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The compositional inverse of a class of bilinear permutation polynomials over finite fields of characteristic 2. (English) Zbl 1286.05005
Summary: Laigle-Chapuy constructed a class of permutation polynomials over a finite field of characteristic 2, which includes several other known classes. In this paper, we determine the compositional inverses of all Laigle-Chapuy’s permutation polynomials. Our method is based on a direct sum decomposition of the finite field.

##### MSC:
 11T06 Polynomials over finite fields
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##### References:
 [1] Akbary, A.; Alaric, S.; Wang, Q., On some classes of permutation polynomials, Int. J. Number Theory, 4, 121-133, (2008) · Zbl 1218.11108 [2] Akbary, A.; Ghioca, D.; Wang, Q., On constructing permutations of finite fields, Finite Fields Appl., 17, 51-67, (2011) · Zbl 1281.11102 [3] Akbary, A.; Wang, Q., On polynomials of the form $$x^r f(x^{(q - 1) / \ell})$$, Int. J. Math. Math. Sci., (2007), Art. ID 23408, 7 pp [4] Blokhuis, A.; Coulter, R. S.; Henderson, M.; O’Keefe, C. M., Permutations amongst the Dembowski-Ostrom polynomials, (Proceedings of the Fifth International Conference on Finite Fields and Applications, Augsburg, 1999, (2001), Springer), 37-42 · Zbl 1009.11064 [5] Charpin, P.; Kyureghyan, G., When does $$G(x) + \gamma \operatorname{Tr}(H(x))$$ permutate $$\mathbb{F}_{p^n}$$?, Finite Fields Appl., 15, 615-632, (2009) · Zbl 1229.11153 [6] Coulter, R. S.; Henderson, M., The compositional inverse of a class of permutation polynomials over a finite field, Bull. Aust. Math. Soc., 65, 521-526, (2002) · Zbl 1023.11061 [7] Dempwolff, U.; Müller, P., Permutation polynomials and translation planes of even order, Adv. Geom., 13, 293-313, (2013) · Zbl 1278.51003 [8] Hou, X.-D., Two classes of permutation polynomials over finite fields, J. Combin. Theory Ser. A, 118, 448-454, (2011) · Zbl 1230.11146 [9] Laigle-Chapuy, Y., A note on a class of quadratic permutation polynomials over $$\mathbb{F}_{2^n}$$, (Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Lecture Notes in Comput. Sci., vol. 4851, (2007), Springer), 130-137 · Zbl 1195.11159 [10] Lidl, R.; Mullen, G. L., When does a polynomial over a finite field permutate the elements of the field?, Amer. Math. Monthly, 95, 243-246, (1988) · Zbl 0653.12010 [11] Lidl, R.; Mullen, G. L., When does a polynomial over a finite field permutate the elements of the field? II, Amer. Math. Monthly, 100, 71-74, (1993) · Zbl 0777.11054 [12] Lidl, R.; Niederreiter, H., Finite fields, Encyclopedia Math. Appl., vol. 20, (1997), Cambridge University Press Cambridge [13] Muratović-Ribić, A., A note on the coefficients of inverse polynomials, Finite Fields Appl., 13, 977-980, (2007) · Zbl 1167.11044 [14] Wan, D.; Lidl, R., Permutation polynomials of the form $$x^r f(x^{(q - 1) / d})$$ and their group structure, Monatsh. Math., 112, 149-163, (1991) · Zbl 0737.11040 [15] Wang, Q., Cyclotomic mapping permutation polynomials over finite fields, (Sequences, Subsequences, and Consequences, International Workshop, SSC 2007, Lecture Notes in Comput. Sci., vol. 4893, (2007), Springer-Verlag Berlin), 119-128 · Zbl 1154.11342 [16] Wang, Q., On inverse permutation polynomials, Finite Fields Appl., 15, 207-213, (2009) · Zbl 1183.11075 [17] Wu, B.; Liu, Z., Linearized polynomials over finite fields revisited, Finite Fields Appl., 22, 79-100, (2013) · Zbl 1345.11084 [18] Wu, B.; Liu, Z., The compositional inverse of a class of linearized permutation polynomials over $$\mathbb{F}_{2^n}$$, n odd, preprint · Zbl 1309.11085 [19] Zha, Z.; Hu, L., Two classes of permutation polynomials over finite fields, Finite Fields Appl., 18, 781-790, (2012) · Zbl 1288.11111 [20] Zieve, M., Some families of permutation polynomials over finite fields, Int. J. Number Theory, 4, 851-857, (2008) · Zbl 1204.11180
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