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Complete mappings and Carlitz rank. (English) Zbl 1408.11115
Summary: The well-known Chowla and Zassenhaus conjecture, proven by S. D. Cohen in [Can. Math. Bull. 33, No. 2, 230–243 (1990; Zbl 0722.11060)], states that for any \(d\geq 2\) 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.

MSC:
11T06 Polynomials over finite fields
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[1] Aksoy, E; Çeşmelioğlu, A; Meidl, W; Topuzoğlu, A, On the Carlitz rank of a permutation polynomial, Finite Fields Appl., 15, 428-440, (2009) · Zbl 1232.11124
[2] Carlitz, L, Permutations in a finite field, Proc. Am. Math. Soc., 4, 538, (1953) · Zbl 0052.03704
[3] Chowla, S; Zassenhaus, H, Some conjectures concerning finite fields, Nor. Vidensk. Selsk. Forh. (Trondheim), 41, 34-35, (1968) · Zbl 0186.09203
[4] Çeşmelioğlu, A; Meidl, W; Topuzoğlu, A, On the cycle structure of permutation polynomials, Finite Fields Appl., 14, 593-614, (2008) · Zbl 1153.11057
[5] Çeşmelioğlu, A; Meidl, W; Topuzoğlu, A, Permutations with prescribed properties, J. Comput. Appl. Math., 259B, 536-545, (2014) · Zbl 1314.05003
[6] Cohen, SD, Proof of a conjecture of chowla and Zassenhaus on permutation polynomials, Can. Math. Bull., 33, 230-234, (1990) · Zbl 0722.11060
[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). · Zbl 1332.11103
[8] Guangkui, X; Cao, X, Complete permutation polynomials over finite fields of odd characteristic, Finite Fields Appl., 31, 228-240, (2015) · Zbl 1320.11121
[9] Işık L.: On complete mappings and value sets of polynomials over finite fields. PhD Thesis. Sabancı University (2015). · Zbl 0495.12018
[10] Laywine C.F., Mullen G.: Discrete Mathematics Using Latin Squares. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (1998). · Zbl 0957.05002
[11] Lidl R., Niederreiter H.: Finite Fields, vol. 20, 2nd edn. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (1997). · Zbl 0866.11069
[12] Mann, HB, The construction of orthogonal Latin squares, Ann. Math. Stat., 13, 418-423, (1942) · Zbl 0060.02706
[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) · Zbl 1302.11096
[14] Niederreiter, H; Robinson, KH, Complete mappings of finite fields, J. Aust. Math. Soc. A, 33, 197-212, (1982) · Zbl 0495.12018
[15] Niederreiter H., Winterhof A.: Cyclotomic \(\cal{R}\)-orthomorphisms of finite fields. Discret. Math. 295, 161-171 (2005). · Zbl 1078.11068
[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). · Zbl 1368.11073
[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). · Zbl 0967.94030
[18] Shaheen, R; Winterhof, A, Permutations of finite fields for check digit systems, Des. Codes Cryptogr., 57, 361-371, (2010) · Zbl 1248.11100
[19] Stănică, P; Gangopadhyay, S; Chaturvedi, A; Gangopadhyay, AK; Maitra, S, Investigations on bent and negabent functions via the nega-Hadamard transform, IEEE Trans. Inf. Theory, 58, 4064-4072, (2012) · Zbl 1365.94684
[20] Topuzoğlu, A, Carlitz rank of permutations of finite fields: a survey, J. Symb. Comput., 64, 53-66, (2014) · Zbl 1332.11106
[21] Tu, Z; Zeng, X; Hu, L, Several classes of complete permutation polynomials, Finite Fields Appl., 25, 182-193, (2014) · Zbl 1284.05012
[22] Winterhof, A, Generalizations of complete mappings of finite fields and some applications, J. Symb. Comput., 64, 42-52, (2014) · Zbl 1332.11107
[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) · Zbl 1314.11073
[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) · Zbl 1320.11123
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