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Polynomialsover Finitefields COMBINATORIALAND ALGEBRAICASPECTS OF POLYNOMIALSOVER FINITEFIELDS Daniel Nelson Panario Rodriguez A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Cornputer Science University of Toront O @ Copyright by Daniel Nelson Panario Rodriguez 1997 National Library Bibliothèque nationale I*I of Canada du Canada Acquisitions and Acquisitions et Bibliographie Services services bibliographiques 395 Wellington Street 395. me Wellington OttawaON KIAON4 Otbwa ON K1A ON4 Canada Canada The author has granted a non- L'auteur a accordé une licence non exclusive licence allowing the exclusive permettant à la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or sell reproduire, prêter, distribuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la fome de microfiche/film, de reproduction sur papier ou sur format électronique. The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantid extracts fiom it Ni la thèse ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. Abstract Combinatorial and Algebraic Aspects of Polynomials over Finite Fields Daniel Nelson Panario Rodriguez Doctor of P hilosophy Graduate Department of Computer Science University of Toronto 1997 This thesis investigates several algebraic algorithms that deal with univariate polynomi- als over finite fields. Our main focus is the factorization problem, but we also consider polynomial irreducibility tests, and constructions of bot h irreducible polynomials and finite fields. A methodology based on generating functions and their asymptotic analysis allows us to study the behavior of the algorithms in question. This analysis reveals pa- rameters of intrinsic interest for understanding the behavior of the algorithms. Moreover, they provide insight into the structure of polynomials over finite fields in a broader sense. First, we summarize and extend known results on counting random polynomials. We investigate properties such as the average number of irreducible factors whose degrees lie in an interval, the average number of polynomials with no factors or with all factors having degrees in an interval, among others. We briefly state problems where these results give useful information. We also survey the known algorithms for factoring polynomials. Second, we give the first average-case analysis of a complete polynomial factorization algorithm. We fully analyze the classical rnethod for factoring polynomials over finite fields consist ing of t hree stages: squarefree factorizat ion, distinct-degree factorizat ion, and equal-degree factorizat ion, including some variants for each stage. Given a partit ion of the possible degrees of the irreducible factors of a polynornial, we study how they split into subintervals. This naturally relates with recent factorization algorithms. It turns out that more computational work is needed when a subinterval contains the degrees of more than one irreducible factor. We study several properties of such intervals. Finally, the methodology is applied to other problems involving polynomials over fi- nite fields. We give two variations of existing ülgorithms for testing the irreducibility of polynomials. in addition, we provide a construction of very sparse irreducible polynomi- ais. We conclude by improving lower bounds for the Euler fuoction for polynomials and for the density of normal elements. Acknowledgements Rudi Mathon supervised me on the final stages of my PhD. His encouragement and advice were essential to the completion of this thesis. Joachim von zur Gathen supervised me on the first years of my PhD. He was influential at several levels. First, he proposed the topic for this thesis. Our numerous discussions at that time greatly increased my understanding of the area. Later, as a co-author. his meticulous way of working was inspiring. 1 hope some day 1 can state sucb interesting questions with such meticulous answers. 1 also benefited enormously from working with Philippe Flajolet, Shuhong Gao, and Xavier Gourdon. Philippe has been a big source of encouragement for me. The several discussions we have had about analytic combinatorics, random processes, and science in general have been enlightening. My visits to INRIA-Rocquencourt have been inspi- rational, and are greatly acknowledged. Shuhong and myself maintained an algebraic seminar that was one of the points of inflexion in my PhD. Our weekly discussions were a starting point for several joint works. Another point of inflexion in my work was Xavierk visit to Toronto. From him 1 learned about the beauty and the difficulties of analytic combinatorics. Lucia Moura and Alfredo Viola read several drafts of this thesis. Their cornments improved my work in al1 possible ways: they detected errors, pointed out results not clearly stated, suggested modifications, etc. David Neto helped with the English. Many people either commented about some partial results in this thesis or offered advice about my PhD in a broad sense. They are: lan Blake, Derek Corneil, Mark Giesbrecht, Ron Mullin, [an Munro, Bruce Richmond, Victor Shoup, Scott Vanstone, and Zeljko Zilic. My cornmittee members gave me useful comments about the thesis. They are: Allan Borodin, Derek Corneil, Rudi Mathon, Eric Meodelsohn, Ron Mullin, Kumar Murty, and Charles Rackoff. 