# Some more functions that are not APN infinitely often. The case of Gold and Kasami exponents

(1)

## The case of Gold and Kasami exponents

(2)

### The Case of Gold and Kasami exponents Eric F´erard, Roger Oyono, and Fran¸cois Rodier

Abstract. We prove a necessary condition for some polynomials of Gold and Kasami degree to be APN overFqnfor largen.

n

q

q

q

q

q

q

q

q

q

q

d

2

i

i

i

### + 1. One calls these exponents exceptional

1991Mathematics Subject Classification. 11T06, 12E05, 14Q10, 11T71.

Key words and phrases. APN functions, algebraic surfaces, finite fields, absolute irreducible polynomials.

1

(3)

q

q

t

q

q

q

q

3

q

qn

qn

j

j

j

22k2k+1

α∈F2k−F2

α

α

k

2k

α

2k+1

### .

(4)

SOME MORE FUNCTIONS THAT ARE NOT APN INFINITELY OFTEN 3

2k+1

4k−2k+1

2

2m

qn

qn

1

2

2

2

4

1

1

4

1

q3

qn

q

qn

3

3

1

3

1

4

2

1+q

1+q2

q+q2

1+q+q2

q3

q

q2

1

k

d

k−1

2k−1+1

j=0

j

j

j

j

i

i

qn

(5)

2k

k

d

2k

k

2k−1

k−1

d

2k−1

k−1

22k−1−2k−1+1

j=0

j

j

j

j

d

j=3

j

j

s

s−1

0

t

t−1

0

j

j

s

t

α∈F2k−F2

α

s

t

2k1

k1

s

t−1

s−1

t

s

s−1

t

s

s−1

s

t

s−1

s−1

s

t−1

s−2

t−2

s−3

t−3

s

0

s−t

t

s

s−t

t

s

s−t

s−t

s

0

t

j

α

2k

2

j

j

j

2

α

2

j

j

d

j

d

d

q

2k−1

k−1

22k−1−2k−1+1

j=0

j

j

### . Suppose

(6)

SOME MORE FUNCTIONS THAT ARE NOT APN INFINITELY OFTEN 5

j

j

q

q

2k−1

k−1

n

d

q

2k−1

k−1

22k−12k−1+2

2

3

22k−1−2k−1+2

2

3

q

22k−1−2k−1+2

j=0

j

j

d

j=3

j

j

s

s−1

0

t

t−1

0

s

t

α∈F2k−F2

α

s

t

2k−1

k−1

s−1

t−1

s−2

t−2

s

1

s−t+1

t

s−t+1

1

2k−1

k−1

s+3

s+3

s

0

s−t

t

s+3

s+3

s−t

0

t

α

2k

2

22k−1−2k−1+2

22k−1−2k−1+2

222k−2

−2k−2+1

α

22k−1−2k−1+2

22k−2−2k−2+1

α

2k+1

22k−2−2k−2+1

22k−2−2k−2+1

22k−2−2k−2+1

22k−2−2k−2+1

22k−2−2k−2+1

22k−2−2k−2+1

(7)

22k−2−2k−2+1

22k−2−2k−2+1

22k−2−2k−2+1

22k−2−2k−2+1

22k−2−2k−2+1

22k−2−2k−2+1

22k−2−2k−2+1

22k−2−2k−2+1

22k−2−2k−2

2k−2

22k−2−2k−1+1

22k−2−2k−2+1

22k−2−2k−2

2k−2

22k−22k−1+1

2k−2+1

2k−1

22k−2−2k−2

2k−2(2k−1)

22k−2−2k−2+1

2k−2

1−2k−2

2k−2

1−2k−2

2k−2+1

2k−2

1−2k−2

1−2k−2

2k−2+1

2k−1

k−1

s

0

0

s

s+3

s+3

0

d

j=3

j

j

s

0

s

d

d

22k−1−2k−1+2

22k−1−2k−1+2

s

t

0

t

0

22k−1−2k−1+2

d

0

0

s

0

s

0

s

s

0

s

s

02

j

d

s+3

s+3

3

3

s

0

s

0

s

0

s

0

d

s+3

s+3

3

3

3

02

0

3

q

s

s

2

s

s

2

0

s

s

s+3

s+3

s

s

s+3

3

s+3

s

s

2k

s+3

2

s

s

q

s

s

2

### . Note that the leading coefficient of

(8)

SOME MORE FUNCTIONS THAT ARE NOT APN INFINITELY OFTEN 7

s

s

2s+3

3

2

s

s

s

s

s

s

s

s

s

s

s

s

s

s

2

s

s

4

2k

2k

4

2

q

57

30

2

3

q

27

27

2

27

β

β2

β4

27

β3

β5

β6

8

2

27

3

9

27

2

3

9

27

27

3

9

2

3

9

30

q

m

r

2n

2n

d

d

2k

k

k

k

q

k

q

d

d

q

q

2k

k

2k

k

q

v

k

k

d

d

q

q

(9)

2k

2

α

t

k

k

2k

2

α

t

k

α

t

t

t

t

α

t

t

2k+1

t

t

t

t

2k+1

t

t

t

i=0

t−i

t−i

i

t

t

t−1

t−1

2k

i=2

t−i

t−i

i

2k+1

t

t

t−1

t−1

t

t−1

t

t

t

t−1

t−1

t−1

2k−1

k

k

t

t

t−1

t−1

t12j

t12j

t−1

2j

t−1

2j

t−1

2j

t−1

2j

t−1

2j

2j

1+2tj

k

k

k

k

q

r

d

i

d

r

d

r

q

### . Then φ is absolutely irreducible.

(10)

SOME MORE FUNCTIONS THAT ARE NOT APN INFINITELY OFTEN 9

d

d

d

d

d

d

v

d

q

α∈F2k−F2

α

d

d

d

d

2k

2

α

d

v

d

2tv

2v1

α

t

### References

1. Y. Aubry, G. McGuire, F. Rodier, A Few More Functions That Are Not APN Infinitely Often, Finite Fields: Theory and applications, Ninth International conference Finite Fields and Applications, McGuire et al. editors, Contemporary Math. n518, AMS, Providence (RI), USA, 2010. Available on arXiv: n0909.2304.

