Triangular polynomial $\Z$-actions on $\F_p^n$ and a cryptographic application

Details Triangular polynomial $\Z$-actions on $\F_p^n$ and a cryptographic application Triangular polynomial $\Z$-actions on $\F_p^n$ and a cryptographic application

2011-06-28

arxiv.org/abs/1106.5800v1

Mathematics

Algebraic Geometry

21 pages

Scientific paper

This article concerns itself with the triangular permutation group, induced by triangular polynomial maps over $\F_p$. 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 $\Z/p^n\Z$.

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