On New Pseudorandom Sequences,Lattices And Subsets Constructed Based On Cyclotomic Classes

Pseudorandom sequences are widely applied in simulation,software testing，spread-spectrum communication systems，pseudo-code range measurement.global positioning systems,channel coding，code-division multiple-access(CDMA)systems,wireless communication systems，digital communication systems,and cryptography such as radar systems and stream cipher cryptosystems.The construction and the randomness properties analysis of pseudorandom sequences are the core problems in cryptography.The theory of cyclotomy is widely applied in cryptography.A typical application is the design of pseudorandom sequences In this paper we design three families of pseudorandom binary sequences by defining the generalized cyclotomic classes modulo pq，pn+1，p+1,q+1,and calculate the autocorrelation values and linear complexity;We completely determine the autocorrelations of a family of pseudorandom quaternary sequences with period 2pm’;We construct pseudorandom binary lattices by using cyclotomic classes in finite fields,and study the pseudorandom measure of order k,family complexity,collision and avalanche effect:We introduce pseudorandom measure for subsets in finite fields and study pseudorandom properties of several subsets in finite fields.The main results can be stated as follows(1)We construct new binary sequences of order two and length pq by using the cyclotomy(Vo,Vi)proposed by Ding,and calculate the autocorrelation values,linear complexity and minimal polynomials.Results indicate that the autocorrelation value of the new sequence δ∞ takes on a few values and the linear complexity takes on one of(pq＋q-p-1)/2,(p-1)(q-1)/2,pq-p-q+1,pq-p,which only depends on the values of p(mod 8).So L(δ∞)pq/2 when p(?) 1,±3(mod 8)with p<q,and the sequence s∞ is considered to be good with respect to its linear complexity(2)By selecting special subsets,we determine the exact values of autocor-relation of generalized cyclotomic binary sequences of period pn+1 proposed by Edemskiy.The method is based on certain identities involving character sums Our results on the autocorrelation values generalize those of Legendre sequences.prime-square sequences,and prime cube sequences(3)We consider a family of Whiteman’s generalized cyclotomic sequences of period pm+1qn+1(m,n≧0)with any order d.The first contribution is to determine their linear complexity,which improves certain results of Hu,Yue and Wang.The second contribution is to compute the autocorrelation values and the result of autocorrelation is new.Results obtained indicate that such sequences are‘good’from the viewpoint of cryptography(4)We completely determine the autocorrelations of the quaternary cyclotomic sequences over F4 of length 2pm constructed by Ke and Zhang in general without the restrictions about e.(5)We construct pseudorandom binary lattices by using cyclotomic classes in finite fields,and study the pseudorandom measure of order k,family complexity,collision and avalanche effect.Results indicate that such binary lattices are’good’,and their families possess a nice structure in terms of family complexity.collision and avalanche effect(6)We first introduce pseudorandom measure for subsets in finite fields and prove lower bounds on pseudorandom measure for any subset in finite fields The pseudorandom properties of supports of Boolean functions and cyclotomic classes in finite fields have been studied.The relationship between the supports of Boolean functions and the pseudorandom measure for subsets in finite fields is analyzed.