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.