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The Exact Formula for the Exponents of the Mixing Digraphs of Register Transformations
Journal of Applied and Industrial Mathematics Pub Date : 2020-07-10 , DOI: 10.1134/s199047892002009x V. M. Fomichev , Ya. E. Avezova
中文翻译:
寄存器变换混合有向图的指数的精确公式
更新日期:2020-07-10
Journal of Applied and Industrial Mathematics Pub Date : 2020-07-10 , DOI: 10.1134/s199047892002009x V. M. Fomichev , Ya. E. Avezova
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:
寄存器变换混合有向图的指数的精确公式