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Permutation and complete permutation polynomials. (English) Zbl 1368.11126
Summary: Polynomials of type $$x^{q + 2} + b x$$ over the field $$\mathbb{F}_{q^2}$$ and of type $$x^{q^2 + q + 2} + b x$$ over $$\mathbb{F}_{q^3}$$, where $$q = p^m > 2$$ is a power of a prime $$p$$ are considered. All cases when these polynomials are permutation polynomials are classified. Therefore, all cases when the polynomials $$b^{- 1} x^{q + 2}$$ over $$\mathbb{F}_{q^2}$$ and $$b^{- 1} x^{q^2 + q + 2}$$ over $$\mathbb{F}_{q^3}$$ are the complete permutation polynomials are enumerated.

MSC:
 11T06 Polynomials over finite fields 11T71 Algebraic coding theory; cryptography (number-theoretic aspects) 12Y05 Computational aspects of field theory and polynomials (MSC2010)
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References:
 [1] L.A. Bassalygo, V.A. Zinoviev, On complete permutation polynomials, in: Proceedings of the Fourteenth International Workshop on Algebraic and Combinatorial Coding Theory, September 7-13, 2014, Svetlogorsk (Kaliningrad region), Russia, pp. 57-62. [2] Charpin, P.; Kyureghyan, G. M., Cubic monomial bent functions: a subclass of $$\mathcal{M}^\ast$$, SIAM J. Discrete Math., 22, 2, 650-665, (2008) · Zbl 1171.11062 [3] Chou, W.-S., Permutation polynomials over finite fields and combinatorial applications, (1990), The Pennsylvania State University, PhD Diss. [4] Lidl, R.; Niederreiter, H., Finite fields, Encycl. Math. Appl., vol. 20, (1983), Addison-Wesley Publishing Company London [5] Mullen, G. L., Permutation polynomials over finite fields, (Finite Fields, Coding Theory with Advances in Comm. and Computing, Proc. of Las Vegas Conference, August, 1991, Lect. Notes Pure Appl. Math., vol. 141, (1993), Marcel Dekker, Inc.), 131-151 · Zbl 0808.11069 [6] Muratovic-Ribic, A.; Pasalic, E., A note on complete polynomials over finite fields and their applications in cryptography, Finite Fields Appl., 25, 306-315, (2014) · Zbl 1302.11096 [7] Niederreiter, H.; Robinson, K. H., Complete mappings of finite fields, J. Aust. Math. Soc. A, 33, 197-212, (1982) · Zbl 0495.12018 [8] Sarkar, S.; Bhattacharya, S.; Cesmelioglu, A., On some permutation binomials of the form $$x^{(2^n - 1) / k + 1} + a x$$ over $$\mathbb{F}_{2^n}$$: existence and count, (International Workshop on the Arithmetic of Finite Fields, WAIFI 2012, Lect. Notes Comput. Sci., vol. 7369, (2012), Springer), 236-246 · Zbl 1292.11132 [9] Tu, Z.; Zeng, X.; Hu, L., Several classes of complete permutation polynomials, Finite Fields Appl., 25, 182-193, (2014) · Zbl 1284.05012 [10] Vinogradov, I. M., Basics of number theory, (1972), Nauka Moscow [11] 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
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