Triangular polynomial maps in characteristic $p$

TL;DR

This paper explores the structure and iteration of triangular polynomial maps over finite fields, introducing $ ext{Z}$-flows and providing efficient evaluation methods with cryptographic applications.

Contribution

It generalizes $ ext{F}_p$-actions to $ ext{Z}$-actions, establishes their relation to polynomial automorphisms, and develops methods for fast iteration of triangular permutations.

Findings

Established equivalence between LFPEs and $ ext{Z}$-flows.

Provided a conjugation method to simplify iterations of triangular permutations.

Applied results to cryptographic fast-forward functions.

Abstract

This paper came to existence out of the desire to understand iterations of strictly triangular polynomial maps over finite fields. This resulted in two connected results: First, we give a generalization of $\F_p$-actions on $\F_p^n$ and their description by "locally iterative higher derivations", namely $\Z$-actions on $\F_p^n$ and show how to describe them by what we call "$\Z$-flows". We prove equivalence between locally finite polynomial automorphisms (LFPEs) over finite fields and $\Z$-flows over finite fields. We elaborate on $\Z$-flows of strictly triangular polynomial maps. Second, we describe how one can efficiently evaluate iterations of triangular polynomial permutations on $\F_p^n$ which have only one orbit. We do this by determining the equivalence classes in the triangular permutation group of such elements. We show how to conjugate them all to the map $z\lp z+1$ on the ring $\Z/p^n\Z$ (which is identified with $\F_{p^n}$), making iterations trivial. An application in the form of fast-forward functions from cryptography is given.