## Cryptology ePrint Archive: Report 2011/383

**A representation of the $p$-sylow subgroup of $\perm(\F_p^n)$ and a cryptographic application**

*Stefan Maubach*

**Abstract: **
This article concerns itself with the triangular permutation group, induced by triangular polynomial maps over $\F_p$, which is a $p$-sylow subgroup of $\perm(\F_p^n)$.
The aim of this article is twofold: on the one hand, we give an alternative to $\F_p$-actions on $\F_p^n$, namely $\Z$-actions on $\F_p^n$ and how to describe them as what we call ``$\Z$-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 $z\lp z+1$ on $\Z/p^n\Z$.

**Category / Keywords: **cryptographic protocols / Diffie-Hellmann session key exchange

**Date: **received 14 Jul 2011

**Contact author: **stefan maubach at gmail com

**Available format(s): **PDF | BibTeX Citation

**Note: **21 pages

**Version: **20110715:112914 (All versions of this report)

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