The Exact Formula for the Exponents of the Mixing Digraphs of Register Transformations

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.