Characteristic Analysis Of Ergodic Matrices Over Finite Field In Cryptography

With the development of internet and electronic commerce, informationsecurity has became more and more important. Information security is based oncryptology , but the main subject studied by cryptology is communication securityand only fit to data communication security. Encryption is the most main methodthat ensures the safety of internet and electronic commerce, and is widely used inCryptography .On exchanging one message, the sender encodes the data, then sendsit to the receiver, the receiver decodes the data, then he gets the original message.Encryption includes two elements: arithmetic and key. The arithmetic is a mathfunction used to combine plain-text with key to create cryptograph that no one canunderstand. The key is a kind of special numbers used to encode and decode themessage. The arithmetic is the kernel of Encryption. A good arithmetic will speedup the development of Encryption. We can use Encryption and managementsystem to ensure the safety of internet communication. In recent years, the burstof computer hardware has a great impact on cryptograph. So we should bedevoted to the researching of arithmetic.In this paper, we simply discussed theory of finite field including propertytheory of finite field, calculation in finite field and construction fromlow-degree finite field to high-degree finite field. A finite field or Galois field(so named in honor of Evariste Galois) is a field that only contains finitelymany elements.There are important uses in ECC and encryption becauseoperation in a finite field has many characteristics such as without carryingbits,equal bits and without rounding error. In this paper, We bring out theconcept of "ergodic matrix"(what is called n-degree "ergodic matrix"in finitefield Fq is the matrix Q with full orders which period is qn-1.If d= qn-1,then theconcourse ?Q?={ Q, Q2,…,Qd =E} made of power of Q constructs a circlinggroup. that is, one given column (row) of { Q, Q2,…,Qd =E} is just ergodic overfrom 1 to qn-1. N-degree "ergodic matrix"in finite field Fq is simply called"ergodic matrix".), and give out the recursion formula(II) which can constructthe 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(thatis to say, construction of ergodic matrix is equal to that of n-power irreduciblepolynomial),at the same time save the complexity of space and time. We bringout arithmetic to construct ergodic matrix according to the recursionformula(II),then expatiate characteristic theorems about "ergodic matrix"by thenumbers, such as ergodicity, the determinant of "ergodic matrix", the number of"ergodic matrix"and so on, we also give particular proofs about the theoryabove.We bring out the concept of strong matrix based on ergodic matrixaccording to a problem and give a way to use the strong matrix to constructone-way function. We give a new plan to realize application pattern incryptology by the one-way function based on ergodic matrix that is widely usedin cryptology. Finally give out many guesses ,we must give a deep study toprove whether which are right or not. According to characteristic analysis ofergodic matrix at chapter above, we can find that the "ergodic matrix"has anumber of good features that can be applied to cryptography, such asrandomicity and ergodicity. So we can realize many application pattern incryptology by ergodic matrix. At the chapter we give the arithmetic to realizecorrespond to application pattern respectively according to special characters ofergodic matrix. These application patterns contain fake random sequence,symmetric algorithm, public-key algorithm, Shamir three tansfering protocoland one-time one-secret identity authentication. So we can know that theconstruction of ergodic matrix is the base to settle the problems above.We givethe arithmetic to construct ergodic matrix. In order to construct (find) the ergodic matrix Q in finite field Fq, for?(gn L g2 g1)∈GF(qn)∧(gn ≠0)defines the following recursion formula: ???aa?1 == ga?a2 (m ≥0) m 1 m?1+ g2am?2 +L+ gn?1am?n+1 + am?n +1 (II) =L= a?n = 0It is clear that the sequence of am(am≥0) is cyclic, and the length of thecycle-segment is only determined by (gn …g2 g1). For any integer i , j (i>j≥0),according to gn ≠0 we know if (ai+n-…ai+1ai)= (aj+n-…aj+1ai),then (ai+n-… 1 1 2aiai )= (aj+n-…ajaj ).The length of the cycle-segment determined by (gn … -1 2 -1g2 g1) is the minimal positive integer L that meets: (aL-aL …aL-n)=(00…0). 1 -2For any n-power irreducible polynomial in finite field Fq g(x) =gnxn+gn-1xn-1+ …+g1x+1, if the cycle-segment determined by (gn …g2 g1) wereconnected end to end, we can get a cycle-segment ring which length is L .