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An asymptotic lower bound on the number of bent functions

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Abstract

A Boolean function f on n variables is said to be a bent function if the absolute value of all its Walsh coefficients is \(2^{n/2}\). Our main result is a new asymptotic lower bound on the number of Boolean bent functions. It is based on a modification of the Maiorana–McFarland family of bent functions and recent progress in the estimation of the number of transversals in latin squares and hypercubes. By-products of our proofs are the asymptotics of the logarithm of the numbers of partitions of the Boolean hypercube into 2-dimensional affine and linear subspaces.

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Acknowledgements

The work of V. N. Potapov and A. A. Taranenko was carried out within the framework of the state contract of the Sobolev Institute of Mathematics (project no. FWNF-2022-0017).

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    V. N. Potapov & A. A. Taranenko

  2. Lomonosov Moscow State University, Moscow, Russia

    Yu. V. Tarannikov

  3. Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia

    Yu. V. Tarannikov

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  1. V. N. Potapov
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  2. A. A. Taranenko
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  3. Yu. V. Tarannikov
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Correspondence to A. A. Taranenko.

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Potapov, V.N., Taranenko, A.A. & Tarannikov, Y.V. An asymptotic lower bound on the number of bent functions. Des. Codes Cryptogr. 92, 639–651 (2024). https://doi.org/10.1007/s10623-023-01239-z

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  • DOI: https://doi.org/10.1007/s10623-023-01239-z

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