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Cryptography and Communications

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27-05-2021

A power sum formula by Carlitz and its applications to permutation rational functions of finite fields

Author: Xiang-dong Hou

Published in: Cryptography and Communications | Issue 5/2021

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Abstract

A formula discovered by L. Carlitz in 1935 finds an interesting application in permutation rational functions of finite fields. It allows us to determine all rational functions of degree three that permute the projective line \(\mathbb {P}^{1}(\mathbb {F}_{q})\) over \(\mathbb {F}_{q}\), a result previously obtained by Ferraguti and Micheli through a different method. It also allows us to determine all rational functions of degree four that permute \(\mathbb {P}^{1}(\mathbb {F}_{q})\) under a certain condition. (A complete determination of all rational functions of degree four that permute \(\mathbb {P}^{1}(\mathbb {F}_{q})\) without any condition will appear in a separate forthcoming paper.)

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Carlitz, L.: On certain functions connected with polynomials in a Galois field. Duke Math. J. 1, 137–168 (1935)

Dickson, L. E.: The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group. Ann. of Math. 11, 65–120 (1896-1897)

Ding, Z., Zieve, M. E.: Low-degree permutation rational functions over finite fields, arXiv:2010.15657 (2020)

Fan, X.: A classification of permutation polynomials of degree 7 over finite fields, arXiv:1812.02080 (2018)

Fan, X.: Permutation polynomials of degree 8 over finite fields of characteristic 2, arXiv:1903.10309 (2019)

Fan, X.: Permutation polynomials of degree 8 over finite fields of odd characteristic, arXiv:1905.04202 (2019)

Ferraguti, A., Micheli, G.: Full Classification of permutation rational functions and complete rational functions of degree three over finite fields. Designs Codes Cryptogr. 88, 867–886 (2020)

Hou, X.: Lectures on Finite Fields, Graduate Studies in Mathematics 190. American Mathematical Society, Providence (2018)

Hou, X.: Rational functions of degree four that permute the projective line over a finite field. Communications in Algebra, Published online 15 April 2021

10.

Hicks, K., Hou, X., Mullen, G. L.: Sums of reciprocals of polynomials over finite fields. Amer. Math. Monthly 119, 313–317 (2012)

11.

Hou, X., Iezzi, A.: An application of the Hasse-Weil bound to rational functions over finite Fields, Acta Arith., Published online 13 May 2020

12.

Lidl, R., Niederreiter, H.: Finite fields. Cambridge University Press, Cambridge (1997)

13.

Li, J., Chandler, D. B., Xiang, Q.: Permutation polynomials of degree 6 or 7 over finite fields of characteristic 2. Finite Fields Appl. 16, 406–419 (2010)

14.

Shallue, C. J., Wanless, I. M.: Permutation polynomials and orthomorphism polynomials of degree six. Finite Fields Appl. 20, 94–92 (2013)

15.

Thakur, D. S.: Power sums with applications to multizeta and zeta zero distribution for \(\mathbb {F}_{q}[t]\). Finite Fields Appl. 15, 534–552 (2009)

16.

Williams, K. S.: Note on cubics over GF(2ⁿ) and GF(3ⁿ). J. Number Theory 7, 361–365 (1975)

17.

Yuan, J., Ding, C., Wang, H., Pieprzyk, J.: Permutation polynomials of the form (x^p − x + δ)^s + L(x). Finite Fields Appl. 14, 482–493 (2008)

Title: A power sum formula by Carlitz and its applications to permutation rational functions of finite fields
Author: Xiang-dong Hou
Publication date: 27-05-2021
Publisher: Springer US
Published in: Cryptography and Communications / Issue 5/2021
Print ISSN: 1936-2447
Electronic ISSN: 1936-2455
DOI: https://doi.org/10.1007/s12095-021-00495-x

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