Abstract
In this paper, we study the action of Dickson polynomials on subsets of finite fields of even characteristic related to the trace of the inverse of an element and provide an alternate proof of a not so well-known result. Such properties are then applied to the study of a family of Boolean functions and a characterization of their hyper-bentness in terms of exponential sums recently proposed by Wang et al.Finally, we extend previous works of Lisoněk and Flori and Mesnager to reformulate this characterization in terms of the number of points on hyperelliptic curves and present some numerical results leading to an interesting problem.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24(3-4), 235–265 (1997); Computational algebra and number theory, London (1993)
Carlet, C., Gaborit, P.: Hyper-bent functions and cyclic codes. J. Comb. Theory, Ser. A 113(3), 466–482 (2006)
Charpin, P., Gong, G.: Hyperbent functions, Kloosterman sums, and Dickson polynomials. IEEE Transactions on Information Theory 54(9), 4230–4238 (2008)
Charpin, P., Helleseth, T., Zinoviev, V.: Divisibility properties of classical binary Kloosterman sums. Discrete Mathematics 309(12), 3975–3984 (2009)
Chou, W.S., Gomez-Calderon, J., Mullen, G.L.: Value sets of Dickson polynomials over finite fields. J. Number Theory 30(3), 334–344 (1988)
Denef, J., Vercauteren, F.: An Extension of Kedlaya’s Algorithm to Artin-Schreier Curves in Characteristic 2. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 308–323. Springer, Heidelberg (2002)
Dillon, J.F.: Elementary Hadamard Difference Sets. ProQuest LLC, Ann Arbor (1974); Thesis (Ph.D.)–University of Maryland, College Park
Dillon, J.F., Dobbertin, H.: New cyclic difference sets with Singer parameters. Finite Fields and Their Applications 10(3), 342–389 (2004)
Dobbertin, H., Felke, P., Helleseth, T., Rosendahl, P.: Niho type cross-correlation functions via dickson polynomials and kloosterman sums. IEEE Transactions on Information Theory 52(2), 613–627 (2006)
Flori, J.P., Mesnager, S.: An efficient characterization of a family of hyperbent functions with multiple trace terms. Cryptology ePrint Archive, Report 2011/373 (2011), http://eprint.iacr.org/
Gong, G., Golomb, S.W.: Transform domain analysis of DES. IEEE Transactions on Information Theory 45(6), 2065–2073 (1999)
Katz, N., Livné, R.: Sommes de Kloosterman et courbes elliptiques universelles en caractéristiques 2 et 3. C. R. Acad. Sci. Paris Sér. I Math. 309(11), 723–726 (1989)
Lachaud, G., Wolfmann, J.: Sommes de Kloosterman, courbes elliptiques et codes cycliques en caractéristique 2. C. R. Acad. Sci. Paris Sér. I Math. 305(20), 881–883 (1987)
Lidl, R., Mullen, G.L., Turnwald, G.: Dickson polynomials. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 65. Longman Scientific & Technical, Harlow (1993)
Lidl, R., Niederreiter, H.: Finite fields, 2nd edn. Encyclopedia of Mathematics and its Applications, vol. 20. Cambridge University Press, Cambridge (1997); with a foreword by P. M. Cohn
Lisoněk, P.: An efficient characterization of a family of hyperbent functions. IEEE Transactions on Information Theory 57(9), 6010–6014 (2011)
Mesnager, S.: Hyper-bent Boolean Functions with Multiple Trace Terms. In: Hasan, M.A., Helleseth, T. (eds.) WAIFI 2010. LNCS, vol. 6087, pp. 97–113. Springer, Heidelberg (2010)
Mesnager, S.: Bent and hyper-bent functions in polynomial form and their link with some exponential sums and Dickson polynomials. IEEE Transactions on Information Theory 57(9), 5996–6009 (2011)
Mesnager, S.: A new class of bent and hyper-bent Boolean functions in polynomial forms. Des. Codes Cryptography 59(1-3), 265–279 (2011)
Ranto, K.: On algebraic decoding of the Z4-linear Goethals-like codes. IEEE Transactions on Information Theory 46(6), 2193–2197 (2000)
Rothaus, O.S.: On ”bent” functions. J. Comb. Theory, Ser. A 20(3), 300–305 (1976)
Schur, I.: Gesammelte Abhandlungen. Band III. Springer, Berlin (1973), Herausgegeben von Alfred Brauer und Hans Rohrbach
Vercauteren, F.: Computing zeta functions of curves over finite fields. Ph.D. thesis, Katholieke Universiteit Leuven (2003)
Wang, B., Tang, C., Qi, Y., Yang, Y., Xu, M.: A new class of hyper-bent Boolean functions in binomial forms. CoRR abs/1112.0062 (2011)
Wang, B., Tang, C., Qi, Y., Yang, Y., Xu, M.: A new class of hyper-bent Boolean functions with multiple trace terms. Cryptology ePrint Archive, Report 2011/600 (2011), http://eprint.iacr.org/
Youssef, A.M., Gong, G.: Hyper-bent Functions. In: Pfitzmann, B. (ed.) EUROCRYPT 2001. LNCS, vol. 2045, pp. 406–419. Springer, Heidelberg (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Flori, JP., Mesnager, S. (2012). Dickson Polynomials, Hyperelliptic Curves and Hyper-bent Functions. In: Helleseth, T., Jedwab, J. (eds) Sequences and Their Applications – SETA 2012. SETA 2012. Lecture Notes in Computer Science, vol 7280. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30615-0_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-30615-0_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-30614-3
Online ISBN: 978-3-642-30615-0
eBook Packages: Computer ScienceComputer Science (R0)