November  2014, 8(4): 497-509. doi: 10.3934/amc.2014.8.497

On the irreducibility of the hyperplane sections of Fermat varieties in $\mathbb{P}^3$ in characteristic $2$

1. 

Équipe GAATI, Université de la Polynésie Française, BP 6570, 98702 FAA'A, Tahiti, Polynésie Française, France

Received  January 2014 Revised  September 2014 Published  November 2014

Let $t$ be an integer $\ge 5$. The absolute irreducibility of the polynomial $\phi_t(x, y) = \frac{x^t + y^t + 1 + (x + y + 1)^t}{(x + y)(x + 1)(y + 1)}$ (over $\mathbb{F}_2$) plays an important role in the study of APN functions. If $t \equiv 5 \bmod{8}$, we give a criterion that ensures that $\phi_t(x, y)$ is absolutely irreducible. We prove that if $\phi_t(x, y)$ is not absolutely irreducible, then it is divisible by $\phi_{13}(x, y)$. We also exhibit an infinite family of integers $t$ such that $\phi_t(x, y)$ is not absolutely irreducible.
Citation: Eric Férard. On the irreducibility of the hyperplane sections of Fermat varieties in $\mathbb{P}^3$ in characteristic $2$. Advances in Mathematics of Communications, 2014, 8 (4) : 497-509. doi: 10.3934/amc.2014.8.497
References:
[1]

Y. Aubry, G. McGuire and F. Rodier, A few more functions that are not APN infinitely often, in Finite Fields: Theory and Applications, Amer. Math. Soc., Providence, 2010, 23-31. doi: 10.1090/conm/518/10193.

[2]

T. Berger, A. Canteaut, P. Charpin and Y. Laigle-Chapuy, On almost perfect nonlinear functions over $F^n_2$, IEEE Trans. Inf. Theory, 52 (2006), 4160-4170. doi: 10.1109/TIT.2006.880036.

[3]

A. W. Bluher, On existence of Budaghyan-Carlet APN hexanomials, Finite Fields Appl., 24 (2013), 118-123. doi: 10.1016/j.ffa.2013.06.003.

[4]

C. Bracken, E. Byrne, N. Markin and G. McGuire, New families of quadratic almost perfect nonlinear trinomials and multinomials, Finite Fields Appl., 14 (2008), 703-714. doi: 10.1016/j.ffa.2007.11.002.

[5]

C. Bracken, C.H. Tan and Y. Tan, On a class of quadratic polynomials with no zeros and its application to APN functions, Finite Fields Appl., 25 (2014), 26-36. doi: 10.1016/j.ffa.2013.08.006.

[6]

L. Budaghyan, C. Carlet, P. Felke and G. Leander, An infinite class of quadratic APN functions which are not equivalent to power mappings, in Proc. IEEE Int. Symp. Inf. Theory, 2006, 2637-2641. doi: 10.1109/ISIT.2006.262131.

[7]

E. Byrne and G. McGuire, On the non-existence of quadratic APN and crooked functions on finite fields, in Proc. Workshop Coding Crypt., 2005, 316-324.

[8]

C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Crypt., 15 (1998), 125-156. doi: 10.1023/A:1008344232130.

[9]

F. Caullery, A new large class of functions not APN infinitely often, Des. Codes Crypt., 73 (2014), 601-614. doi: 10.1007/s10623-014-9956-2.

[10]

M. Delgado and H. Janwa, On the conjecture on APN functions, preprint, arXiv:1207.5528[cs.IT]

[11]

J. F. Dillon, APN Polynomials: An update, invited talk at 9th Int. Conf. Finite Fields Appl., 2009.

[12]

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

[13]

E. Férard, R. Oyono and F. Rodier, Some more functions that are not APN infinitely often. The case of Gold and Kasami exponents, in Arithmetic, Geometry, Cryptography and Coding Theory, Amer. Math. Soc., Providence, 2012, 27-36. doi: 10.1090/conm/574/11423.

