Research On Some Issues In Error-correcting Codes Theory Over Rings And Their Applications In Information Security

Error-correcting codes theory and stream ciphers are the theoretical basis of information security. In coding theory, error-correcting codes theory over finite fields has been not only developed perfecting but also applied widely to the productive practice. With the successive deeply researching on theory, research the theory and practical value on error-correcting codes theory over finite rings or group Algebra have been received much attention. In stream ciphers, the sequence of pseudo-random properties directly affect the security of cryptography systems. Linear complexity and autocorrelation are two of important criterions for measuring the pseudo-random properties of sequences. This dissertation studies the MacWilliams identities for linear codes and the Chinese product codes over finite rings or group Algebra. Furthermore, the study of error-correcting codes theory applied in stream ciphers. The details are given as follows:1、The MacWilliams type identity for the m-ply Lee weight enumerator for linear codes over F2+uF2is determined. As an application of this identity, the authors obtain a MacWilliams type identity on Lee weight for linear codes over F2m+uF2m. Furthermore, the authors prove a duality for the m-ply Lee weight distributions by taking advantage of the Krawtchouk polynomials.2、The generalized MacWilliams identities between the support weight distributions of a Fp+uFp linear code of type p and that of its dual is obtained.3、The relationship of the support weight of linear codes over Z8between that of their subcodes is determined.4、The Gray images of the Chinese product of constacyclic and cyclic codes over a finite ring are studied. The Chinese product of constacyclic and cyclic codes over the finite ring are introduced. A Gray map between codes over the finite ring and a finite field is defined. The Gray image of the Chinese product of constacyclic codes over the finite ring is a distance-invariant quasi-cyclic code over the finite field are proved. Each code over the finite field, which is the Gray image of the Chinese product of cyclic codes over the finite ring, is permutation equivalent to a quasi-cyclic code is also proved.5、Let k=(Πi-1spi)m and m≥1. Define R(k,r)=GR(p1m,r)(?)GR(p2m,r)(?)…(?)GR(psm,r), where GR(pim,r) denotes the Galois ring of degree r, and the code length n can not be divided by pi. Then, the Chinese Remainder Theorem is described for studying cyclic codes and their duals over the ring R(k, r). The structures of the Chinese product of cyclic codes and their duals over the ring R(k, r) are established. Furthermore, we study the Chinese product of cyclic self-dual codes over the ring R(k, r). Necessary and sufficient conditions for the existence of nontrivial such cyclic self-dual codes are given. The generating idempotents of the Chinese product of cyclic codes are studied.6、The Chinese Remainder Theorem for studying Abelian codes of length N over the ring Zm is described, where m=Πi=1spi,k=Πi=1qpit2, N=kn, pi are distinct primes, s≥2is a positive integer, ti are positive integers and n is a positive integer prime to k. The structure theorems for Abelian codes and their duals in ZmG are obtained, where G=Ck×H, Ck denotes the cyclic group of order k and H denotes a group of order n. The existence of self-orthogonal and the nonexistence of self-dual Abelian codes over Zm are studied.7、Let Fp+μFp, where p is an odd prime. A class of new linear codes over Fp+uFp is constructed by means of the trace map. Then a kind of optimal codes over Fp is obtained via the Gray map from the punctured new linear codes. Furthermore, a class of new linear cyclic codes over Fp+uFp is also constructed by means of the trace map. A kind of low correlation linear sequences overis observed via the generalized Nechaev-Gray map from the class of new linear cyclic codes, which are regarded as a class of linear periodic sequences.8、By taking advantage of a generalized discrete Fourier transform in the study of coding theory, the linear complexity of r-dimensional n-periodic sequences overis studied. Lower bounds on the linear complexity for these sequences are obtained. Furthermore, an algorithm for lower bounds of linear complexity is proposed.9、Let S(t)=(S1,S2,...,St) be a linear recurring sequence over Fqm, where each Si is a linear recurring sequence over Fqm with minimal polynomial∫i (x) over Fqmt and the finite field Fqmi is a subfield of finite field Fqm. Let T be a linear transformation of Fqm over Fq. Denote T(S(t))=(T1(S1),T2(S2),...,Tt(St)), where Ti is a linear transformation of Fqnover Fqmi. We study the minimal polynomial and linear complexity of linear recurring sequence S(t). If the canonical factorization of each∫i(x) over Fqm is known, then the minimal polynomial and linear complexity of linear recurring sequence S(t) over Fqm is determined. Furthermore, the minimal polynomial of T(S(t)) is determined, if the minimal polynomial of each Si without multiple roots is known.