Abstract
In this paper we consider APN functions \({f:\mathcal{F}_{2^m}\to \mathcal{F}_{2^m}}\) of the form f(x) = x −1 + g(x) where g is any non \({\mathcal{F}_{2}}\)-affine polynomial. We prove a lower bound on the degree of the polynomial g. This bound in particular implies that such a function f is APN on at most a finite number of fields \({\mathcal{F}_{2^m}}\). Furthermore we prove that when the degree of g is less than 7 such functions are APN only if m ≤ 3 where these functions are equivalent to x 3.
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Bracken C., Byrne E., Markin N., McGuire G.: A few more quadratic APN functions. Cryptogr. Commun. (to appear).
Bracken C., Byrne E., Markin N., McGuire G.: New families of quadratic almost perfect nonlinear trinomials and multinomials. Finite Fields Appl. 14(3), 703–714 (2008)
Budaghyan L., Carlet C.: Classes of quadratic APN trinomials and hexanomials and related structures. IEEE Trans. Inform. Theory 54(5), 2354–2357 (2008)
Budaghyan L., Carlet C., Leander G.: Constructing new APN functions from known ones. Finite Fields Appl. (in press).
Budaghyan L., Carlet C., Leander G.: Two classes of quadratic APN binomials inequivalent to power functions. IEEE Trans. Inform. Theory 54(9), 4218–4229 (2008)
Carlet C., Charpin P., Zinoviev V.: Codes, bent functions and permutations suitable for DES-like cryptosystems. Des. Codes Cryptogr. 15(2), 125–156 (1998)
Deligne P.: La conjecture de Weil: I. Publications Mathematiques of l’IHES 43, 273–307 (1974)
Edel Y., Kyureghyan G., Pott A.: A new APN function which is not equivalent to a power map**. IEEE Trans. Inform. Theory 52(2), 744–747 (2006)
Ghorpade S.R., Lachaud G.: Etale cohomology Lefschetz theorems and the number of points of singular varieties over finite fields. Mosc. Math. J. 2, 589–631 (2002)
Hernando F., McGuire G.: Proof of a conjecture on the sequence of exceptional numbers, classifying cyclic codes and APN functions. Preprint ar**v:0903.2016.
Janwa H., McGuire G., Wilson R.M.: Double-error-correcting cyclic codes and absolutely irreducible polynomials over GF(2). J. Algebra 178(2), 665–676 (1995)
Jedlicka D.: APN monomials over GF(2n) for infinitely many n. Finite Fields Appl. 13(4), 1006–1028 (2007)
Lang S., Weil A.: Number of points of varieties in finite fields. Am. J. Math. 76(4), 819–827 (1954)
Rodier F.: Borne sur le degré des polynômes presque parfaitement non-linéaires. Arxiv preprint math.AG/0605232, to be published with the proceedings of the conference AGCT-11 (2006).
Rodier F.: Bounds on the degrees of APN polynomials. To be published with the proceedings of the workshop BFCA08, Copenhagen, 2008 (2006).
Serre J.P.: Lettre à M. Tsfasman. Asterisque 198–200, 351–353 (1991)
Voloch F.: Symmetric cryptography and algebraic curves. In: Proceedings of the First SAGA Conference, Papeete, France (2007).
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Leander, G., Rodier, F. Bounds on the degree of APN polynomials: the case of x −1 + g(x). Des. Codes Cryptogr. 59, 207–222 (2011). https://doi.org/10.1007/s10623-010-9456-y
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DOI: https://doi.org/10.1007/s10623-010-9456-y