Stronger bounds on the cost of computing Groebner bases for HFE systems

TL;DR

This paper improves bounds on the computational complexity of solving HFE cryptosystem equations, revealing new insights into the relationship between solving degree and last fall degree, and showing independence from coordinate changes in some cases.

Contribution

It provides tighter bounds on solving degree and last fall degree for HFE systems and explores their connection, including conditions for independence from coordinate transformations.

Findings

New upper bounds for solving degree and last fall degree

Demonstration of independence of solving degree from coordinate changes in certain cases

Enhanced understanding of the algebraic complexity of HFE systems

Abstract

We give upper bounds for the solving degree and the last fall degree of the polynomial system associated to the HFE (Hidden Field Equations) cryptosystem. Our bounds improve the known bounds for this type of systems. We also present new results on the connection between the solving degree and the last fall degree and prove that, in some cases, the solving degree is independent of coordinate changes.