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    Please use this identifier to cite or link to this item: https://nccur.lib.nccu.edu.tw/handle/140.119/84947


    Title: 有限体上的排列多項式之判斷準則的各種證明方法
    Various Proofs of PP`s Criteria over Finite Fields
    Authors: 解創智
    Hsieh, Chuang-Chih
    Contributors: 陳永秋
    解創智
    Hsieh, Chuang-Chih
    Keywords: 排列多項式
    有限体
    PP
    Finite fields
    Permutation polynomials
    Hermite-Dickson`s Criterion
    Wan-Turnwald`s Criterion
    Date: 2001
    Issue Date: 2016-04-15 16:02:49 (UTC+8)
    Abstract: In this paper, we provide a complete survey of the important criteria for permutation polynomials over finite fields, including the classical Hermite-Dickson`s Criterion and the recent Wan-Turnwald`s Criterion. We review the various proofs of these criteria and give new proofs of them.
    封面頁
    證明書
    致謝詞
    論文摘要
    目錄
    1 Introduction
    2 Hermite-Dickson`s Criterion for Permutation Polynomials
    2.1 Dickson`s Proof of Hermite-Dickson`s Criterion
    2.2 Carlitz and Lutz`s Proof of Hermite-Dickson`s Criterion
    2.3 Lidl and Niederreiter`s Proof of Hermite-Dickson`s Criterion
    2.4 Wan and Turnwald`s Proof of Hermite-Dickson`s Criterion
    2.5 A New Proof of Hermite-Dickson`s Criterion
    3 Wan-Turnwald`s Criterion for Permutation Polynomials
    3.1 Wan`s Proof of Wan-Turnwald`s Criterion
    3.2 Turnwald`s Proof of Wan-Turnwald`s Criterion
    3.3 Generalization for Turnwald`s Proof by Aitken
    3.4 A Proof of Wan-Turnwald`s Criterion a la Hermite-Dickson
    3.5 An Application to Prove Cohen`s Theorem
    4 Equivalent Conditions for Permutation Polynomials
    4.1 Fundamental Relations among the Invariants
    4.2 New Proofs for Some Inequalities about the Invariants
    4.3 Turnwald`s Equivalent Conditions for Permutation Polynomials
    5 Further Directions of Research
    References
    Reference: [1] Wayne Aitken, On value sets of polynomials over a finite field, Preprint (1997).
    [2] L. Carlitz, D. J. Lewis, W. H. Mills and E. G. Straus, Polynomials over finite fields with minimal value set, Mathematika 8(1961), 121-130.
    [3] L. Carlitz and J. A. Lutz, A characterization of permutation polynomials over a finite field, American Mathematical Monthly 85(1978), 746-748.
    [4] S. D. Cohen, The distribution of polynomials over finite fields, Acta Arithmetica 17(1970), 255-271.
    [5] H. Davenport and D. J. Lewis, Note on congruences (I), Quarterly Journal of Mathematics 14(1963), 51-60.
    [6] L. E. Dickson, The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group, Annals of Mathematics 11(1897), 65-120, 161-183.
    [7] J. Gomez-Calderon, A note on polynomials with minimal value set over finite field, Mathematika 35(1988), 144-148.
    [8] J. Gomez-Calderon, and D. J. Madden, Polynomials with small value set over finite fields, Journal of Number Theory 28(1988), 167-188.
    [9] J. von zur Gathen, Values of polynomials over finite fields, Bulletin of Australian Mathematical Society 43(1991), 141-146.
    [10] D. R. Hayes, A geometric approach to permutation polynomials over a finite field, Duke Mathematical Journal 34(1967), 293-305.
    [11] C. Hermite, Sur les fonctions de sept letters, Comptes Rendus de L` Academie des Sciences, Paris, 57(1863), 750-757; Oeuvres, vol 2, Gauthier-Villars, Paris, 1908, 280-288.
    [12] N. Koblitz, p-adic analysis: a short course on recent work, Cambridge University Press, 1980.
    [13] S. Lang, Linear Algebra, third edition by Springer-Verlag New York Inc. 1987.
    [14] J. Levine and J. V. Brawley, Some cryptographic applications of permutation polynomials, Cryptologia 1(1977), 76-92.
    [15] R. Lidl and G. L. Mullen, When does a polynomial over a finite field permute the elements of the field, American Mathematical Monthly 95(1988), 243-246.
    [16] R. Lidl and G. L. Mullen, When does a polynomial over a finite field permute the elements of the field? 2, American Mathematical Monthly 100(1993), 71-74.
    [17] R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of mathematics and its applications, Vol 20, first published by Addison-Wesley Publishing Inc. 1983, and second edition published by Cambridge University Press 1997.
    [18] C. R. MacCluer, On a conjecture of Davenport and Lewis concerning exceptional polynomials, Acta Arithmetica 12(1967), 289-299.
    [19] W. H. Mills, Polynomials with minimal value sets, Pacific Journal of Mathematics 14(1964), 225-241.
    [20] G. L. Mullen, Permutation polynomials over finite fields, Proceedings of international conference on finite fields, coding theory and advences in communications and computing, lecture notes in Pure and Applied Mathematika, vol 141, Marcel Dekker, New York, 1992, 131-151.
    [21] G. L. Mullen, Permutation polynomials: A matrix analogue of Schur`s conjecture and a survey of recent result, Finite Fields and Their Applications 1, 1995, 242-258.
    [22] G. Turnwald, A new criterion for permutation polynomials, Finite Fields and Their Applications 1, 1995, 64-82.
    [23] D. Wan, On a conjecture of Carlitz, Journal of Australian Mathematical Society, Series A 43(1987), 375-384.
    [24] D. Wan, A p-adic lifting lemma and its applications to permutation polynomials, Proceedings of the international conference on finite fields, coding theory and advances in communication and computing, lecture notes in Pure and Applied Mathematika, vol 141, Marcel Dekker, New York, 1993, 209-216.
    [25] D. Wan, A generalization of Carlitz conjecture, Proceedings of the international conference on finite fields, coding theory and advances in communication and computing, lecture notes in Pure and Applied Mathematika, vol 141, Marcel Dekker, New York, 1993, 431-432.
    [26] D. Wan, G. L. Mullen and P. J.-S. Shiue, The number of permutation polynomials of the form f(x)+cx over a finite field, Proceedings Edinburgh Mathematical Society, 38(1995), 133-149.
    [27] D. Wan, P. J.-S. Shiue, and C.-S. Chen, Value sets of polynomial over finite fields, Proceedings of the American Mathematical Society 119(1993), 711-717.
    [28] K. S. Williams, On exceptional polynomials, Canadian Mathematical Bulletin 11(1968), 279-282.
    Description: 碩士
    國立政治大學
    應用數學系
    Source URI: http://thesis.lib.nccu.edu.tw/record/#A2002001138
    Data Type: thesis
    Appears in Collections:[Department of Mathematical Sciences] Theses

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