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Interval Partitions and Polynomial Factorization

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Abstract

The fastest algorithms for factoring a univariate polynomial f of degree n over a finite field use a baby-step/giant-step approach. The set {1,…,n} of potential factor degrees is partitioned into intervals. In a first stage, for each interval the product of all irreducible factors with degree in the interval is determined, generalizing the method of Cantor & Zassenhaus. In a second stage, each polynomial corresponding to a multi-factor interval—containing two or more irreducible factors—is completely factored. The goal in this work is to analyze the behavior of this algorithm on uniformly random squarefree input polynomials, for various partitions. To this end, we study several parameters such as the expected number of multi-factor intervals, the expected number of irreducible factors with degrees lying in multi-factor intervals, the number of gcds executed in the factoring process, the expected total degree among the irreducible factors with degrees in multi-factor intervals, and the probability of a polynomial to have no multi-factor interval. We concentrate on partitions with polynomially growing interval sizes, and determine the partition that minimizes the expected number of gcds.

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Authors and Affiliations

  1. B-IT Computer Security, Universität Bonn, 53113, Bonn, Germany

    Joachim von zur Gathen

  2. School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, K1S 5B6, Canada

    Daniel Panario

  3. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

    Bruce Richmond

Authors
  1. Joachim von zur Gathen
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  2. Daniel Panario
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  3. Bruce Richmond
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Correspondence to Daniel Panario.

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Dedicated to the memory of Philippe Flajolet.

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von zur Gathen, J., Panario, D. & Richmond, B. Interval Partitions and Polynomial Factorization. Algorithmica 63, 363–397 (2012). https://doi.org/10.1007/s00453-011-9537-y

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  • DOI: https://doi.org/10.1007/s00453-011-9537-y

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