Abstract
Bent functions are maximally nonlinear Boolean functions with an even number of variables. These combinatorial objects, with fascinating properties, are rare. The class of bent functions contains a subclass of functions the so-called hyper-bent functions whose properties are still stronger and whose elements are still rarer. In fact, hyper-bent functions seem still more difficult to generate at random than bent functions and many problems related to the class of hyper-bent functions remain open. (Hyper)-bent functions are not classified. A complete classification of these functions is elusive and looks hopeless.
In this paper, we contribute to the knowledge of the class of hyper-bent functions on finite fields \(\mathbb{F}_{2^n}\) (where n is even) by studying a subclass \(\mathfrak {F}_n\) of the so-called Partial Spreads class PS − (such functions are not yet classified, even in the monomial case). Functions of \(\mathfrak {F}_n\) have a general form with multiple trace terms. We describe the hyper-bent functions of \(\mathfrak {F}_n\) and we show that the bentness of those functions is related to the Dickson polynomials. In particular, the link between the Dillon monomial hyper-bent functions of \(\mathfrak {F}_n\) and the zeros of some Kloosterman sums has been generalized to a link between hyper-bent functions of \(\mathfrak {F}_n\) and some exponential sums where Dickson polynomials are involved. Moreover, we provide a possibly new infinite family of hyper-bent functions. Our study extends recent works of the author and is a complement of a recent work of Charpin and Gong on this topic.
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Canteaut, A., Charpin, P., Kyureghyan, G.: A New Class of Monomial Bent Functions. Finite Fields and Their Applications 14(1), 221–241 (2008)
Carlet, C.: Boolean Functions for Cryptography and Error Correcting Codes. In: Crama, Y., Hammer, P.L. (eds.) Chapter of the monography, Boolean Models and Methods in Mathematics, Computer Science, and Engineering, pp. 257–397. Cambridge University Press, Cambridge (2010)
Carlet, C., Gaborit, P.: Hyperbent functions and cyclic codes. Journal of Combinatorial Theory, Series A 113(3), 466–482 (2006)
Carlitz, L.: Explicit evualation of certain exponential sums. Math. Scand. 44, 5–16 (1979)
Charpin, P., Gong, G.: Hyperbent functions, Kloosterman sums and Dickson polynomials. IEEE Trans. Inform.Theory 9(54), 4230–4238 (2008)
Charpin, P., Helleseth, T., Zinoviev, V.: Divisibility properties of Kloosterman sums over finite fields of characteristic two. In: ISIT 2008, Toronto, Canada, July 6–11, 2008, pp. 2608–2612 (2008)
Charpin, P., Kyureghyan, G.: Cubic monomial bent functions: A subclass of \(\mathcal {M}\). SIAM, J. Discr. Math. 22(2), 650–665 (2008)
Dillon, J.: Elementary Hadamard difference sets. PhD dissertation, University of Maryland (1974)
Dillon, J.F., Dobbertin, H.: New cyclic difference sets with Singer parameters. Finite Fields and Their Applications 10(3), 342–389 (2004)
Dobbertin, H., Leander, G., Canteaut, A., Carlet, C., Felke, P., Gaborit, P.: Construction of bent functions via Niho Power Functions. Journal of Combinatorial therory, Serie A 113, 779–798 (2006)
Gold, R.: Maximal recursive sequences with 3-valued recursive crosscorrelation functions. IEEE Trans. Inform. Theory 14(1), 154–156 (1968)
Gologlu, F.: Almost Bent and Almost Perfect Nonlinear Functions, Exponential Sums, Geometries ans Sequences. PhD dissertation, University of Magdeburg (2009)
Lachaud, G., Wolfmann, J.: The weights of the orthogonals of the extended quadratic binary Goppa codes. IEEE Trans. Inform. Theory 36(3), 686–692 (1990)
Leander, G.: Monomial Bent Functions. IEEE Trans. Inform. Theory 2(52), 738–743 (2006)
Mesnager, S.: A new class of bent and hyper-bent boolean functions in polynomial forms. Journal Design, Codes and Cryptography (in press)
Mesnager, S.: A new class of bent boolean functions in polynomial forms. In: Proceedings of international Workshop on Coding and Cryptography, WCC 2009, pp. 5–18 (2009)
Mesnager, S.: A new family of hyper-bent boolean functions in polynomial form. In: Parker, M.G. (ed.) IMACC 2009. LNCS, vol. 5921, pp. 402–417. Springer, Heidelberg (2009)
Rothaus, O.: On ”bent” functions. J. Combin. Theory Ser. A 20, 300–305 (1976)
Youssef, A.M., Gong, G.: Hyper-bent functions. In: Pfitzmann, B. (ed.) EUROCRYPT 2001. LNCS, vol. 2045, pp. 406–419. Springer, Heidelberg (2001)
Yu, N.Y., Gong, G.: Construction of quadratic Bent functions in polynomial forms. IEEE Trans. Inform. Theory 7(52), 3291–3299 (2006)
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Mesnager, S. (2010). Hyper-bent Boolean Functions with Multiple Trace Terms. 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_8
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