On the exceptionality of rational APN functions. (English) Zbl 1530.11097
A function \(f\) on the finite field \(\mathbb{F}_q\), \(q=2^n\), is called an APN function (almost perfect nonlinear) if for all nonzero \(a\in\mathbb{F}_q\), the derivative \(D_af(x) = f(x+a)+f(x)\) is \(2\)-to-\(1\). An APN function \(f\) (not necessarily represented as a polynomial) is called exceptional, if \(f\) is APN over infinitely many extensions \(\mathbb{F}_{q^m}\) of \(\mathbb{F}_q\). The authors consider functions, which can be represented by rational maps. As main result, they present non-existence results for exceptional APN functions given as \(\varphi = \frac{f}{g}\), \(f,g\in\mathbb{F}_q[x]\), \(\gcd(f,g) = 1\), \(g(x)\ne 0\) for all \(x\in\mathbb{F}_q\).
Reviewer: Wilfried Meidl (Klagenfurt)
References:
| [1] | Aubry, Y.; McGuire, G.; Rodier, F., A few more functions that are not APN infinitely often, finite fields theory and applications, Contemp. Math., 518, 23-31 (2010) · Zbl 1206.94025 · doi:10.1090/conm/518/10193 |
| [2] | Bartoli, D.; Zhou, Y., Exceptional scattered polynomials, J. Algebra, 509, 507-534 (2018) · Zbl 1393.51002 · doi:10.1016/j.jalgebra.2018.03.010 |
| [3] | Biham, E.; Shamir, A., Differential cryptanalysis of DES-like cryptosystems, finite fields: theory and applications, Contemp. Math., 4, 3-72 (1991) · Zbl 0729.68017 |
| [4] | Browning, KA; Dillon, J.; Kibler, R.; McQuistan, MT, APN polynomials and related codes, J. Combin. Inf. Syst. Sci., 34, 1-4, 135-159 (2009) · Zbl 1269.94035 |
| [5] | Budaghyan, L.; Carlet, C.; Pott, A., New classes of almost bent and almost perfect nonlinear polynomials, IEEE Tran. Inf. Theory, 52, 1141-1152 (2006) · Zbl 1177.94136 · doi:10.1109/TIT.2005.864481 |
| [6] | Budaghyan, L.; Carlet, C., Classes of quadratic APN trinomials and hexanomials and related structures, IEEE Tran. Inf. Theory, 54, 5, 2354-2357 (2008) · Zbl 1177.94134 · doi:10.1109/TIT.2008.920246 |
| [7] | Budaghyan, L.; Calderini, M.; Carlet, C.; Coulter, RS; Villa, I., Constructing APN functions through isotopic shifts, IEEE Tran. Inf. Theory, 66, 8, 5299-5309 (2020) · Zbl 1446.94085 · doi:10.1109/TIT.2020.2974471 |
| [8] | Budaghyan, L.; Calderini, M.; Villa, I., On equivalence between known families of quadratic APN functions, Finite Fields Appl., 66 (2020) · Zbl 1458.94217 · doi:10.1016/j.ffa.2020.101704 |
| [9] | Budaghyan, L.; Calderini, M.; Carlet, C.; Coulter, R.; Villa, I., Generalized isotopic shift construction for APN functions, Des. Codes Cryptogr., 89, 1, 19-32 (2021) · Zbl 1458.94201 · doi:10.1007/s10623-020-00803-1 |
| [10] | Cafure, A.; Matera, G., Improved explicit estimates on the number of solutions of equations over a finite field, Finite Fields Appl., 12, 2, 155-185 (2006) · Zbl 1163.11329 · doi:10.1016/j.ffa.2005.03.003 |
| [11] | Carlet, C., Boolean Functions for Cryptography and Coding Theory (2021), Cambridge: Cambridge University Press, Cambridge · Zbl 1512.94001 |
| [12] | Carlet, C.; Charpin, P.; Zinoviev, V., Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Cryptogr., 15, 125-156 (1998) · Zbl 0938.94011 · doi:10.1023/A:1008344232130 |
| [13] | Coulter, RS; Henderson, M., A class of functions and their application in constructing semi-biplanes and association schemes, Discret. Math., 202, 1-3, 21-31 (1999) · Zbl 0938.51011 · doi:10.1016/S0012-365X(98)00345-8 |
