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Designs, Codes and Cryptography

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01.11.2014

A new large class of functions not APN infinitely often

verfasst von: Florian Caullery

Erschienen in: Designs, Codes and Cryptography | Ausgabe 2/2014

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Abstract

We show that there is no vectorial Boolean function of degree \(4e\), with some conditions on \(e\), which is APN over infinitely many extensions of its field of definition. It is a new step in the proof of the conjecture of Aubry, McGuire and Rodier.

Vorheriger Artikel A matrix approach for constructing quadratic APN functions

Nächster Artikel On the rank of incidence matrices in projective Hjelmslev spaces

Aubry Y., McGuire G., Rodier F.: A few more functions that are not APN infinitely often. In: Finite Fields: Theory and Applications. Contemporary Mathematics, vol. 518, pp. 23–31. American Mathematical Society, Providence (2010). doi:10.1090/conm/518/10193.

Berger T.P., Canteaut A., Charpin P., Laigle-Chapuy Y.: On almost perfect nonlinear functions over \(f^{n}_{2}\). IEEE Trans. Inform. Theory 52(9), 4160–4170 (2006). doi:10.1109/TIT.2006.880036.

Biham E., Shamir A.: Differential cryptanalysis of DES-like cryptosystems. J. Cryptol. 4(1), 3–72 (1991). doi:10.1007/BF00630563.

Bourbaki N.: Éléments de mathématique. Masson, Paris (1985). Algèbre commutative. Chapitres 5 à 7.

Byrne E., McGuire G.: Quadratic binomial APN functions and absolutely irreducible polynomials. ArXiv e-prints (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). doi:10.1023/A:1008344232130.

Delgado M., Janwa H.: On the conjecture on APN functions. CoRR abs/1207.5528 (2012).

Dillon J.F.: APN polynomials (2009). http://mathsci.ucd.ie/~gmg/Fq9Talks/Dillon.pdf. Accessed 18 July 2013.

Dobbertin H., Mills D., Muller E.N., Pott A., Willems W.: APN functions in odd characteristic. Discret. Math. 267(1–3), 95–112 (2003). Combinatorics 2000 (Gaeta).

10.

Edel Y., Pott A.: A new almost perfect nonlinear function which is not quadratic. Adv. Math. Commun. 3(1), 59–81 (2009). doi:10.3934/amc.2009.3.59.

11.

Edel Y., Kyureghyan G., Pott A.: A new APN function which is not equivalent to a power mapping. IEEE Trans. Inf. Theory 52(2), 744–747 (2006). doi:10.1109/TIT.2005.862128.

12.

Férard E., Oyono R., Rodier F.: Some more functions that are not APN infinitely often. The case of Gold and Kasami exponents. In: Arithmetic, Geometry, Cryptography and Coding Theory. Contemporary Mathematics, vol. 574, pp. 27–36. American Mathematical Society, Providence (2012). doi:10.1090/conm/574/11423.

13.

Hernando F., McGuire G.: Proof of a conjecture on the sequence of exceptional numbers, classifying cyclic codes and APN functions. J. Algebra 343, 78–92 (2011). doi:10.1016/j.jalgebra.2011.06.019.

14.

Janwa H., Wilson R.M.: Hyperplane sections of Fermat varieties in \(P^3\) in char 2 and some applications to cyclic codes. In: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (San Juan, PR, 1993). Lecture Notes in Computer Science, vol. 673, pp. 180–194. Springer, Berlin (1993). doi:10.1007/3-540-56686-4_43.

15.

Janwa H., McGuire G.M., Wilson R.M.: Double-error-correcting cyclic codes and absolutely irreducible polynomials over GF(2). J. Algebra 178(2), 665–676 (1995). doi:10.1006/jabr.1995.1372.

16.

Leducq E.: New families of APN functions in characteristic 3 or 5. In: Arithmetic, Geometry, Cryptography and Coding Theory. Contemporary Mathematics, vol. 574, pp. 115–123. American Mathematical Society, Providence (2012). doi:10.1090/conm/574/11419.

17.

Ness G.J., Helleseth T.: A new family of ternary almost perfect nonlinear mappings. IEEE Trans. Inf. Theory 53(7), 2581–2586 (2007). doi:10.1109/TIT.2007.899508.

18.

Nyberg K.: Differentially uniform mappings for cryptography. In: Advances in Cryptology–EUROCRYPT ’93 (Lofthus, 1993). Lecture Notes in Computer Science, vol. 765, pp. 55–64. Springer, Berlin (1994). doi:10.1007/3-540-48285-7_6.

19.

Poinsot L., Pott, A.: Non-boolean almost perfect nonlinear functions on non-abelian groups. Int. J. Found. Comput. Sci. 22(6), 1351–1367 (2011). doi:10.1142/S0129054111008751.

20.

Rodier F.: Borne sur le degré des polynômes presque parfaitement non-linéaires. In: Arithmetic, Geometry, Cryptography and Coding theory. Contemporary Mathematics, vol. 487, pp. 169–181. American Mathematical Society, Providence (2009). doi:10.1090/conm/487/09531.

21.

Rodier F.: Functions of degree 4e that are not APN infinitely often. Cryptogr. Commun. 3(4), 227–240 (2011). doi:10.1007/s12095-011-0050-6.

22.

Zha Z., Wang X.: Power functions with low uniformity on odd characteristic finite fields. Sci. China Math. 53(8), 1931–1940 (2010). doi:10.1007/s11425-010-3149-x.

23.

Zha Z., Wang X.: Almost perfect nonlinear power functions in odd characteristic. IEEE Trans. Inf. Theory 57(7), 4826–4832 (2011). doi:10.1109/TIT.2011.2145130.

Titel: A new large class of functions not APN infinitely often
verfasst von: Florian Caullery
Publikationsdatum: 01.11.2014
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 2/2014
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-014-9956-2

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