[14]

E. Férard and F. Rodier, Non linéarité des fonctions booléennes données par des traces de polynômes de degré binaire 3 [Nonlinearity of Boolean functions given by traces of polynomials of binary degree 3], in Algebraic Geometry and its Applications, World Sci. Publ., Hackensack, 2008, 388-409. doi: 10.1142/9789812793430_0021.

[15]

E. Férard and F. Rodier, Non linéarité des fonctions booléennes données par des polynômes de degré binaire 3 définies sur $\mathbbF_{2^m}$ avec $m$ pair [Nonlinearity of Boolean functions given by polynomials of binary degree 3 defined on $\mathbbF_{2^m}$ with $m$ even], in Arithmetic, Geometry, Cryptography and Coding Theory, Amer. Math. Soc., Providence, 2010, 41-53. doi: 10.1090/conm/521/10272.

[16]

W. Fulton, Algebraic Curves, Benjamin, New York, 1969.

[17]

B. Hassett, Introduction to Algebraic Geometry, Cambridge Univ. Press, 2007. doi: 10.1017/CBO9780511755224.

[18]

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

[19]

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

[20]

H. Janwa and R. M. Wilson, 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 (eds. G. Cohen, T. Mora and O. Moreno), Springer-Verlag, NewYork, 1993, 180-194. doi: 10.1007/3-540-56686-4_43.

[21]

D. Jedlicka, APN monomials over $GF(2^n)$ for infinitely many $n$, Finite Fields Appl., 13 (2007), 1006-1028. doi: 10.1016/j.ffa.2007.04.004.

[22]

E. Leducq, Autour des Xodes de Reed-Muller Généralisés, Ph.D thesis, Université Paris 7, 2011; available online at http://www.math.u-psud.fr/ leducq/these.pdf

[23]

E. Lucas, Théorie des fonctions numériques simplement périodiques, Amer. J. Math., 1 (1878), 197-240 and 289-321. doi: 10.2307/2369373.

[24]

K. Nyberg, Differentially uniform mappings for cryptography, in Adv. Crypt.- Eurocrypt '93, Springer, Berlin, 1994, 55-64. doi: 10.1007/3-540-48285-7_6.

[25]

F. Rodier, Bornes sur le degré des polynômes presque parfaitement non-linéaires, in Arithmetic, Geometry, Cryptography and Coding Theory, Amer. Math. Soc., Providence, 2009, 169-181. doi: 10.1090/conm/487/09531.

[26]

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

[27]

The Sage Development Team, Sage Mathematics Software (Version 4.8), http://www.sagemath.org

show all references

References:
[1]

Y. Aubry, G. McGuire and F. Rodier, A few more functions that are not APN infinitely often, in Finite Fields: Theory and Applications, Amer. Math. Soc., Providence, 2010, 23-31. doi: 10.1090/conm/518/10193.

[2]

T. Berger, A. Canteaut, P. Charpin and Y. Laigle-Chapuy, On almost perfect nonlinear functions over $F^n_2$, IEEE Trans. Inf. Theory, 52 (2006), 4160-4170. doi: 10.1109/TIT.2006.880036.

[3]

A. W. Bluher, On existence of Budaghyan-Carlet APN hexanomials, Finite Fields Appl., 24 (2013), 118-123. doi: 10.1016/j.ffa.2013.06.003.

[4]

C. Bracken, E. Byrne, N. Markin and G. McGuire, New families of quadratic almost perfect nonlinear trinomials and multinomials, Finite Fields Appl., 14 (2008), 703-714. doi: 10.1016/j.ffa.2007.11.002.

[5]

C. Bracken, C.H. Tan and Y. Tan, On a class of quadratic polynomials with no zeros and its application to APN functions, Finite Fields Appl., 25 (2014), 26-36. doi: 10.1016/j.ffa.2013.08.006.

