Computing Frobenius maps and factoring polynomials

A new probabilistic algorithm for factoring univariate polynomials over finite fields is presented whose asymptotic running time improves upon previous results. To factor a polynomial of degree <italic>n</italic> over <bold>F<subscrpt>q</subscrpt></bold>, the algorithm uses <italic>O</italic>((<italic>n</italic><supscrpt>2</supscrpt> + <italic>n</italic> log <italic>q</italic>)•(log <italic>n</italic>)<supscrpt>2</supscrpt> log log <italic>n</italic>) arithmetic operations in <bold>F<subscrpt>q</subscrpt></bold>. The main technical innovation is a new way to compute Frobenius and trace maps in the ring of polynomials modulo the polynomial to be factored.

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