# American Institute of Mathematical Sciences

doi: 10.3934/amc.2021048
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## Vandermonde sets, hyperovals and Niho bent functions

 1 UAE University, PO Box 15551, Al Ain, UAE, and Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan 2 UAE University, PO Box 15551, Al Ain, UAE

* Corresponding author: Kanat Abdukhalikov

Received  June 2021 Revised  September 2021 Early access October 2021

We consider relationships between Vandermonde sets and hyperovals. Hyperovals are Vandermonde sets, but in general, Vandermonde sets are not hyperovals. We give necessary and sufficient conditions for a Vandermonde set to be a hyperoval in terms of power sums. Therefore, we provide purely algebraic criteria for the existence of hyperovals. Furthermore, we give necessary and sufficient conditions for the existence of hyperovals in terms of $g$-functions, which can be considered as an analog of Glynn's Theorem for $o$-polynomials. We also get some important applications to Niho bent functions.

Citation: Kanat Abdukhalikov, Duy Ho. Vandermonde sets, hyperovals and Niho bent functions. Advances in Mathematics of Communications, doi: 10.3934/amc.2021048
##### References:
 [1] K. Abdukhalikov, Bent functions and line ovals, Finite Fields Appl., 47 (2017), 94-124.  doi: 10.1016/j.ffa.2017.06.002. [2] K. Abdukhalikov, Hyperovals and bent functions, European J. Combin., 79 (2019), 123-139.  doi: 10.1016/j.ejc.2019.01.003. [3] K. Abdukhalikov, Short description of the Lunelli-Sce hyperoval and its automorphism group, J. Geom., 110 (2019), 8pp. doi: 10.1007/s00022-019-0509-8. [4] K. Abdukhalikov, Equivalence classes of Niho bent functions, Des. Codes Cryptogr., 89 (2021), 1509-1534.  doi: 10.1007/s10623-021-00885-5. [5] S. Ball, Polynomials in finite geometries, Surveys in Combinatorics, 1999, (Canterbury), London Math. Soc. Lecture Note Ser., 267, Cambridge Univ. Press, Cambridge, 1999, 17–35. [6] S. Ball and M. Lavrauw, Arcs in finite projective spaces, EMS Surv. Math. Sci., 6 (2019), 133-172.  doi: 10.4171/emss/33. [7] L. Budaghyan, A. Kholosha, C. Carlet and T. Helleseth, Univariate Niho bent functions from $o$-polynomials, IEEE Trans. Inform. Theory, 62 (2016), 2254-2265.  doi: 10.1109/TIT.2016.2530083. [8] C. Carlet and S. Mesnager, On Dillon's class $H$ of bent functions, Niho bent functions and $o$-polynomials, J. Combin. Theory Ser. A, 118 (2011), 2392-2410.  doi: 10.1016/j.jcta.2011.06.005. [9] C. Carlet, T. Helleseth, A. Kholosha and S. Mesnager, On the duals of bent functions with $2^r$ Niho exponents, IEEE International Symposium on Information Theory, (2011), 703–707. [10] C. Carlet and S. Mesnager, Four decades of research on bent functions, Des. Codes Cryptogr., 78 (2016), 5-50.  doi: 10.1007/s10623-015-0145-8. [11] W. Cherowitzo, Hyperovals in Desarguesian planes of even order, Combinatorics '86 (Trento, 1986), Ann. Discrete Math., 37 (1988), 87-94.  doi: 10.1016/S0167-5060(08)70228-0. [12] W. Cherowitzo, Hyperovals in Desarguesian planes: An update, Discrete Mathematics, 155 (1996), 31-38.  doi: 10.1016/0012-365X(94)00367-R. [13] P. A. DeOrsey, Hyperovals and Cyclotomic Sets in $AG(2, q)$, , Ph.D Thesis, University of Colorado at Denver, 2015. [14] H. Dobbertin, G. Leander, A. Canteaut, C. Carlet, P. Felke and P. Gaborit, Construction of bent functions via Niho power functions, J. Combin. Theory Ser. A, 113 (2006), 779-798.  doi: 10.1016/j.jcta.2005.07.009. [15] J. C. Fisher and B. Schmidt, Finite Fourier series and ovals in PG$(2, 2^h)$, J. Aust. Math. Soc., 81 (2006), 21-34.  doi: 10.1017/S1446788700014610. [16] A. Gács and Z. Weiner, On $(q+t, t)$-arcs of type $(0, 2, t)$, Des. Codes Cryptogr., 29 (2003), 131-139.  doi: 10.1023/A:1024152424893. [17] G. Glynn, A condition for the existence of ovals in $\rm{PG}$(2, q), $q$ even, Geom. Dedicata, 32 (1989), 247-252.  doi: 10.1007/BF00147433. [18] T. Helleseth, A. Kholosha and S. Mesnager, Niho bent functions and Subiaco hyperovals, Theory and Applications of Finite Fields, Contemp. Math., Amer. Math. Soc., Providence, RI, 579 (2012), 91-101.  doi: 10.1090/conm/579/11522. [19] J. W. P. Hirschfeld, Projective Geometries Over Finite Fields, 2$^nd$ edition, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. [20] J. W. P. Hirschfeld and L. Storme, The packing problem in statistics, coding theory and finite projective spaces: Update 2001, Finite Geometries, Dev. Math., 3, Kluwer Acad. Publ., Dordrecht, 2001,201–246. doi: 10.1007/978-1-4613-0283-4_13. [21] R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, 1997. [22] C. M. O'Keefe and T. Penttila, Polynomials for hyperovals of Desarguesian planes, J. Austral. Math. Soc. Ser. A, 51 (1991), 436-447.  doi: 10.1017/S1446788700034601. [23] P. Sziklai and M. Takáts, Vandermonde sets and super-Vandermonde sets, Finite Fields Appl., 14 (2008), 1056-1067.  doi: 10.1016/j.ffa.2008.06.004. [24] P. Sziklai, Polynomials in finite geometry, https://web.cs.elte.hu/sziklai/polynom/poly08feb.pdf.

