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.  Google Scholar

[2]

K. Abdukhalikov, Hyperovals and bent functions, European J. Combin., 79 (2019), 123-139.  doi: 10.1016/j.ejc.2019.01.003.  Google Scholar

[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.  Google Scholar

[4]

K. Abdukhalikov, Equivalence classes of Niho bent functions, Des. Codes Cryptogr., 89 (2021), 1509-1534.  doi: 10.1007/s10623-021-00885-5.  Google Scholar

[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.  Google Scholar

[6]

S. Ball and M. Lavrauw, Arcs in finite projective spaces, EMS Surv. Math. Sci., 6 (2019), 133-172.  doi: 10.4171/emss/33.  Google Scholar

[7]

L. BudaghyanA. KholoshaC. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

W. Cherowitzo, Hyperovals in Desarguesian planes: An update, Discrete Mathematics, 155 (1996), 31-38.  doi: 10.1016/0012-365X(94)00367-R.  Google Scholar

[13]

P. A. DeOrsey, Hyperovals and Cyclotomic Sets in $AG(2, q)$, , Ph.D Thesis, University of Colorado at Denver, 2015.  Google Scholar

[14]

H. DobbertinG. LeanderA. CanteautC. CarletP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

T. HellesethA. 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.  Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[21] R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, 1997.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[24]

P. Sziklai, Polynomials in finite geometry, https://web.cs.elte.hu/sziklai/polynom/poly08feb.pdf. Google Scholar

show all references

References:
[1]

K. Abdukhalikov, Bent functions and line ovals, Finite Fields Appl., 47 (2017), 94-124.  doi: 10.1016/j.ffa.2017.06.002.  Google Scholar

[2]

K. Abdukhalikov, Hyperovals and bent functions, European J. Combin., 79 (2019), 123-139.  doi: 10.1016/j.ejc.2019.01.003.  Google Scholar

[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.  Google Scholar

[4]

K. Abdukhalikov, Equivalence classes of Niho bent functions, Des. Codes Cryptogr., 89 (2021), 1509-1534.  doi: 10.1007/s10623-021-00885-5.  Google Scholar

[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.  Google Scholar

[6]

S. Ball and M. Lavrauw, Arcs in finite projective spaces, EMS Surv. Math. Sci., 6 (2019), 133-172.  doi: 10.4171/emss/33.  Google Scholar

[7]

L. BudaghyanA. KholoshaC. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

W. Cherowitzo, Hyperovals in Desarguesian planes: An update, Discrete Mathematics, 155 (1996), 31-38.  doi: 10.1016/0012-365X(94)00367-R.  Google Scholar

[13]

P. A. DeOrsey, Hyperovals and Cyclotomic Sets in $AG(2, q)$, , Ph.D Thesis, University of Colorado at Denver, 2015.  Google Scholar

[14]

H. DobbertinG. LeanderA. CanteautC. CarletP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

T. HellesethA. 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.  Google Scholar

[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.   Google Scholar
[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.  Google Scholar

[21] R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, 1997.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[24]

P. Sziklai, Polynomials in finite geometry, https://web.cs.elte.hu/sziklai/polynom/poly08feb.pdf. Google Scholar

Table 1.  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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