双素数周期的六阶二元广义割圆序列的随机性研究

Pseudo-Random Properties of Binary Sextic Generalized Cyclotomic Sequences with Two-Prime Periods

剩余类环上的割圆理论在序列设计和通信码的构造方面有着广泛的应用.根据前人(目前)的研究成果可知,剩余类环上2阶和4阶二元Whiteman割圆序列有很多好的随机性质.本文基于双素数剩余类环Z_(n_1n_2)上的割圆理论和中国剩余定理,对剩余类环Z_(n_1n_2)作二元分割,并利用特征集法构造了一类周期为(n_1n_2)的6阶二元Whiteman广义割圆序列.进而根据有限域上的多项式理论,通过构造多项式x^(n_1n_2)-1的分裂域和讨论n_1和n_2的不同取值,计算了这些序列的线性复杂度.计算结果表明这类序列线性复杂度的最小值是(n_1-1)(n_2-1)/2,符合密码学要求.另外,利用6阶Whiteman割圆数和差分函数计算了部分6阶二元Whiteman广义割圆序列的自相关值,其它的情形也可以同理得到.

Cyclotomy theory on residue class ring has widely been used in the design of sequences and the construction of communication codes.According to the known research achievements,binary Whiteman generalized cyclotomic sequences of orders 2 and 4 over the two-prime residue ring Z_(n_1n_2) have a number of good randomness properties.Based on cyclotomy on the two-prime residue ring Z_(n_1n_2) and the Chinese Remainder Theorem,by making a binary segmentation of the residue class ring and applying characteristic set method,this paper constructs a class of binary Whiteman generalized cyclotomic sequences of order 6 of two-prime period(n_1n_2).Then,according to the polynomial theory on finite fields,by constructing the splitting field of the polynomial x^(n_1n_2)-1 and discussing different values of n_1 and n_2,the corresponding linear complexities(linear span) of these sequences are calculated.Our results show that the least value of their linear complexities is(n_1-1)(n_2-1)/2,which meet the cryptographic requirement.In addition,we calculate the autocorrelation values of some binary generalized cyclotomic sequences of order 6 constructed in this paper by making use of the Whiteman cyclotomic numbers of order 6 and the differential function.Other situations can also be obtained in the same way.