Abstract
We prove a necessary condition for some polynomials of degree 4e (e an odd number) to be APN over \(\mathbb{F}_{q^n}\) for large n, and we investigate the polynomials f of degree 12.
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Aubry, Y., McGuire, G., Rodier, F.: A few more functions that are not APN infinitely often. In: McGuire, G., et al. (eds.) Finite Fields: Theory and Applications, Ninth International conference Finite Fields and Applications, Contemporary Math. n° 518, pp. 23–31. AMS, Providence (2010)
Berger, T., Canteaut, A., Charpin, P., Laigle-Chapuy, Y.: On almost perfect nonlinear functions over \(F_2^n\). IEEE Trans. Inf. Theory 52(9), 4160–4170 (2006)
Budaghyan, L., Carlet, C., Felke, P., Leander, G.: An infinite class of quadratic APN functions which are not equivalent to power mappings. Cryptology ePrint Archive, n° 2005/359
Byrne, E., McGuire, G.: On the non-existence of quadratic APN and crooked functions on finite fields. WCC 2005 Conference Extended Abstracts (2005)
Carlet, C., Charpin, P., Zinoviev, V.: Codes, bent functions and permutations suitable for DES-like cryptosystems. Designs Codes Cryptogr. 15(2), 125–156 (1998)
Dillon, J.: APN Polynomials: An Update. Fq9, International Conference on Finite Fields and their Applications, July 2009
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)
Edel, Y., Pott, A.: A new almost perfect nonlinear function which is not quadratic. Adv. Math. Commun. 3(1), 59–81 (2009)
Hernando, F., McGuire, G.: Proof of a conjecture on the sequence of exceptional numbers, classifying cyclic codes and APN functions. J. Algebra (2009). arXiv:0903.2016v3 [cs.IT]
Janwa, H., Wilson, R.M.: Hyperplane sections of Fermat varieties in P 3 in char. 2 and some applications to cyclic codes. In: Cohen, G., Mora, T., Moreno, O. (eds.) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Proceedings AAECC-10, Lecture Notes in Computer Science, vol. 673, pp. 180–194. Springer, New York (1993)
Jedlicka, D.: APN monomials over GF(2n) for infinitely many n. Finite Fields Appl. 13(4), 1006–1028 (2007)
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)
Rodier, F.: Bornes sur le degré des polynômes presque parfaitement non-linéaires. In: Lachaud, G., Ritzenthaler, C., Tsfasman, M. (eds.) Arithmetic, Geometry, Cryptography and Coding Theory, Contemporary Math. no. 487, pp. 169–181. AMS, Providence. arXiv:math/0605232v3 [math.AG] (2009)
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Rodier, F. Functions of degree 4e that are not APN infinitely often. Cryptogr. Commun. 3, 227–240 (2011). https://doi.org/10.1007/s12095-011-0050-6
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DOI: https://doi.org/10.1007/s12095-011-0050-6