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##### References:
 [1] K. Abdukhalikov, Bent functions and line ovals, Finite Fields Appl., 47 (2017), 94-124.  doi: 10.1016/j.ffa.2017.06.002. [2] K. Abdukhalikov, Hyperovals and bent functions, European J. Combin., 79 (2019), 123-139.  doi: 10.1016/j.ejc.2019.01.003. [3] K. Abdukhalikov, Short description of the Lunelli-Sce hyperoval and its automorphism group, J. Geom., 110 (2019), 8pp. doi: 10.1007/s00022-019-0509-8. [4] K. Abdukhalikov, Equivalence classes of Niho bent functions, Des. Codes Cryptogr., 89 (2021), 1509-1534.  doi: 10.1007/s10623-021-00885-5. [5] S. Ball, Polynomials in finite geometries, Surveys in Combinatorics, 1999, (Canterbury), London Math. Soc. Lecture Note Ser., 267, Cambridge Univ. Press, Cambridge, 1999, 17–35. [6] S. Ball and M. Lavrauw, Arcs in finite projective spaces, EMS Surv. Math. Sci., 6 (2019), 133-172.  doi: 10.4171/emss/33. [7] L. Budaghyan, A. Kholosha, C. Carlet and T. Helleseth, Univariate Niho bent functions from $o$-polynomials, IEEE Trans. Inform. Theory, 62 (2016), 2254-2265.  doi: 10.1109/TIT.2016.2530083. [8] C. Carlet and S. Mesnager, On Dillon's class $H$ of bent functions, Niho bent functions and $o$-polynomials, J. Combin. Theory Ser. A, 118 (2011), 2392-2410.  doi: 10.1016/j.jcta.2011.06.005. [9] C. Carlet, T. Helleseth, A. Kholosha and S. Mesnager, On the duals of bent functions with $2^r$ Niho exponents, IEEE International Symposium on Information Theory, (2011), 703–707. [10] C. Carlet and S. Mesnager, Four decades of research on bent functions, Des. Codes Cryptogr., 78 (2016), 5-50.  doi: 10.1007/s10623-015-0145-8. [11] W. Cherowitzo, Hyperovals in Desarguesian planes of even order, Combinatorics '86 (Trento, 1986), Ann. Discrete Math., 37 (1988), 87-94.  doi: 10.1016/S0167-5060(08)70228-0. [12] W. Cherowitzo, Hyperovals in Desarguesian planes: An update, Discrete Mathematics, 155 (1996), 31-38.  doi: 10.1016/0012-365X(94)00367-R. [13] P. A. DeOrsey, Hyperovals and Cyclotomic Sets in $AG(2, q)$, , Ph.D Thesis, University of Colorado at Denver, 2015. [14] H. Dobbertin, G. Leander, A. Canteaut, C. Carlet, P. Felke and P. Gaborit, Construction of bent functions via Niho power functions, J. Combin. Theory Ser. A, 113 (2006), 779-798.  doi: 10.1016/j.jcta.2005.07.009. [15] J. C. Fisher and B. Schmidt, Finite Fourier series and ovals in PG$(2, 2^h)$, J. Aust. Math. Soc., 81 (2006), 21-34.  doi: 10.1017/S1446788700014610. [16] A. Gács and Z. Weiner, On $(q+t, t)$-arcs of type $(0, 2, t)$, Des. Codes Cryptogr., 29 (2003), 131-139.  doi: 10.1023/A:1024152424893. [17] G. Glynn, A condition for the existence of ovals in $\rm{PG}$(2, q), $q$ even, Geom. Dedicata, 32 (1989), 247-252.  