# On properties of translation groups in the affine general linear group with applications to cryptography

## TL;DR

This paper studies the structure of translation groups within the affine general linear group, providing new representations, counting formulas, and classifications to aid cryptanalysis in cryptography.

## Contribution

It introduces a new representation of elementary abelian regular subgroups and classifies their conjugacy classes, enhancing cryptanalysts' ability to analyze block ciphers.

## Key findings

- Developed a convenient representation for subgroup elements
- Derived combinatorial counting formulas for subgroup properties
- Classified conjugacy classes of these subgroups

## Abstract

The affine general linear group acting on a vector space over a prime field is a well-understood mathematical object. Its elementary abelian regular subgroups have recently drawn attention in applied mathematics thanks to their use in cryptography as a way to hide or detect weaknesses inside block ciphers. This paper is focused on building a convenient representation of their elements which suits better the purposes of the cryptanalyst. Several combinatorial counting formulas and a classification of their conjugacy classes are given as well.