# American Institute of Mathematical Sciences

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] C. Bracken, C.H. Tan and Y. 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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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 Google Scholar [23] E. Lucas, Théorie des fonctions numériques simplement périodiques, Amer. J. Math., 1 (1878), 197-240 and 289-321. doi: 10.2307/2369373.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [27] The Sage Development Team, Sage Mathematics Software (Version 4.8),, , ().   Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [10] M. Delgado and H. Janwa, On the conjecture on APN functions,, preprint, ().   Google Scholar [11] J. F. Dillon, APN Polynomials: An update, invited talk at 9th Int. Conf. Finite Fields Appl., 2009. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] W. Fulton, Algebraic Curves, Benjamin, New York, 1969.  Google Scholar [17] B. Hassett, Introduction to Algebraic Geometry, Cambridge Univ. Press, 2007. doi: 10.1017/CBO9780511755224.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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 Google Scholar [23] E. Lucas, Théorie des fonctions numériques simplement périodiques, Amer. J. Math., 1 (1878), 197-240 and 289-321. doi: 10.2307/2369373.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [27] The Sage Development Team, Sage Mathematics Software (Version 4.8),, , ().   Google Scholar
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