# zbMATH — the first resource for mathematics

Reversed Dickson polynomials over finite fields. (English) Zbl 1228.11174
Summary: Reversed Dickson polynomials over finite fields are obtained from Dickson polynomials $$D_n(x,a)$$ over finite fields by reversing the roles of the indeterminate $$x$$ and the parameter $$a$$. We study reversed Dickson polynomials with emphasis on their permutational properties over finite fields. We show that reversed Dickson permutation polynomials (RDPPs) are closely related to almost perfect nonlinear (APN) functions. We find several families of nontrivial RDPPs over finite fields; some of them arise from known APN functions and others are new. Among RDPPs on $$\mathbb F_q$$ with $$q<200$$, with only one exception, all belong to the RDPP families established in this paper.

##### MSC:
 11T06 Polynomials over finite fields
Full Text:
##### References:
 [1] Budaghyan, L.; Carlet, C.; Leander, G., Two classes of quadratic APN binomials inequivalent to power functions, IEEE trans. inform. theory, 54, 4218-4229, (2008) · Zbl 1177.94135 [2] Chou, W.-S.; Gomez-Calderon, J.; Mullen, G.L., Value sets of dickson polynomials over finite fields, J. number theory, 30, 334-344, (1988) · Zbl 0689.12012 [3] Cohen, S.D.; Matthews, R.W., A class of exceptional polynomials, Trans. amer. math. soc., 345, 897-909, (1994) · Zbl 0812.11070 [4] Coulter, R.S., Explicit evaluation of some Weil sums, Acta arith., 83, 241-251, (1998) · Zbl 0924.11098 [5] 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), 161-183 · JFM 28.0135.03 [6] Dobbertin, H., Almost perfect nonlinear power functions on $$\mathit{GF}(2^n)$$: the welsh case, IEEE trans. inform. theory, 45, 1271-1275, (1999) · Zbl 0957.94021 [7] Dobbertin, H., Almost perfect nonlinear power functions on $$\mathit{GF}(2^n)$$: A new case for n divisible by 5, (), 113-121 · Zbl 1010.94550 [8] Helleseth, T.; Rong, C.; Sandberg, D., New families of almost perfect nonlinear power mappings, IEEE trans. inform. theory, 45, 474-485, (1999) · Zbl 0960.11051 [9] Kang, S.W., Remarks on finite fields, Bull. Korean math. soc., 20, 2, 81-85, (1983) · Zbl 0599.12021 [10] Kang, S.W., Remarks on finite fields II, Bull. Korean math. soc., 22, 1, 37-41, (1985) · Zbl 0586.12015 [11] Kang, S.W., Remarks on finite fields III, Bull. Korean math. soc., 23, 2, 103-111, (1986) · Zbl 0634.12013 [12] Lidl, R.; Mullen, G.L.; Turnwald, G., Dickson polynomials, (1993), Longman Scientific and Technical Essex, United Kingdom · Zbl 0823.11070 [13] Lidl, R.; Niederreiter, H., Finite fields, (1997), Cambridge Univ. Press Cambridge [14] Macdonald, I.G., Symmetric functions and orthogonal polynomials, (1998), Amer. Math. Soc. Providence, RI · Zbl 0887.05053 [15] Nyberg, K., Differentially uniform mappings for cryptography, (), 55-64 · Zbl 0951.94510 [16] I. Schur, Über den Zusammenhang zwischen einen Problem der Zahlentheorie und einen Satz über algebraische Funktionen, in: Sitzungsber. Preuss. Akad. Wiss. Berlin, 1923, pp. 123-134 · JFM 49.0093.02
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.