Factoring Polynomials over Finite Fields using Balance Test

We study the problem of factoring univariate polynomials over finite fields. Under the assumption of the Extended Riemann Hypothesis (ERH), (Gao, 2001) designed a polynomial time algorithm that fails to factor only if the input polynomial satisfies a strong symmetry property, namely square balance. In this paper, we propose an extension of Gao’s algorithm that fails only under an even stronger symmetry property. We also show that our property can be used to improve the time complexity of best deterministic algorithms on most input polynomials. The property also yields a new randomized polynomial time algorithm.

💡 Research Summary

The paper addresses the classic problem of factoring a univariate polynomial f over a finite field Fₚ, assuming the Extended Riemann Hypothesis (ERH). Gao’s 2001 deterministic algorithm runs in polynomial time but fails precisely when f exhibits a strong symmetry called “square‑balanced”. The author introduces a stricter symmetry notion—“k‑cross‑balanced”—and builds an extended algorithm that only fails under this more restrictive condition.

The construction starts with a monic, square‑free, completely splitting polynomial f(x)=∏_{i=1}^n (x−ξ_i) in Fₚ