[6]

L. Budaghyan, C. Carlet, P. Felke and G. Leander, An infinite class of quadratic APN functions which are not equivalent to power mappings, in Proc. IEEE Int. Symp. Inf. Theory, 2006, 2637-2641. doi: 10.1109/ISIT.2006.262131.

[7]

E. Byrne and G. McGuire, On the non-existence of quadratic APN and crooked functions on finite fields, in Proc. Workshop Coding Crypt., 2005, 316-324.

[8]

C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Crypt., 15 (1998), 125-156. doi: 10.1023/A:1008344232130.

[9]

F. Caullery, A new large class of functions not APN infinitely often, Des. Codes Crypt., 73 (2014), 601-614. doi: 10.1007/s10623-014-9956-2.

[10]

M. Delgado and H. Janwa, On the conjecture on APN functions, preprint, arXiv:1207.5528[cs.IT]

[11]

J. F. Dillon, APN Polynomials: An update, invited talk at 9th Int. Conf. Finite Fields Appl., 2009.

[12]

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

[13]

E. Férard, R. Oyono and F. Rodier, Some more functions that are not APN infinitely often. The case of Gold and Kasami exponents, in Arithmetic, Geometry, Cryptography and Coding Theory, Amer. Math. Soc., Providence, 2012, 27-36. doi: 10.1090/conm/574/11423.

[14]

E. Férard and F. Rodier, Non linéarité des fonctions booléennes données par des traces de polynômes de degré binaire 3 [Nonlinearity of Boolean functions given by traces of polynomials of binary degree 3], in Algebraic Geometry and its Applications, World Sci. Publ., Hackensack, 2008, 388-409. doi: 10.1142/9789812793430_0021.

[15]

E. Férard and F. Rodier, Non linéarité des fonctions booléennes données par des polynômes de degré binaire 3 définies sur $\mathbbF_{2^m}$ avec $m$ pair [Nonlinearity of Boolean functions given by polynomials of binary degree 3 defined on $\mathbbF_{2^m}$ with $m$ even], in Arithmetic, Geometry, Cryptography and Coding Theory, Amer. Math. Soc., Providence, 2010, 41-53. doi: 10.1090/conm/521/10272.

[16]

W. Fulton, Algebraic Curves, Benjamin, New York, 1969.

[17]

B. Hassett, Introduction to Algebraic Geometry, Cambridge Univ. Press, 2007. doi: 10.1017/CBO9780511755224.

[18]

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

[19]

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

[20]

H. Janwa and R. M. Wilson, 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 (eds. G. Cohen, T. Mora and O. Moreno), Springer-Verlag, NewYork, 1993, 180-194. doi: 10.1007/3-540-56686-4_43.

[21]

D. Jedlicka, APN monomials over $GF(2^n)$ for infinitely many $n$, Finite Fields Appl., 13 (2007), 1006-1028. doi: 10.1016/j.ffa.2007.04.004.

[22]

E. Leducq, Autour des Xodes de Reed-Muller Généralisés, Ph.D thesis, Université Paris 7, 2011; available online at http://www.math.u-psud.fr/ leducq/these.pdf

[23]

E. Lucas, Théorie des fonctions numériques simplement périodiques, Amer. J. Math., 1 (1878), 197-240 and 289-321. doi: 10.2307/2369373.

[24]

K. Nyberg, Differentially uniform mappings for cryptography, in Adv. Crypt.- Eurocrypt '93, Springer, Berlin, 1994, 55-64. doi: 10.1007/3-540-48285-7_6.

[25]

F. Rodier, Bornes sur le degré des polynômes presque parfaitement non-linéaires, in Arithmetic, Geometry, Cryptography and Coding Theory, Amer. Math. Soc., Providence, 2009, 169-181. doi: 10.1090/conm/487/09531.

[26]

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

[27]

The Sage Development Team, Sage Mathematics Software (Version 4.8), http://www.sagemath.org

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