Feedback linearly extended discrete functions

TL;DR

This paper introduces a novel linear extension technique for non-linear functions, ensuring bijectivity, with applications in cryptography, and analyzes the properties and solvability of related equations through linear algebra.

Contribution

It presents a new method to linearly extend non-linear functions to bijections, connecting linear coding theory and cryptography, and characterizes the parameter space for solvability of related equations.

Findings

The extension method guarantees bijectivity for non-linear functions.

Characterization of parameter spaces for solvability of equations.

Relations between composition count and vector space dimensions.

Abstract

We study a new flexible method to extend linearly the graph of a non-linear, and usually not bijective, function so that the resulting extension is a bijection. Our motivation comes from cryptography. Examples from symmetric cryptography are given as how the extension was used implicitly in the construction of some well-known block ciphers. The method heavily relies on ideas brought from linear coding theory and secret sharing. We are interested in the behaviour of the composition of many extensions, and especially the space of parameters that defines a family of equations based on finite differences or linear forms. For any linear extension, we characterize entirely the space of parameters for which such equations are solvable in terms of the space of parameters that render those equations for the corresponding non-linear extended functions solvable. Conditions are derived to assess the solvability of those kind of equations in terms of the number of compositions or iterations. We prove a relation between the number of compositions and the dimensions of vector spaces that appear in our results. The proofs of those properties rely mostly on tools from linear algebra.