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 ` a 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´ e 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

8

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 ` a 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´ e 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

8