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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 inputpolynomial satisfies a strong symmetry property, namely square balance. In thispaper, we propose an extension of Gao's algorithm that fails only under an evenstronger symmetry property. We also show that our property can be used toimprove the time complexity of best deterministic algorithms on most inputpolynomials. The property also yields a new randomized polynomial timealgorithm.

Factoring Polynomials over Finite Fields using Balance Test