This paper gives an algorithm to factor a polynomial f (in one variable) over rings like Z /rZ for r ∈ Z or F q [y]/rF q [y] for r ∈ F q [y]. The Chinese Remainder Theorem reduces our problem to the case where r is a prime power. Then factorization is not unique, but if r does not divide the discriminant of f , our (probabilistic) algorithm produces a description of all (possibly exponentially many) factorizations into irreducible factors in polynomial time. If r divides the discriminant, we only know how to factor by exhaustive search, in exponential time.
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