Paper 2011/383

A representation of the $p$-sylow subgroup of $\text{perm}({\text{F}}_p^n)$ and a cryptographic application

Stefan Maubach

Abstract

This article concerns itself with the triangular permutation group, induced by triangular polynomial maps over ${\text{F}}_p$, which is a $p$-sylow subgroup of $\text{perm}({\text{F}}_p^n)$. The aim of this article is twofold: on the one hand, we give an alternative to ${\text{F}}_p$-actions on ${\text{F}}_p^n$, namely $\text{Z}$-actions on ${\text{F}}_p^n$ and how to describe them as what we call ``$\text{Z}$-flows''. On the other hand, we describe how the triangular permutation group can be used in applications, in particular we give a cryptographic application for session-key generation. The described system has a certain degree of information theoretic security. We compute its efficiency and storage size. To make this work, we give explicit criteria for a triangular permutation map to have only one orbit, which we call ``maximal orbit maps''. We describe the conjugacy classes of maximal orbit maps, and show how one can conjugate them even further to the map $$ on $$.

Note: 21 pages