Word Linear Complexity of sequences and Local Inversion of maps over finite fields

• URL: http://arxiv.org/abs/2311.06574v1
• Date: Sat, 11 Nov 2023 14:06:23 GMT
• Title: Word Linear Complexity of sequences and Local Inversion of maps over finite fields
• Authors: Virendra Sule,
• Abstract summary: This paper develops the notion of vector valued sequences over finite fields $ff$ as an extension of Linear Complexity$ of their ensembles. It shows that the matrix minimal can be used with iteratively generated vector valued by maps $F:ffn at a given $ffn$ in $ffn$ for solving the unique local inverse $x$ in $ffn$ when the sequence is periodic.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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