Abstract
In this paper we characterize the d-dimensional dual hyperovals in PG(2d + 1, 2) that can be obtained by Yoshiara’s construction (Innov Incid Geom 8:147–169, 2008) from quadratic APN functions and state a one-to-one correspondence between the extended affine equivalence classes of quadratic APN functions and the isomorphism classes of these dual hyperovals.
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References
Bierbrauer J.: A family of crooked functions. Des. Codes Cryptogr. 50, 235–241 (2009)
Brinkmann M., Leander G.: On the classification of APN functions up to dimension five. Des. Codes Cryptogr. 49, 273–288 (2008)
Browning K., Dillon J., Kibler R., McQuistan M.: APN polynomials and related codes. Submitted (2008).
Budaghyan L., Carlet C., Leander G.: Two classes of quadratic APN binomials inequivalent to power functions. IEEE Trans. Inform. Theory 54, 4218–4229 (2008)
Budaghyan L., Carlet C., Leander G.: Constructing new APN functions from known ones. Finite Fields Appl. 15, 150–159 (2009)
Carlet C.: Vectorial Boolean functions for cryptography. In: Crama Y., Hammer P. (eds.) Boolean Methods and Models. Cambridge University Press (to appear).
Carlet C., Charpin P., Zinoviev V.: Codes, bent functions and permutations suitable for DES-like cryptosystems. Des. Codes Cryptogr. 15, 125–156 (1998)
Chabaud F., Vaudenay S.: Links between differential and linear cryptanalysis. In: Santis, A.D. (eds) Advances in Cryptology – EUROCRYPT 94, vol. 950 of Lecture Notes in Computer Science, pp. 356–365. Springer, New York (1995)
Cooperstein B.N., Thas J.A.: On generalized k-arcs in PG(2n, q). Ann. Comb. 5, 141–152 (2001)
Del Fra A.: On d-dimensional dual hyperovals. Geom. Dedicata 79, 157–178 (2000)
Edel Y., Pott A.: A new almost perfect nonlinear function which is not quadratic. Adv. Math. Commun. 3, 59–81 (2009)
Edel Y., Kyureghyan G., Pott A.: A new APN function which is not equivalent to a power mapping. IEEE Trans. Inform. Theory 52, 744–747 (2006)
Gold R.: Maximal recursive sequences with 3-valued recursive cross-correlation function. IEEE Trans. Inform. Theory 14, 154–156 (1968)
Göloglu F., Pott A.: Almost perfect nonlinear functions: a possible geometric approach. In: Proceedings of the Contact Forum Coding Theory and Cryptography II, pp. 75–100. Royal Flemish Academy of Belgium for Science and the Arts, Belgium (2008).
Huybrechts C., Pasini A.: Flag-transitive extensions of dual affine spaces. Bull. Belg. Math. Soc. Simon Stevin 5, 341–353 (1998)
Nakagawa N., Yoshiara S.: A construction of differentially 4-uniform functions from commutative semifields of characteristic 2. Lect. Notes Comput. Sci. 4547, 134–146 (2007)
Nyberg K.: Differentially uniform mappings for cryptography. In: Advances in Cryptography. EUROCRYPT’93, volume 765 of Lecture Notes in Computer Science, pp. 55–64. Springer-Verlag, New York (1994).
Taniguchi H.: A family of dual hyperovals over GF(q) with q even. Eur. J. Combin. 26, 95–99 (2005)
Yoshiara S.: Dimensional dual arcs a survey. In: Hulpke, A., Liebler, B., Penttila, T., Seress, A. (eds) Finite Geometires, Groups, and Computation, pp. 247–266. Walter de Gruyter, Berlin (2006)
Yoshiara S.: Notes on Taniguchi’s dimensional dual hyperovals. Eur. J. Combin. 28, 674–684 (2007)
Yoshiara S.: Dimensional dual hyperovals associated with quadratic APN functions. Innov. Incid. Geom. 8, 147–169 (2008)
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Communicated by Victor A. Zinoviev.
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Edel, Y. On quadratic APN functions and dimensional dual hyperovals. Des. Codes Cryptogr. 57, 35–44 (2010). https://doi.org/10.1007/s10623-009-9347-2
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DOI: https://doi.org/10.1007/s10623-009-9347-2