Abstract
We introduce two infinite families of perfect nonlinear Dembowski-Ostrom multinomials over \(\textbf{F}_{p^{2k}}\) where p is any odd prime. We prove that in general these functions are CCZ-inequivalent to previously known PN mappings. One of these families has been constructed by extension of a known family of APN functions over \(\textbf{F}_{2^{2k}}\). This shows that known classes of APN functions over fields of even characteristic can serve as a source for further constructions of PN mappings over fields of odd characteristics.
Besides, we supply results indicating that these PN functions define new commutative semifields. After the works of Dickson (1906) and Albert (1952), these are the firstly found infinite families of commutative semifields which are defined for all odd primes p.
This work was supported by the Norwegian Research Council.
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References
Albert, A.A.: On nonassociative division algebras. Trans. Amer. Math. Soc. 72, 296–309 (1952)
Albert, A.A.: Generalized twisted fields. Pacific J. Math. 11, 1–8 (1961)
Biham, E., Shamir, A.: Differential Cryptanalysis of DES-like Cryptosystems. Journal of Cryptology 4(1), 3–72 (1991)
Bracken, C., Byrne, E., Markin, N., McGuire, G.: New families of quadratic almost perfect nonlinear trinomials and multinomials. Finite Fields and Applications (to appear, 2008)
Budaghyan, L., Carlet, C., Pott, A.: New Classes of Almost Bent and Almost Perfect Nonlinear Functions. IEEE Trans. Inform. Theory 52(3), 1141–1152 (2006)
Carlet, C., Charpin, P., Zinoviev, V.: Codes, bent functions and permutations suitable for DES-like cryptosystems. Des., Codes Cryptogr. 15(2), 125–156 (1998)
Cohen, S.D., Ganley, M.J.: Commutative semifields, two-dimensional over their middle nuclei. J. Algebra 75, 373–385 (1982)
Coulter, R.S., Matthews, R.W.: Planar functions and planes of Lenz-Barlotti class II. Des., Codes Cryptogr. 10, 167–184 (1997)
Coulter, R.S., Henderson, M.: Commutative presemifields and semifields. Advances in Math. 217, 282–304 (2008)
Coulter, R.S., Henderson, M., Kosick, P.: Planar polynomials for commutative semifields with specified nuclei. Des. Codes Cryptogr. 44, 275–286 (2007)
Dembowski, P., Ostrom, T.: Planes of order n with collineation groups of order n 2. Math. Z. 103, 239–258 (1968)
Dickson, L.E.: On commutative linear algebras in which division is always uniquely possible. Trans. Amer. Math. Soc. 7, 514–522 (1906)
Dickson, L.E.: Linear algebras with associativity not assumed. Duke Math. J. 1, 113–125 (1935)
Ding, C., Yuan, J.: A new family of skew Paley-Hadamard difference sets. J. Comb. Theory Ser. A. 133, 1526–1535 (2006)
Ganley, M.J.: Central weak nucleus semifields. European J. Combin. 2, 339–347 (1981)
Helleseth, T., Rong, C., Sandberg, D.: New families of almost perfect nonlinear power mappings. IEEE Trans. Inf. Theory 45, 475–485 (1999)
Helleseth, T., Sandberg, D.: Some power mappings with low differential uniformity. Applic. Alg. Eng., Commun. Comput. 8, 363–370 (1997)
Minami, K., Nakagawa, N.: On planar functions of elementary abelian p-group type (submitted)
Nakagawa, N.: On functions of finite fields, http://www.math.is.tohoku.ac.jp/~taya/sendaiNC/2006/report/nakagawa.pdf
Ness, G.J.: Correlation of sequences of different lengths and related topics. PhD dissertation. University of Bergen, Norway (2007)
Nyberg, K.: Differentially uniform mappings for cryptography. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 55–64. Springer, Heidelberg (1994)
Penttila, T., Williams, B.: Ovoids of parabolic spaces. Geom. Dedicata 82, 1–19 (2000)
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Budaghyan, L., Helleseth, T. (2008). New Perfect Nonlinear Multinomials over F \(_{p^{2k}}\) for Any Odd Prime p . In: Golomb, S.W., Parker, M.G., Pott, A., Winterhof, A. (eds) Sequences and Their Applications - SETA 2008. SETA 2008. Lecture Notes in Computer Science, vol 5203. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85912-3_35
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DOI: https://doi.org/10.1007/978-3-540-85912-3_35
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