1,tE2,tE3,...,tEqn-1},which can take over each element of∈Fq1×n{\0}. So we can get an hard problem:when E1,E2∈Fqn×n are two ergodic matrices and M∈Fqn×n don't equal to zero matrix,how to get x and y under known(E,E1x,E2y).We can consider the difficulties with this kind of hard problem to design one-way trapdoor function.The difficulty of this hard problem,which can be deduced into the BMQ problem,depends on the strong matrix M∈Fqn×n .Suppose Fq[E1]={E10,...,E1n-1} and Fq[E2]={E20,...,E2n-1)are both bases established. Expanding the equation, MQ equation E as follows can be got:In the equation E, the number of equations is m and the number of variables is 2n. When m=2n, it is hard to resolve the BMQ equation. Only just Rank(M)=m=2n is it hard problem. from the above analysis, it is critical that how select the M. we must select a strong M to make E1xME2y to be an problem of MQ. when Rank(M)=m equal m, Solving the equation is difficult as problem of MQ,which include 2n independent variables and m equation. When the number of variables is close to the number of equation, problem of MQ can't solving by polynomial. So the above question is difficult.the key is to construct one-way function to find specific data structure of the mathematical difficulties. The one-way functions can be constructed with different password for different systems. DH algorithm is based on integer discrete logarithm problem is difficult to break down the difficult, Elliptic curve encryption algorithm is based on the elliptic curve discrete logarithm problem is difficult to solve mathematical problems. Therefore, the difficult problem construct the different cryptographic techniques. According to nature of Ergodic Matrices and the two-side-multiply set, proposing the following severe difficulty problem based on Ergodic Matrices:Problem 1:let E∈Fqn×n be a ergodic matrix and x∈{1,...qn-1},.Given (E,Ex) to calculate xProblem 2:let E1,E2∈Fqn×n be ergodic matrix and M∈Fqn×n \{0} and x∈{1,...qn-1}, Given (E,M,E1xME2y) to calculate x,yProblem 3:let E1,E2,E3∈Fqn×n be ergodic matrix and M1∈Fqn×n \{0} and x∈{1,...qn-1},.Given (E1,E2,M1,E1xM1E2yM1E1x) to calculate x,y.Problem 4:let E1,E2∈Fqn×n be a ergodic matrix and M1,M2∈Fqn×n \{0} and x∈{1,...qn-1},.Given (E1,E2,M1,M2,E1xM1E2yM2E1x) to calculate x,y.Problem 5:let E1,E2,E3∈Fqn×n be a ergodic matrix and M,,M2∈Fqn×n \{0} and x∈{1,...qn-1},.Given (E1,E2,E3,M1,M2,E1xM1E2yM2E34) to calculate x,y,z.Through analyzing the difficulty of problem,we find that problem 1 is more difficult than exchange-algebraic-group on the discrete logarithm problem. But problem 2 is difficult as problem of MQ. it is Non-deterministic Polynomial problem because that problem of MQ is Non-deterministic Polynomial problem. Problem 3 and problem 4 and problem 5 have more discrete logarithm than problem 2, so they are more difficult.We can construct one-way function with the above problem. using the one-way-function, we can implement two protocol:key-exchange of two discrete logarithm based on the ergodic matrix, key-exchange of three discrete logarithm based on the ergodic matrix. And choose a double discrete logarithm and three discrete logarithm of a typical program, the program is given on the analysis, and program verification.