Giesbrecht's algorithm, the HFE cryptosystem and Ore's $p^s$-polynomials

TL;DR

This paper discusses Giesbrecht's algorithm for skew-polynomial factorization, its historical context, its relation to $p^s$-polynomials over finite fields, and implications for the security of the HFE cryptosystem.

Contribution

It provides a recent implementation of Giesbrecht's algorithm, explores its theoretical equivalence with $p^s$-polynomial decomposition, and analyzes security implications for HFE.

Findings

Implementation of Giesbrecht's algorithm demonstrated

Equivalence between skew-polynomial factorization and $p^s$-polynomial decomposition established

Observations on HFE cryptosystem security based on $p$-polynomials

Abstract

We report on a recent implementation of Giesbrecht's algorithm for factoring polynomials in a skew-polynomial ring. We also discuss the equivalence between factoring polynomials in a skew-polynomial ring and decomposing $p^s$-polynomials over a finite field, and how Giesbrecht's algorithm is outlined in some detail by Ore in the 1930's. We end with some observations on the security of the Hidden Field Equation (HFE) cryptosystem, where $p$-polynomials play a central role.