Deterministic versus probabilistic factorization of integral polynomials
We have shown that the Cantor-Zassenhaus probabilistic step to find a factor over GF(p) of a polynomial being the product of equal degree factors takes O(n3L2(p)Lβ(p)2) units of time. This is, using classical arithmetic, the cost in Rabin's algorithm to find a root of an irreducible factor of degree d in the extension field GF(pd). The constants involved in Rabin's algorithm seem to be higher than in the simple Cantor-Zassenhaus test, which is the most promising candidate of a probabilistic algorithmic to be compared with the deterministic Berlekamp-Hensel algorithm. A careful analysis, backed up by measurements using current technology in computer algebra, demonstrates that the time has not yet come to beat Berlekamp-Hensel. This is mainly due to the cost of exponentiation both in Berlekamp's Q-matrix and in the probabilistic test of Cantor-Zassenhaus which makes both algorithms for large primes intractable. In contradistinction : the restriction of the Berlekamp-Hensel algorithm to small primes is computationally its greatest strength.
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