doi: 10.1007/BF00147433. [18] T. Helleseth, A. Kholosha and S. Mesnager, Niho bent functions and Subiaco hyperovals, Theory and Applications of Finite Fields, Contemp. Math., Amer. Math. Soc., Providence, RI, 579 (2012), 91-101.  doi: 10.1090/conm/579/11522. [19] J. W. P. Hirschfeld, Projective Geometries Over Finite Fields, 2$^nd$ edition, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. [20] J. W. P. Hirschfeld and L. Storme, The packing problem in statistics, coding theory and finite projective spaces: Update 2001, Finite Geometries, Dev. Math., 3, Kluwer Acad. Publ., Dordrecht, 2001,201–246. doi: 10.1007/978-1-4613-0283-4_13. [21] R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, 1997. [22] C. M. O'Keefe and T. Penttila, Polynomials for hyperovals of Desarguesian planes, J. Austral. Math. Soc. Ser. A, 51 (1991), 436-447.  doi: 10.1017/S1446788700034601. [23] P. Sziklai and M. Takáts, Vandermonde sets and super-Vandermonde sets, Finite Fields Appl., 14 (2008), 1056-1067.  doi: 10.1016/j.ffa.2008.06.004. [24] P. Sziklai, Polynomials in finite geometry, https://web.cs.elte.hu/sziklai/polynom/poly08feb.pdf.
Set $\mathcal{D}$ in small fields
 $q$ Elements in $\mathcal{D}$ $\mid \mathcal{D} \mid$ 4 1 1 8 1, 3, 5 3 16 1, 3, 5, 7, 9, 11, 13, 37 8 32 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 69, 73, 77, 85, 89,147 21 64 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61,133,137,141,145,149,153,157,165,169,173,177,181,185,275,281,283,291,297,299,307,313,409,425,661 55 128 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99,101,103,105,107,109,111,113,115,117,119,121,123,125,261,265,269,273,277,281,285,289,293,297,301,305,309,313,317,325,329,333,337,341,345,349,353,357,361,365,369,373,377,529,531,537,539,547,553,555,561,563,569,571,579,585,587,593,595,601,603,611,617,619,625,627,633,785,793,809,817,825,841,849,857,873,881, 1093, 1095, 1107, 1109, 1111, 1123, 1125, 1127, 1139, 1141, 1301, 1317, 1333, 1365, 1381, 1587, 1619, 2341, 2349, 2381, 2405 147
 $q$ Elements in $\mathcal{D}$ $\mid \mathcal{D} \mid$ 4 1 1 8 1, 3, 5 3 16 1, 3, 5, 7, 9, 11, 13, 37 8 32 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 69, 73, 77, 85, 89,147 21 64 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61,133,137,141,145,149,153,157,165,169,173,177,181,185,275,281,283,291,297,299,307,313,409,425,661 55 128 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99,101,103,105,107,109,111,113,115,117,119,121,123,125,261,265,269,273,277,281,285,289,293,297,301,305,309,313,317,325,329,333,337,341,345,349,353,357,361,365,369,373,377,529,531,537,539,547,553,555,561,563,569,571,579,585,587,593,595,601,603,611,617,619,625,627,633,785,793,809,817,825,841,849,857,873,881, 1093, 1095, 1107, 1109, 1111, 1123, 1125, 1127, 1139, 1141, 1301, 1317, 1333, 1365, 1381, 1587, 1619, 2341, 2349, 2381, 2405 147
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