# 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, (2010), 23. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (2012), 27. 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, (2008), 388. 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, (2010), 41. doi: 10.1090/conm/521/10272. Google Scholar [16] W. Fulton, Algebraic Curves,, Benjamin, (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. 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. 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, (1993), 180. 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. doi: 10.1016/j.ffa.2007.04.004. Google Scholar [22] E. Leducq, Autour des Xodes de Reed-Muller Généralisés,, Ph.D thesis, (2011). Google Scholar [23] E. Lucas, Théorie des fonctions numériques simplement périodiques,, Amer. J. Math., 1 (1878), 197. doi: 10.2307/2369373. Google Scholar [24] K. Nyberg, Differentially uniform mappings for cryptography,, in Adv. Crypt.- Eurocrypt '93, (1994), 55. 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, (2009), 169. 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. 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, (2010), 23. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (2012), 27. 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, (2008), 388. 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, (2010), 41. doi: 10.1090/conm/521/10272. Google Scholar [16] W. Fulton, Algebraic Curves,, Benjamin, (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. 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. 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, (1993), 180. 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. doi: 10.1016/j.ffa.2007.04.004. Google Scholar [22] E. Leducq, Autour des Xodes de Reed-Muller Généralisés,, Ph.D thesis, (2011). Google Scholar [23] E. Lucas, Théorie des fonctions numériques simplement périodiques,, Amer. J. Math., 1 (1878), 197. doi: 10.2307/2369373. Google Scholar [24] K. Nyberg, Differentially uniform mappings for cryptography,, in Adv. Crypt.- Eurocrypt '93, (1994), 55. 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, (2009), 169. 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. 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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