Designs, Codes and Cryptography

, Volume 85, Issue 1, pp 121–128 | Cite as

Complete mappings and Carlitz rank



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.


Permutation polynomials Complete mappings Carlitz rank Value sets of polynomials 

Mathematics Subject Classification




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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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Sabancı University, MDBFTuzlaTurkey
  2. 2.Johann Radon Institute for Computational and Applied MathematicsAustrian Academy of SciencesLinzAustria

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