Abstract

A digraph is primitive if some positive degree of it is a complete digraph, i.e. has all possible arcs. The smallest degree of this kind is called the exponent of the digraph. Given a primitive digraph, the elementary local exponent for some vertices \(u\) and \(v \) is the least positive integer \(\gamma \) such that there exists a path from \(u \) to \(v\) of any length not less than \(\gamma \). For the transformation on the binary \(n \)-dimensional vector space which is given by a set of \(n \) coordinate functions, there corresponds some \(n \) vertex digraph such that a pair \((u,v) \) is an arc if the \(v \)th coordinate component of the transformation depends essentially on the \(u\)th variable. Such a digraph we call a mixing digraph of the transformation. Under study are the mixing digraphs of \(n\)-bit shift registers with a nonlinear Boolean feedback function (NFSR), \(n>1 \), which are widely used in cryptography. We find the exact formulas for the exponent and elementary local exponents of the \(n \)-vertex primitive mixing digraph associated to an NFSR. The above results can be applied to evaluating the length of the blank run of the pseudo-random sequences generators based on NFSRs.

中文翻译：

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寄存器变换混合有向图的指数的精确公式

摘要

如果有向图是肯定的，则该图是原始图，即具有所有可能的弧。这种最小的程度称为有向图的指数。给定一个基本图，某些顶点\（u \）和 \（v \）的基本局部指数是最小正整数\（\ gamma \），因此存在从\（u \）到\（v的路径 \）的任何长度不小于\（\伽马\） 。对于由一组\（n \）坐标函数给定的二元\（n \）维向量空间 的变换 ，对应着一些 \（n \）如果变换的第\（v \）个坐标分量基本上取决于第\（u \）个变量，那么顶点对有向图，使得一对\（（u，v）\）是弧。这样的图我们称为变换的混合图。正在研究的是具有非线性布尔反馈函数（NFSR）\（n> 1 \）的\（n \）位移位寄存器的混合图，它们在密码学中得到了广泛使用。我们找到与NFSR相关的\（n \）-顶点原始混合有向图的指数和基本局部指数的精确公式。以上结果可用于评估基于NFSR的伪随机序列生成器的空白游程长度。