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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.
In: Journal of Applied and Industrial Mathematics, Vol. 17, No. 3, 09.2023, p. 507-520.Research output: Contribution to journal › Article › peer-review
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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