Deterministic polynomial factoring under the assumption of the Extended Riemann Hypothesis (ERH)

TL;DR

This paper advances deterministic polynomial factoring over finite fields under ERH by strengthening graph-based methods and leveraging Weisfeiler-Leman algorithms, revealing new structural properties of the associated graphs.

Contribution

It introduces stronger graph relations and regularity conditions, and applies higher-dimensional Weisfeiler-Leman algorithms to improve polynomial factoring methods under ERH.

Findings

Graphs associated with unfactorable polynomials are strongly regular.

The set of adjacency matrices forms an association scheme.

Higher-dimensional Weisfeiler-Leman algorithms can be implicitly computed for these graphs.

Abstract

We consider the problem of deterministically factoring a univariate polynomial over a finite field under the assumption of the Extended Riemann Hypothesis (ERH). This work builds upon the line of approach first explored by Gao in $2001$. The general approach has been to implicitly construct a graph with the roots as vertices and the edges formed by some polynomial time computable relation. The algorithm then fails to factor a polynomial if this associated graph turns out to be \emph{regular}. In the first part of our work we strengthen the edge relation so that the resulting set of graphs we obtain are subgraphs of Gao's graphs, all of which must be \emph{regular}. In the second part of our work we strengthen the regularity condition of these graphs. This is accomplished by finding a parallel between their algorithms and the $1$-dimensional Weisfeiler-Leman algorithm for solving the Graph Isomorphism problem. We observe that the general principle behind their algorithms is to separate the roots by computing the $1$-dimensional Weisfeiler-Leman approximation to the orbits of this graph. This leads us to the natural question of whether this approximation may be improved. We then go on to show how to implicitly compute the $2$-dimensional Weisfeiler-Leman approximation of the orbits of these graphs. The polynomials that our algorithm fails to factor form graphs that are \emph{strongly regular} and their set of adjacency matrices forms a combinatorial structure called an \emph{Association scheme}.