| [14] | Delgado, M., The state of the art on the conjecture of exceptional APN functions, Note Mat., 37, 1, 41-51 (2017) · Zbl 1407.11141 |
| [15] | Dembowski, P.; Ostrom, TG, Planes of order \(n\) with collineation groups of order \(n^2\), Math. Z., 103, 3, 239-258 (1968) · Zbl 0163.42402 · doi:10.1007/BF01111042 |
| [16] | Dempwolff, U.; Edel, Y., Dimensional dual hyperovals and APN functions with translation groups, J. Algebraic Combin., 39, 2, 457-496 (2014) · Zbl 1292.05068 · doi:10.1007/s10801-013-0454-9 |
| [17] | Fine, NJ, Binomial coefficients modulo a prime, Am. Math. Mon., 54, 589-592 (1947) · Zbl 0030.11102 · doi:10.2307/2304500 |
| [18] | Fulton W.: Algebraic Curves. Advanced Book Classics. Addison-Wesley Publishing Company Advanced Book Program, Redwood City (1989). · Zbl 0681.14011 |
| [19] | Ghorpade, S.; Lachaud, G., Etale cohomology, Lefschetz theorem and number of points of singular varieties over finite fields, Mosc. Math. J., 2, 589-631 (2002) · Zbl 1101.14017 · doi:10.17323/1609-4514-2002-2-3-589-631 |
| [20] | Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics. Springer, New York. (1977) · Zbl 0367.14001 |
| [21] | Hernando, F.; McGuire, G., Proof of a conjecture on the sequence of exceptional numbers classifying cyclic codes and APN functions, J. Algebra, 343, 78-92 (2011) · Zbl 1244.94046 · doi:10.1016/j.jalgebra.2011.06.019 |
| [22] | Hirschfeld, J.W.P., Korchmáros, G., Torres, F.: Algebraic Curves over a Finite Field. Princeton Series in Applied Mathematics, Princeton (2008) · Zbl 1200.11042 |
| [23] | Janwa, H.; McGuire, G.; Wilson, R., Double-error correcting cyclic codes and absolutely irreducible polynomials over \(GF(2)\), J. Algebra, 178, 665-676 (1995) · Zbl 0853.94021 · doi:10.1006/jabr.1995.1372 |
| [24] | Jedlicka, D., APN monomials over \(GF(2^n)\) for infinitely many \(n\), Finite Fields Appl., 13, 1006-1028 (2007) · Zbl 1133.11068 · doi:10.1016/j.ffa.2007.04.004 |
| [25] | Lang, S.; Weil, A., Number of points of varieties in finite fields, Am. J. Math., 76, 819-827 (1954) · Zbl 0058.27202 · doi:10.2307/2372655 |
| [26] | Mullen, GL; Panario, D., Handbook of Finite Fields (2013), Boca Raton: Chapman and Hall/CRC, Boca Raton · Zbl 1319.11001 · doi:10.1201/b15006 |
| [27] | Nyberg K.: Differentially uniform mappings for cryptography. In: Advances in Cryptography, EUROCRYPT’93. Lecture Notes in Computer Science, vol. 765, pp. 55-64. Springer, New York (1994). · Zbl 0951.94510 |
| [28] | Pott, A., Almost perfect and planar functions, Des. Codes Cryptogr., 78, 1, 141-195 (2016) · Zbl 1351.51004 · doi:10.1007/s10623-015-0151-x |
| [29] | Rodier, F., Bornes sur le degré des polynômes presque parfaitement non-linéaires, Contemp. Math., 487, 169-181 (2009) · Zbl 1257.94033 · doi:10.1090/conm/487/09531 |
| [30] | Schmidt, K-U; Zhou, Y., Planar functions over fields of characteristic two, J. Algebraic Combin., 40, 503-526 (2014) · Zbl 1319.51008 · doi:10.1007/s10801-013-0496-z |
| [31] | Taniguchi, H., On some quadratic APN functions, Des. Codes Cryptogr., 87, 9, 1973-1983 (2019) · Zbl 1419.11138 · doi:10.1007/s10623-018-00598-2 |
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