Computer Science > Symbolic Computation
[Submitted on 26 Oct 2020 (v1), last revised 9 Apr 2022 (this version, v4)]
Title:On Linear Representation, Complexity and Inversion of maps over finite fields
View PDFAbstract:The paper primarily addressed the problem of linear representation, invertibility, and construction of the compositional inverse for non-linear maps over finite fields. Though there is vast literature available for the invertibility of polynomials and construction of inverses of permutation polynomials over $\mathbb{F}$, this paper explores a completely new approach using the dual map defined through the Koopman operator. This helps define the linear representation of the non-linear map,, which helps translate the map's non-linear compositions to a linear algebraic framework. The linear representation, defined over the space of functions, naturally defines a notion of linear complexity for non-linear maps, which can be viewed as a measure of computational complexity associated with such maps. The framework of linear representation is then extended to parameter dependent maps over $\mathbb{F}$, and the conditions on parametric invertibility of such maps are established, leading to a construction of a parametric inverse map (under composition). It is shown that the framework can be extended to multivariate maps over $\mathbb{F}^n$, and the conditions are established for invertibility of such maps, and the inverse is constructed using the linear representation. Further, the problem of linear representation of a group generated by a finite set of permutation maps over $\mathbb{F}^n$ under composition is also solved by extending the theory of linear representation of a single map.
Submission history
From: Ramachandran Anantharaman [view email][v1] Mon, 26 Oct 2020 12:34:27 UTC (12 KB)
[v2] Thu, 5 Nov 2020 09:21:02 UTC (17 KB)
[v3] Sun, 20 Feb 2022 14:49:15 UTC (27 KB)
[v4] Sat, 9 Apr 2022 10:52:03 UTC (26 KB)
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