Abstract
A function \(f: \mathbb{F}_{p^n}\rightarrow\mathbb{F}_{p^n}\) is planar, if f(x + a) − f(x) = b has precisely one solution for all \(a,b\in\mathbb{F}_{p^n}\), a ≠ 0. In this paper, we discuss possible extensions of the switching idea developed in [1] to the case of planar functions. We show that some of the known planar functions can be constructed from each other by switching.
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References
Edel, Y., Pott, A.: A new almost perfect nonlinear function which is not quadratic. Advances in Mathematics of Communications 3(1), 59–81 (2009)
Dickson, L.: On commutative linear algebras in which division is always uniquely possible. Transaction of the American Mathematical Society 7, 514–522 (1906)
Knuth, D.: Finite semifields and projective planes. PhD thesis, California Institute of Technology, Pasadena, California (1963)
Hughes, D., Piper, F. (eds.): Projective Planes. Springer, Berlin (1973)
Albert, A.: Finite division algebras and finite planes. In: Combinatorial Analysis: Proceedings of the 10th Symposium in Appled Mathematics Symposia in Appl. Math., vol. 10, pp. 53–70. American Mathematical Society, Providence (1960)
Lidl, R., Niederreiter, H.: Finite fields, 2nd edn. Cambridge University Press, Cambridge (1997)
Dembowski, P., Ostrom, T.: Planes of order n with collineation groups of order n 2. Mathematische Zeitschrift 103(3), 239–258 (1968)
Carlet, C.: Vectorial boolean functions for cryptography. In: Crama, Y., Hammer, P. (eds.) Boolean Methods and Models., vol. 2. Cambridge University Press, Cambridge (in press), http://www-roc.inria.fr/secret/Claude.Carlet/chap-vectorial-fcts.pdf
Carlet, C., Ding, C.: Highly nonlinear mappings. J. Complexity 20(2-3), 205–244 (2004)
Pott, A.: Nonlinear functions in abelian groups and relative difference sets. Discrete Applied Mathematics 138(1-2), 177–193 (2004)
Coulter, R.S., Henderson, M.: Commutative presemifields and semifields. Advances in Mathematics 217(1), 282–304 (2008)
Budaghyan, L., Carlet, C., Pott, A.: New classes of almost bent and almost perfect nonlinear polynomials. IEEE Transactions on Information Theory 52(3), 1141–1152 (2006)
Kyureghyan, G.M., Pott, A.: Some theorems on planar mappings. In: von zur Gathen, J., Imaña, J.L., Koç, Ç.K. (eds.) WAIFI 2008. LNCS, vol. 5130, pp. 117–122. Springer, Heidelberg (2008)
Albert, A.: On nonassociative division algebras. Transaction of the American Mathematical Society 72, 292–309 (1952)
Zha, Z., Kyureghyan, G.M., Wang, X.: Perfect nonlinear binomials and their semifields. Finite Fields and Their Applications 15(2), 125–133 (2009)
Bierbrauer, J.: New commutative semifields and their nuclei. In: Bras-Amorós, M., Høholdt, T. (eds.) AAECC-18. LNCS, vol. 5527, pp. 179–185. Springer, Heidelberg (2009)
Bierbrauer, J.: New semifields, PN and APN functions. Des. Codes Cryptography 54(3), 189–200 (2010)
Budaghyan, L., Helleseth, T.: New perfect nonlinear multinomials over \(\mathbb{F}_{p^{2k}}\) for any odd prime p. In: Golomb, S.W., Parker, M.G., Pott, A., Winterhof, A. (eds.) SETA 2008. LNCS, vol. 5203, pp. 403–414. Springer, Heidelberg (2008)
Bierbrauer, J.: New commutative semifields from projection mappings. Submitted, also presented at the Colloquium on Combinatorics 2009, Magdeburg, Germany (2009)
Lunardon, G., Marino, G., Polverino, O., Trombetti, R.: Symplectic spreads and quadric veroneseans. Manuscript, also presented at Finite Fields 2009, Dublin, Ireland (2009)
Coulter, R.S., Matthews, R.W.: Planar functions and planes of Lenz-Barlotti class II. Des. Codes Cryptography 10(2), 167–184 (1997)
Ding, C., Yuan, J.: A family of skew Hadamard difference sets. J. Comb. Theory Ser. A 113(7), 1526–1535 (2006)
Cohen, S., Ganley, M.: Commutative semifields, two-dimensional over their middle nuclei. Journal of Algebra 75, 373–385 (1982)
Ganley, M.: Central weak nucleus semifields. European Journal of Combinatorics 2, 339–347 (1981)
At, N., Cohen, S.D.: A new tool for assurance of perfect nonlinearity. In: Golomb, S.W., Parker, M.G., Pott, A., Winterhof, A. (eds.) SETA 2008. LNCS, vol. 5203, pp. 415–419. Springer, Heidelberg (2008)
Weng, G.: A new planar function. Private communication (2007)
Coulter, R.S., Henderson, M., Kosick, P.: Planar polynomials for commutative semifields with specified nuclei. Des. Codes Cryptography 44(1-3), 275–286 (2007)
Penttila, T., Williams, B.: Ovoids of parabolic spaces. Geometriae Dedicata 82(1-3), 1–19 (2004)
Nakagawa, N.: On functions of finite fields (2006), http://www.math.is.tohoku.ac.jp/~taya/sendaiNC/2006/report/nakagawa.pdf
Kantor, W.M.: Commutative semifields and symplectic spreads. Journal of Algebra 270(1), 96–114 (2003)
Bosma, W., Cannon, J., Playoust, C.: The MAGMA algebra system I: the user language. J. Symb. Comput. 24(3-4), 235–265 (1997)
Li, C., Qu, L., Ling, S.: On the covering structures of two classes of linear codes from perfect nonlinear functions. IEEE Transactions on Information Theory 55(1), 70–82 (2009)
Pless, V.S., Huffman, W.C., Brualdi, R.A.: Handbook of Coding Theory, Vol. I, II. Elsevier Science Inc., New York (1998)
Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003)
Kasami, T., Lin, S., Peterson, W.W.: New generalizations of the Reed-Muller codes part I: primitive codes. IEEE Transactions on Information Theory 14(2), 189–199 (1968)
Berger, T., Charpin, P.: The permutation group of affine-invariant extended cyclic codes. IEEE Transactions on Information Theory 42(6), 2194–2209 (1996)
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Pott, A., Zhou, Y. (2010). Switching Construction of Planar Functions on Finite Fields. In: Hasan, M.A., Helleseth, T. (eds) Arithmetic of Finite Fields. WAIFI 2010. Lecture Notes in Computer Science, vol 6087. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13797-6_10
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DOI: https://doi.org/10.1007/978-3-642-13797-6_10
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