Linear recurrent cryptography: golden-like cryptography for higher order linear recurrences

TL;DR

This paper introduces a matrix cryptography scheme based on linear recurrent sequences of any order, enhancing security against brute force and plaintext attacks, and generalizing error detection for golden cryptography.

Contribution

It generalizes error detection and correction algorithms for linear recurrent cryptography beyond special cases, utilizing properties of the golden Q-matrix and Pisot polynomials.

Findings

Cryptography based on linear recurrences can be secured against brute force attacks.

Error detection algorithms are extended to higher-order recurrences.

Algorithms for generating suitable recurrences are proposed.

Abstract

We develop matrix cryptography based on linear recurrent sequences of any order that allows securing encryption against brute force and chosen plaintext attacks. In particular, we solve the problem of generalizing error detection and correction algorithms of golden cryptography previously known only for recurrences of a special form. They are based on proving the checking relations (inequalities satisfied by the ciphertext) under the condition that the analog of the golden $Q$-matrix has the strong Perron-Frobenius property. These algorithms are proved to be especially efficient when the characteristic polynomial of the recurrence is a Pisot polynomial. Finally, we outline algorithms for generating recurrences that satisfy our conditions.