This paper develops the notion of \emph{Word Linear Complexity} ($WLC$) of vector valued sequences over finite fields $\ff$ as an extension of Linear Complexity ($LC$) of sequences and their ensembles. This notion of complexity extends the concept of the minimal polynomial of an ensemble (vector valued) sequence to that of a matrix minimal polynomial and shows that the matrix minimal polynomial can be used with iteratively generated vector valued sequences by maps $F:\ff^n\rightarrow\ff^n$ at a given $y$ in $\ff^n$ for solving the unique local inverse $x$ of the equation $y=F(x)$ when the sequence is periodic. The idea of solving a local inverse of a map in finite fields when the iterative sequence is periodic and its application to various problems of Cryptanalysis is developed in previous papers \cite{sule322, sule521, sule722,suleCAM22} using the well known notion of $LC$ of sequences. $LC$ is the degree of the associated minimal polynomial of the sequence. The generalization of $LC$ to $WLC$ considers vector valued (or word oriented) sequences such that the word oriented recurrence relation is obtained by matrix vector multiplication instead of scalar multiplication as considered in the definition of $LC$. Hence the associated minimal polynomial is matrix valued whose degree is called $WLC$. A condition is derived when a nontrivial matrix polynomial associated with the word oriented recurrence relation exists when the sequence is periodic. It is shown that when the matrix minimal polynomial exists $n(WLC)=LC$. Finally it is shown that the local inversion problem is solved using the matrix minimal polynomial when such a polynomail exists hence leads to a word oriented approach to local inversion.

翻译：本文发展了有限域$\ff$上向量值序列的\emph{词线性复杂度} ($WLC$)概念，作为序列及其集合的线性复杂度($LC$)的推广。该复杂度概念将集合（向量值）序列的最小多项式扩展为矩阵最小多项式，并证明当迭代生成的向量值序列（由映射$F:\ff^n\rightarrow\ff^n$在$\ff^n$中给定$y$处生成）具有周期性时，该矩阵最小多项式可用于求解方程$y=F(x)$的唯一局部逆$x$。在先前的论文中，利用广为人知的序列$LC$概念，已发展了在有限域中迭代序列具有周期性时求解映射局部逆的思想，并将其应用于密码分析中的多种问题。$LC$是序列关联最小多项式的次数。从$LC$到$WLC$的推广考虑了向量值（即面向词）序列，使得面向词递推关系通过矩阵向量乘法（而非$LC$定义中采用的标量乘法）获得。因此，关联的最小多项式为矩阵值，其次数被称为$WLC$。本文推导出当序列为周期性时，存在面向词递推关系非平凡矩阵多项式的条件。证明当矩阵最小多项式存在时，有$n(WLC)=LC$。最后证明，当此类多项式存在时，利用矩阵最小多项式可求解局部逆问题，从而提出一种面向词的局部逆求解方法。