Complete mappings and Carlitz rank

Abstract

The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990, states that for any \(d\ge 2\) and any prime \(p>(d^2-3d+4)^2\) there is no complete mapping polynomial in \(\mathbb {F}_p[x]\) of degree d. For arbitrary finite fields \(\mathbb {F}_q\), we give a similar result in terms of the Carlitz rank of a permutation polynomial rather than its degree. We prove that if \(n<\lfloor q/2\rfloor \), then there is no complete mapping in \(\mathbb {F}_q[x]\) of Carlitz rank n of small linearity. We also determine how far permutation polynomials f of Carlitz rank \(n<\lfloor q/2\rfloor \) are from being complete, by studying value sets of \(f+x.\) We provide examples of complete mappings if \(n=\lfloor q/2\rfloor \), which shows that the above bound cannot be improved in general.

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References

  1. 1.

    Aksoy E., Çeşmelioğlu A., Meidl W., Topuzoğlu A.: On the Carlitz rank of a permutation polynomial. Finite Fields Appl. 15, 428–440 (2009).

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Carlitz L.: Permutations in a finite field. Proc. Am. Math. Soc. 4, 538 (1953).

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Chowla S., Zassenhaus H.: Some conjectures concerning finite fields. Nor. Vidensk. Selsk. Forh. (Trondheim) 41, 34–35 (1968).

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Çeşmelioğlu A., Meidl W., Topuzoğlu A.: On the cycle structure of permutation polynomials. Finite Fields Appl. 14, 593–614 (2008).

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Çeşmelioğlu A., Meidl W., Topuzoğlu A.: Permutations with prescribed properties. J. Comput. Appl. Math. 259B, 536–545 (2014).

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Cohen S.D.: Proof of a conjecture of Chowla and Zassenhaus on permutation polynomials. Can. Math. Bull. 33, 230–234 (1990).

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Gomez-Perez D., Ostafe A., Topuzoğlu A.: On the Carlitz rank of permutations of \({\mathbb{F}}_q\) and pseudorandom sequences. J. Complex. 30, 279–289 (2014).

  8. 8.

    Guangkui X., Cao X.: Complete permutation polynomials over finite fields of odd characteristic. Finite Fields Appl. 31, 228–240 (2015).

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Işık L.: On complete mappings and value sets of polynomials over finite fields. PhD Thesis. Sabancı University (2015).

  10. 10.

    Laywine C.F., Mullen G.: Discrete Mathematics Using Latin Squares. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (1998).

  11. 11.

    Lidl R., Niederreiter H.: Finite Fields, vol. 20, 2nd edn. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (1997).

  12. 12.

    Mann H.B.: The construction of orthogonal Latin squares. Ann. Math. Stat. 13, 418–423 (1942).

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Muratovic-Ribic A., Pasalic E.: A note on complete mapping polynomials over finite fields and their applications in cryptography. Finite Fields Appl. 25, 306–315 (2014).

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Niederreiter H., Robinson K.H.: Complete mappings of finite fields. J. Aust. Math. Soc. A 33, 197–212 (1982).

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Niederreiter H., Winterhof A.: Cyclotomic \(\cal{R}\)-orthomorphisms of finite fields. Discret. Math. 295, 161–171 (2005).

  16. 16.

    Pausinger F., Topuzoğlu A.: Permutations of finite fields and uniform distribution modulo 1. In: Niederrreiter H., Ostafe A., Panario D., Winterhof A. (eds.) Algebraic Curves and Finite Fields, vol. 16, pp. 145–160. Radon Series on Applied and Computational Mathematics. De Gruyter, Berlin (2014).

  17. 17.

    Schulz R.-H.: On check digit systems using anti-symmetric mappings. In: Numbers Information and Complexity (Bielefeld, 1998), pp. 295–310. Kluwer Academic, Boston, MA (2000).

  18. 18.

    Shaheen R., Winterhof A.: Permutations of finite fields for check digit systems. Des. Codes Cryptogr. 57, 361–371 (2010).

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Stănică P., Gangopadhyay S., Chaturvedi A., Gangopadhyay A.K., Maitra S.: Investigations on bent and negabent functions via the nega-Hadamard transform. IEEE Trans. Inf. Theory 58, 4064–4072 (2012).

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Topuzoğlu A.: Carlitz rank of permutations of finite fields: a survey. J. Symb. Comput. 64, 53–66 (2014).

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Tu Z., Zeng X., Hu L.: Several classes of complete permutation polynomials. Finite Fields Appl. 25, 182–193 (2014).

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Winterhof A.: Generalizations of complete mappings of finite fields and some applications. J. Symb. Comput. 64, 42–52 (2014).

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Wu G., Li N., Helleseth T., Zhang Y.: Some classes of monomial complete permutation polynomials over finite fields of characteristic two. Finite Fields Appl. 28, 148–165 (2014).

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Zha Z., Hu L., Cao X.: Constructing permutations and complete permutations over finite fields via subfield-valued polynomials. Finite Fields Appl. 31, 162–177 (2015).

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgments

L.I. and A.T. were supported by TUBITAK Project Number 114F432. A.W. is partially supported by the Austrian Science Fund FWF Project F5511-N26 which is part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”.

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Correspondence to Leyla Işık.

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Communicated by C. Mitchell.

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Işık, L., Topuzoğlu, A. & Winterhof, A. Complete mappings and Carlitz rank. Des. Codes Cryptogr. 85, 121–128 (2017). https://doi.org/10.1007/s10623-016-0293-5

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Keywords

  • Permutation polynomials
  • Complete mappings
  • Carlitz rank
  • Value sets of polynomials

Mathematics Subject Classification

  • 11T06