On a Lower Bound for the Number of Bent Functions at the Minimum Distance from a Bent Function in the Maiorana

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On a Lower Bound for the Number of Bent Functions at the Minimum Distance from a Bent Function in the Maiorana–McFarland Class. / Bykov, D. A.; Kolomeec, N. A.

в: Journal of Applied and Industrial Mathematics, Том 17, № 3, 09.2023, стр. 507-520.

Результаты исследований: Научные публикации в периодических изданиях › статья › Рецензирование

BibTeX

@article{70c41a88c47e41fc9424280221ae440f,

title = "On a Lower Bound for the Number of Bent Functions at the Minimum Distance from a Bent Function in the Maiorana–McFarland Class",

abstract = "Bent functions at the minimum distance 2^n from a given bent function of 2n variables belonging to the Maiorana–McFarland class \mathcal {M}_{2n} are investigated. We provide a criterion for a function obtained using theaddition of the indicator of an n -dimensional affine subspace to a given bent function from \mathcal {M}_{2n} to be a bent function as well. In other words, all bent functions at theminimum distance from a Maiorana–McFarland bent function are characterized. It is shown thatthe lower bound 2^{2n+1}-2^n for the number of bent functions at the minimum distance from f \in \mathcal {M}_{2n} is not attained if the permutation used for constructing f is not an APN function. It is proved that for any prime n\geq 5 there exist functions in \mathcal {M}_{2n} for which this lower bound is accurate. Examples of such bent functions arefound. It is also established that the permutations of EA-equivalent functions in \mathcal {M}_{2n} are affinely equivalent if the second derivatives of at least one of thepermutations are not identically zero.",

keywords = "bent function, Boolean function, minimum distance, Maiorana–McFarland class, lowerbound, affine equivalence",

author = "Bykov, {D. A.} and Kolomeec, {N. A.}",

note = "This research was carried out within the framework of the state contract of the Sobolev Institute of Mathematics, project no. FWNF–2022–0018. Публикация для корректировки.",

year = "2023",

month = sep,

doi = "10.1134/S1990478923030055",

language = "English",

volume = "17",

pages = "507--520",

journal = "Journal of Applied and Industrial Mathematics",

issn = "1990-4789",

publisher = "Maik Nauka-Interperiodica Publishing",

number = "3",

}

RIS

TY - JOUR

T1 - On a Lower Bound for the Number of Bent Functions at the Minimum Distance from a Bent Function in the Maiorana–McFarland Class

AU - Bykov, D. A.

AU - Kolomeec, N. A.

N1 - This research was carried out within the framework of the state contract of the Sobolev Institute of Mathematics, project no. FWNF–2022–0018. Публикация для корректировки.

PY - 2023/9

Y1 - 2023/9

N2 - Bent functions at the minimum distance 2^n from a given bent function of 2n variables belonging to the Maiorana–McFarland class \mathcal {M}_{2n} are investigated. We provide a criterion for a function obtained using theaddition of the indicator of an n -dimensional affine subspace to a given bent function from \mathcal {M}_{2n} to be a bent function as well. In other words, all bent functions at theminimum distance from a Maiorana–McFarland bent function are characterized. It is shown thatthe lower bound 2^{2n+1}-2^n for the number of bent functions at the minimum distance from f \in \mathcal {M}_{2n} is not attained if the permutation used for constructing f is not an APN function. It is proved that for any prime n\geq 5 there exist functions in \mathcal {M}_{2n} for which this lower bound is accurate. Examples of such bent functions arefound. It is also established that the permutations of EA-equivalent functions in \mathcal {M}_{2n} are affinely equivalent if the second derivatives of at least one of thepermutations are not identically zero.

AB - Bent functions at the minimum distance 2^n from a given bent function of 2n variables belonging to the Maiorana–McFarland class \mathcal {M}_{2n} are investigated. We provide a criterion for a function obtained using theaddition of the indicator of an n -dimensional affine subspace to a given bent function from \mathcal {M}_{2n} to be a bent function as well. In other words, all bent functions at theminimum distance from a Maiorana–McFarland bent function are characterized. It is shown thatthe lower bound 2^{2n+1}-2^n for the number of bent functions at the minimum distance from f \in \mathcal {M}_{2n} is not attained if the permutation used for constructing f is not an APN function. It is proved that for any prime n\geq 5 there exist functions in \mathcal {M}_{2n} for which this lower bound is accurate. Examples of such bent functions arefound. It is also established that the permutations of EA-equivalent functions in \mathcal {M}_{2n} are affinely equivalent if the second derivatives of at least one of thepermutations are not identically zero.

KW - bent function, Boolean function, minimum distance, Maiorana–McFarland class, lowerbound, affine equivalence

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85175856956&origin=inward&txGid=0355e1a2916c307dccee65656a7e117a

UR - https://www.mendeley.com/catalogue/ee9354d3-beff-3c4a-9d60-3d32fba7ff21/

U2 - 10.1134/S1990478923030055

DO - 10.1134/S1990478923030055

M3 - Article

VL - 17

SP - 507

EP - 520

JO - Journal of Applied and Industrial Mathematics

JF - Journal of Applied and Industrial Mathematics

SN - 1990-4789

IS - 3

ER -

ID: 59553786

On a Lower Bound for the Number of Bent Functions at the Minimum Distance from a Bent Function in the Maiorana–McFarland Class

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