1 am grateful to the Department of Cornputer Science for the opportunity I had, for the financial support, and for the teaching possibilities, first as a teaching assistant and then as a lecturer. My understanding of the dynamics of the Department was enhanced by this experience, and particularly by several talks with Allan Borodin, Jim Clarke, Derek Corneil, amoog others. 1 count myself as an extremely lucky person. My passage in the University of Toronto just increased t his belief. Not a minor component of this increase is the large number of new friends 1 have made. Finally, I'd Iike to thank Lucia for al1 these years together and for her encouragement. Natan for the new colours he brought to my life, and my family in Uruguay for their long-standing support. Contents 1 Introduction 1 1.1 Overview of the thesis ............................ 1 1.2 Arithmetic in finite fields ........................... 5 1.2.1 Polynornial bases ........................... 5 1.2.2 Normal bases ............................. 7 1.3 Basic methodology .............................. 8 1.3.1 Generating functions ......................... 9 1.3.2 Parameters .............................. 11 1.3.3 Asymptotic aoalysis ......................... 12 1.3.4 The permutation mode1 ....................... lB 1.4 Mathematical results ............................. 15 2 Counting Polynomials over Finite Fields 16 3.1 Motivation and results ............................ 16 2.2 The number of irreducible factors of a polynornial ............. 18 2.2.1 The number of irreducible factors of a polynomial with degrees in an interval ............................... 20 2.2.2 The number of factors of fixed degree in a random polynomial . 23 2.3 The number of polynomials without factors with degrees in a fixed interval 24 2.4 The number of polynomials with al1 factors in a fixed interval ...... 27 3 Factoring Polynomials over Finite Fields 31 3.1 Introduction .................................. 31 3.2 Ageneralfactoringalgorithm ........................ 33 3.2.1 Squarefree factorization ....................... 33 3.2.2 Distinct-degree factorization ..................... 34 3.2.3 Equal-degree factorization ...................... 37 3.3 Algorithms based on linear algebra ..................... 40 3.4 Polynomial factorization algorit hms ..................... 43 3.4.1 Probabilistic algorithms ....................... 43 3.4.2 Deterministic algorithms ....................... 44 3.5 Average-case analysis ............................. 44 4 Average-case Analysis of Polynomial Factorization Algorithms 46 4.1 Introduction .................................. 46 4.1.1 An application of factoring random polynomials .......... 48 4.1.2 Summary of results .......................... 49 4.2 Elirnination of repeated factors (ERF) ................... 50 4.3 Distinct-degree factorization (DDF) ..................... 53 4.3.1 The basic algorithm ......................... 54 4.3.2 Stopping at 7212 ............................ 57 4.3.3 Early abort strategy ......................... 59 4.4 TheoutputconfigurationofDDF ...................... 62 4.5 Equal-degree factorization (EDF) ...................... 67 4.5.1 Irreducible factors of each degree .................. 69 4.5.2 Equal-degree and tries ........................ 69 4.5.3 Cornplete analysis ........................... 73 4.5.4 Equal-degree factorizat ion in characterist ic 2 ............ 75 4.6 Algorithmicvariants ............................. 76 5 Polynomial Factorization and Analysis of Intervals 78 5.1 Motivation and results ............................ 79 5.2 Distinct-degree factorization with growing interval sizes .......... 80 5.3 Analysis of interval parameters for DDF .................. 54 5.31 Probability of a polynomial having no multi-factor intervals ... 84 vii 5.3.2 Number of multi-factor intervals for a polynomial ......... 88 5.3.3 Number of factors in any multi-factor
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