On Linear Representation, Complexity and Inversion of maps over finite fields

10/26/2020
βˆ™
by   Ramachandran Anantharaman, et al.
βˆ™
0
βˆ™

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 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 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 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 𝔽^n under composition is also solved by extending the theory of linear representation of a single map.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
βˆ™ 07/08/2019

Further results on some classes of permutation polynomials over finite fields

Let F_q denote the finite fields with q elements. The permutation behavi...
research
βˆ™ 12/01/2020

Image sets of perfectly nonlinear maps

We present a lower bound on the image size of a d-uniform map, dβ‰₯ 1, of ...
research
βˆ™ 06/01/2021

Extended retraction maps: a seed of geometric integrators

The classical notion of retraction map used to approximate geodesics is ...
research
βˆ™ 11/22/2019

Parametric Models Analysed with Linear Maps

Parametric entities appear in many contexts, be it in optimisation, cont...
research
βˆ™ 01/17/2023

Weighted and Branching Bisimilarities from Generalized Open Maps

In the open map approach to bisimilarity, the paths and their runs in a ...
research
βˆ™ 12/25/2022

A general construction of regular complete permutation polynomials

Let rβ‰₯ 3 be a positive integer and 𝔽_q the finite field with q elements....
research
βˆ™ 10/08/2018

Trilinear maps for cryptography II

We continue to study the construction of cryptographic trilinear maps in...

Please sign up or login with your details

Forgot password? Click here to reset