# zbMATH — the first resource for mathematics

On the difference between permutation polynomials. (English) Zbl 1374.11089
Summary: The well-known Chowla and Zassenhaus conjecture, proved by Cohen in 1990, states that if $$p>(d^2-3d+4)^2$$, then there is no complete mapping polynomial $$f$$ in $$\mathbb{F}_p[x]$$ of degree $$d\geq 2$$. For arbitrary finite fields $$\mathbb{F}_q$$, a similar non-existence result was obtained recently by Işık, Topuzoğlu and Winterhof in terms of the Carlitz rank of $$f$$.
Cohen, Mullen and Shiue generalized the Chowla-Zassenhaus-Cohen Theorem significantly in 1995, by considering differences of permutation polynomials. More precisely, they showed that if $$f$$ and $$f+g$$ are both permutation polynomials of degree $$d\geq 2$$ over $$\mathbb{F}_p$$, with $$p>(d^2-3d+4)^2$$, then the degree $$k$$ of $$g$$ satisfies $$k\geq 3d/5$$, unless $$g$$ is constant. In this article, assuming $$f$$ and $$f+g$$ are permutation polynomials in $$\mathbb{F}_q[x]$$, we give lower bounds for the Carlitz rank of $$f$$ in terms of $$q$$ and $$k$$. Our results generalize the above mentioned result of Işık et al. We also show for a special class of permutation polynomials $$f$$ of Carlitz rank $$n\geq 1$$ that if $$f+x^k$$ is a permutation over $$\mathbb{F}_q$$, with $$\operatorname{gcd}(k+1,q-1)=1$$, then $$k\geq(q-n)/(n+3)$$.
##### MSC:
 11T06 Polynomials over finite fields 14H05 Algebraic functions and function fields in algebraic geometry
Full Text:
##### References:
 [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] Cohen, S. D., Proof of a conjecture of chowla and Zassenhaus on permutation polynomials, Can. Math. Bull., 33, 230-234, (1990) · Zbl 0722.11060 [5] Cohen, S. D.; Mullen, G. L.; Shiue, P. J.-S., The difference between permutation polynomials over finite fields, Proc. Am. Math. Soc., 123, 2011-2015, (1995) · Zbl 0827.11074 [6] Hirschfeld, J. W.P.; Korchmáros, G.; Torres, F., Algebraic curves over a finite field, (2013), Princeton University Press · Zbl 1200.11042 [7] L. Işık, A. Topuzoğlu, A note on value set of polynomials over finite fields, preprint. [8] Işık, L.; Topuzoğlu, A.; Winterhof, A., Complete mappings and Carlitz rank, Des. Codes Cryptogr., 85, 121-128, (2017) · Zbl 1408.11115 [9] Laywine, C. F.; Mullen, G., Discrete mathematics using Latin squares, Wiley-Interscience Series in Discrete Mathematics and Optimization, A Wiley-Interscience Publication, (1998), John Wiley & Sons, Inc. New York · Zbl 0957.05002 [10] 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 [11] Niederreiter, H.; Robinson, K. H., Complete mappings of finite fields, J. Aust. Math. Soc. A, 33, 197-212, (1982) · Zbl 0495.12018 [12] Schulz, R.-H., On check digit systems using anti-symmetric mappings, (Numbers, Information and Complexity, Bielefeld, 1998, (2000), Kluwer Acad. Publ. Boston, MA), 295-310 · Zbl 0967.94030 [13] Shaheen, R.; Winterhof, A., Permutations of finite fields for check digit systems, Des. Codes Cryptogr., 57, 361-371, (2010) · Zbl 1248.11100 [14] 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) · Zbl 1365.94684 [15] Stichtenoth, H., Algebraic function fields and codes, Graduate Texts in Mathematics, vol. 254, (2009), Springer-Verlag · Zbl 1155.14022 [16] Topuzoğlu, A., Carlitz rank of permutations of finite fields: a survey, J. Symb. Comput., 64, 53-66, (2014) · Zbl 1332.11106 [17] Winterhof, A., Generalizations of complete mappings of finite fields and some applications, J. Symb. Comput., 64, 42-52, (2014) · Zbl 1332.11107
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.