8 YVES AUBRY AND FRANC ¸OIS RODIER References [1] Y. Aubry and M. Perret, A Weil theorem for singular curves, Arithmetic, Geometry and Coding Theory, (Luminy, 1993), Walter de Gruyter, 1-7, Berlin - New-York 1996. [2] Y. Aubry and M. Perret, On the characteristic polynomials of the Frobenius endomorphism for projective curves over finite fields, Finite Fields and Their Applications, 10 (2004), no. 3, 412-431. [3] T.P. Berger, A. Canteaut, P. Charpin and Y. Laigle-Chapuy, On almost perfect nonlinear functions over F2n, IEEE Trans. Inform. Theory 52 (2006), no. 9, 4160-4170. [4] C. Bracken and G. Leander, New families of functions with differential uniformity of 4, to be published with the proceedings of the workshop BFCA08, Copenhague, 2008. [5] C. Bracken and G. Leander, A highly nonlinear differentially 4-uniform power mapping that permutes fields of even degree, preprint, arXiv:0901.1824v1. [6] L. Budaghyan, C. Carlet and G. Leander, Two classes of quadratic APN binomials inequivalent to power functions, IEEE Trans. Inform. Theory, vol. 54, pp. 4218-4229, 2008. [7] L. Budaghyan, C. Carlet and A. Pott, New constructions of almost perfect nonlinear and almost bent functions. Proceedings of the Workshop on Coding and Cryptography 2005, P. Charpin and Ø. Ytrehus eds, pp. 306-315, 2005. [8] A. Canteaut, Differential cryptanalysis of Feistel ciphers and differentially δ-uniform mappings, In Selected Areas on Cryptography, SAC’97, pp. 172-184, Ottawa, Canada, 1997. [9] C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Designs, Codes and Cryptography, 15(2), pp. 125-156, 1998. [10] F. Hernando and G. McGuire, Proof of a conjecture on the sequence of exceptional numbers, classifying cyclic codes and APN functions, arXiv:0903.2016v1, [cs.IT] (math.AG), 11 march 2009. [11] S. R. Ghorpade and G. Lachaud, Etale cohomology, Lefschetz theorems and number of points of singular varieties over finite fields, Mosc. Math. J., 2 (2002), n. 3, 589-631. [12] R. Harshorne, Algebraic geometry, Graduate Texts in Math., 52 (1977), Springer-Verlag. [13] H. Janwa and R. M. Wilson, Hyperplane sections of Fermat varieties in P 3 in char. 2 and some applications to cyclic codes, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Proceedings AAECC-10 (G Cohen, T. Mora and O. Moreno Eds.), Lecture Notes in Computer Science, Vol. 673, Springer-Verlag, NewYork/Berlin 1993. [14] H. Janwa, G. McGuire and R. M. Wilson, Double-error-correcting cyclic codes and absolutely irreducible polynomials over GF(2), Applied J. of Algebra, 178, 665-676 (1995). [15] S. Lang and A. Weil, Number of points of varieties in finite fields, Amer. J. Math. 76, (1954), pp. 819-827. [16] K. Nyberg, Differentially uniform mappings for cryptography, Advances in cryptology— Eurocrypt ’93 (Lofthus, 1993), 55–64, Lecture Notes in Comput. Sci., n◦ 765, Springer, Berlin, 1994. [17] F. Rodier, Bornes sur le degr´ e des polynˆ omes presque parfaitement non-lin´eaires, Contempo- rary Math., Vol. 487, 169-181 2009) arXiv:math/0605232v3 [math.AG], 2 may 2008. [18] F. Rodier, Bounds on the degrees of APN polynomials, to be published with the proceedings of the workshop BFCA08, Copenhague, 2008. [19] J. -P. Serre, Lettre ` M. Tsfasman, Ast´ erisque 198-199-200 (1991), 351-353. [20] J. F. Voloch, Symmetric cryptography and algebraic curves, Algebraic Geometry and its Applications, Ser. Number Theory Appl., 5, World Sci. Publ., Hackensack, NJ, 135-141 (2008). Institut de Math´ ematiques de Toulon, Universit´ du Sud Toulon-Var, France, and, Institut de Math´ ematiques de Luminy, Marseille, France E-mail address: yves.aubry@univ-tln.fr and rodier@iml.univ-mrs.fr

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