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Planarity of products of two linearized polynomials. (English) Zbl 1271.11112

Summary: Let \(L_1(x)\) and \(L_2(x)\) be linearized polynomials over \(\mathbb F_{q^n}\). We give conditions when the product \(L_1(x)\cdot L_2(x)\) defines a planar mapping on \(\mathbb F_{q^n}\). For a polynomial \(L\) over \(\mathbb F_{q^n}\), let \(M(L)=\{\alpha \in \mathbb F_{q^n}: L(x)+\alpha \cdot x\}\) is bijective on \(\mathbb F_{q^n}\). We show that the planarity of the product \(L_1(x)\cdot L_2(x)\) is linked with the set \(M(L)\) of a suitable linearized polynomial \(L\). We use this relation to describe families of such planar mappings as well as to obtain nonexistence results.

MSC:

11T06 Polynomials over finite fields
12E10 Special polynomials in general fields
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[1] Ball, S., The number of directions determined by a function over a finite field, J. Combin. Theory Ser. A, 104, 341-350 (2003) · Zbl 1045.51004
[2] Blokhuis, A.; Brouwer, A. E.; Szőnyi, T., The number of directions determined by a function \(f\) on a finite field, J. Combin. Theory Ser. A, 70, 349-353 (1995) · Zbl 0823.51013
[3] Blokhuis, A.; Coulter, R. S.; Henderson, M.; OʼKeefe, Ch. M., Permutations amongst the Dembowski-Ostrom polynomials, (Finite Fields and Applications. Finite Fields and Applications, Augsburg, 1999 (2001), Springer), 37-42 · Zbl 1009.11064
[4] Berger, Th. P.; Canteaut, A.; Charpin, P.; Laigle-Chapuy, Y., On almost perfect nonlinear functions over \(F_2^n\), IEEE Trans. Inform. Theory, 52, 9, 4160-4170 (2006) · Zbl 1184.94224
[5] Coulter, R. S.; Henderson, M., Commutative presemifields and semifields, Adv. Math., 217, 282-304 (2008) · Zbl 1194.12007
[6] Coulter, R. S.; Matthews, R. W., Planar functions and planes of Lenz-Barlotti class II, Des. Codes Cryptogr., 10, 167-184 (1997) · Zbl 0872.51007
[7] Dembowski, P.; Ostrom, T., Planes of order \(n\) with collineation groups of order \(n^2\), Math. Z., 103, 239-258 (1968) · Zbl 0163.42402
[8] Ding, C.; Yin, J., Signal sets from functions with optimum nonlinearity, IEEE Trans. Commun., 55, 936-940 (2007)
[9] Ding, C.; Yuan, J., A family of optimal constant-composition codes, IEEE Trans. Inform. Theory, 51, 3668-3671 (2005) · Zbl 1181.94129
[10] Helleseth, T.; Kyureghyan, G.; Ness, G. J.; Pott, A., On a family of perfect nonlinear binomials, (Preenel, B.; Logachev, O. A., Boolean Functions in Cryptology and Information Security (2008), IOS Press), 126-139 · Zbl 1198.94098
[11] Hou, X.-D.; Sze, Ch., On certain diagonal equations over finite fields, Finite Fields Appl., 15, 633-643 (2009) · Zbl 1228.11090
[12] Laigle-Chapuy, Y., A note on a class of quadratic permutations over \(F_{2^n}\), (Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Lecture Notes in Comput. Sci., vol. 4851 (2007), Springer: Springer Berlin), 130-137 · Zbl 1195.11159
[13] Kyureghyan, G., Constructing permutations of finite fields via linear translators, J. Combin. Theory Ser. A, 118, 3, 1052-1061 (2011) · Zbl 1241.11136
[14] Nyberg, K., Differentially uniform mappings for cryptography, (Advances in Cryptology—EUROCRYPT 93. Advances in Cryptology—EUROCRYPT 93, Lecture Notes in Comput. Sci., vol. 765 (1994), Springer-Verlag: Springer-Verlag New York), 134-144 · Zbl 0951.94510
[15] Rédei, L., Lückenhafte Polynome über endlichen Körpern (1970), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 0321.12028
[16] Stichtenoth, H., Algebraic Function Fields and Codes (2009), Springer-Verlag: Springer-Verlag Berlin · Zbl 1155.14022
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