2. T. Berger, A. Canteaut, P. Charpin and Y. Laigle-Chapuy, On almost perfect nonlinear func- tions overF2n, IEEE Trans. Inform. Theory, 52(9), pp. 4160–4170, 2006.

3. C. Bracken, E. Byrne, N. Markin, G. McGuire, New families of quadratic almost perfect nonlinear trinomials and multinomials, Finite Fields and their Applications, 14 (2008), pp.

703–714.

4. L. Budaghyan, C. Carlet, P. Felke and G. Leander, An infinite class of quadratic APN functions which are not equivalent to power mappings, Cryptology ePrint Archive, 2005/359.

5. E. Byrne and G. McGuire On the Non-Existence of Quadratic APN and Crooked Functions on Finite Fields, WCC 2005.

6. A. Canteaut, Differential cryptanalysis of Feistel ciphers and differentiallyδ-uniform mappings, In Selected Areas on Cryptography, SAC’97, pp. 172–184, Ottawa, Canada, 1997.

7. 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.

8. R. S. Coulter and R. W. Matthews, Planar functions and planes of Lenz-Barlotti class II, Des.

Codes Cryptogr., 10(2), pp. 167–184, 1997.

9. J. F. Dillon, APN Polynomials: An update, Fq9 International conference on finite fields and their Applications, July 2009.

10. C. Ding and J. Yuan, A family of skew Hadamard difference sets, J. Combin. Theory Ser. A, 113(7), pp. 1526–1535, 2006.

11. Y. Edel, G. Kyureghyan and A. Pott, A new APN function which is not equivalent to a power mapping. IEEE Trans. Inform. Theory 52, (2006), n2, pp. 744–747.

12. E. F´erard and F. Rodier, Non lin´earit´e des fonctions bool´eennes donn´ees par des polynˆomes de degr´e binaire 3 d´efinies surF2m avecm pair [Nonlinearity of Boolean functions given by polynomials of binary degree 3 defined onF2m withmeven]. Arithmetic, geometry, cryptogra- phy and coding theory 2009, pp. 41–53, Contemp. Math., 521, Amer. Math. Soc., Providence, RI, 2010.

13. E. F´erard and F. Rodier, Non lin´earit´e des fonctions bool´eennes donn´ees par des traces de polynˆomes de degr´e binaire 3 [Nonlinearity of Boolean functions given by traces of polynomials of binary degree 3]. Algebraic geometry and its applications, pp. 388–409, Ser. Number Theory Appl., 5, World Sci. Publ., Hackensack, NJ, 2008.

14. F. Hernando and G. McGuire, Proof of a conjecture on the sequence of exceptional numbers, classifying cyclic codes and APN functions, Available onarXiv:0903.2016v3[cs.IT], 2009.

15. H. Janwa and R. M. Wilson, Hyperplane sections of Fermat varieties in P3 in char. 2 and some applications to cyclic codes,Applied Algebra, Algebraic Algorithms and Error-Correcting

(11)

Codes, Proceedings AAECC-10 (G Cohen, T. Mora and O. Moreno Eds.), pp. 180-194, Lec- ture Notes in Computer Science, Vol. 673, Springer-Verlag, NewYork/Berlin 1993.

16. 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, pp. 665–676 (1995).

17. D. Jedlicka, APN monomials overGF(2n) for infinitely manyn, Finite Fields Appl. 13 (2007), n4, pp. 1006–1028.

18. E. Leducq,Autour des codes de Reed-Muller g´en´eralis´es, PhD thesis, Universit´e Paris 7, 2011.

Available on http://www.math.jussieu.fr/∼elodieleducq/these.pdf

19. E. Lucas, Th´eorie des fonctions num´eriques simplement p´eriodiques, American Journal of Mathematics, vol. 1, pp. 197-240 and pp. 289–321, 1878.

20. K. Nyberg, Differentially uniform mappings for cryptography, Advances in cryptology—

Eurocrypt ’93(Lofthus, 1993), pp. 55–64, Lecture Notes in Comput. Sci., Vol. 765, Springer, Berlin, 1994.

21. F. Rodier, Bornes sur le degr´e des polynˆomes presque parfaitement non-lin´eaires, in Arith- metic, Geometry, Cryptography and Coding Theory, G. Lachaud, C. Ritzenthaler and M.Tsfasman editors, Contemporary Math. n487, AMS, Providence (RI), USA, pp. 169-181, 2009. Available onarXiv:math/0605232v3[math.AG].

22. F. Rodier, Functions of degree 4e that are not APN Infinitely Often, Cryptography and Communications, 1–14, 2011.

23. F. Voloch, Symmetric cryptography and Algebraic curves,Algebraic geometry and its appli- cations, Proceedings of the first SAGA conference, Ser. Number theory and its applications, World Sci. Publ., Hackensack, NJ, pp. 135–141, 2008.

Equipe GAATI, Universit´´ e de la Polyn´esie Franc¸aise E-mail address: eric.ferard@upf.pf

Equipe GAATI, Universit´´ e de la Polyn´esie Franc¸aise E-mail address: roger.oyono@upf.pf

Institut de Math´ematiques de Luminy, CNRS, Universit´e de la M´editerran´ee, Mar- seille