In this paper we study pseudorandomness of a family of sequences in terms of two measures, the family complexity ($f$-complexity) and the cross-correlation measure of order $\ell$. We consider sequences not only on binary alphabet but also on $k$-symbols ($k$-ary) alphabet. We first generalize some known methods on construction of the family of binary pseudorandom sequences. We prove a bound on the $f$-complexity of a large family of binary sequences of Legendre-symbols of certain irreducible polynomials. We show that this family as well as its dual family have both a large family complexity and a small cross-correlation measure up to a rather large order. Next, we present another family of binary sequences having high $f$-complexity and low cross-correlation measure. Then we extend the results to the family of sequences on $k$-symbols alphabet.
翻译:在本文中,我们从两个衡量标准,即家庭复杂性(ff$-complexity)和交叉关系测量标准(cross-conclation)的角度研究一个家庭序列的假随机性。我们不仅考虑二进制字母的序列,而且还考虑以美元-symbols(k$-mary)字母的序列。我们首先对二进制假冒序列的家庭结构的一些已知方法进行概括化研究。我们证明,一个大家庭的一成二进制($-f$-commonnicals)和若干不可复制的多式传说-符号的二进制序列($-commballs)。我们表明,这个家庭及其双家族既具有很大的家庭复杂性,又具有与相当大顺序的小型交叉关系测量标准。接下来,我们提出了另一个二进制组合的二进制组合,其美元-comcol-commol-cormols 度值很高。然后我们将结果推广到以美元-symolms字母的序列的家庭。