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

Bent functions at the minimum distance from a given bent function of variables belonging to the Maiorana–McFarland class are investigated. We provide a criterion for a function obtained using the addition of the indicator of an -dimensional affine subspace to a given bent function from to be a bent...

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Bibliographic Details
Published in:Journal of applied and industrial mathematics 2023-09, Vol.17 (3), p.507-520
Main Authors: Bykov, D. A., Kolomeec, N. A.
Format: Article
Language:English
Subjects:
Codes
Equivalence
Lower bounds
Mathematical analysis
Mathematics
Mathematics and Statistics
Permutations
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Summary:Bent functions at the minimum distance from a given bent function of variables belonging to the Maiorana–McFarland class are investigated. We provide a criterion for a function obtained using the addition of the indicator of an -dimensional affine subspace to a given bent function from to be a bent function as well. In other words, all bent functions at the minimum distance from a Maiorana–McFarland bent function are characterized. It is shown that the lower bound for the number of bent functions at the minimum distance from is not attained if the permutation used for constructing is not an APN function. It is proved that for any prime there exist functions in for which this lower bound is accurate. Examples of such bent functions are found. It is also established that the permutations of EA-equivalent functions in are affinely equivalent if the second derivatives of at least one of the permutations are not identically zero.
ISSN:1990-4789
1990-4797
DOI:10.1134/S1990478923030055