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The graph of minimal distances of bent functions and its properties

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Abstract

A notion of the graph of minimal distances of bent functions is introduced. It is an undirected graph (V, E) where V is the set of all bent functions in 2k variables and \((f, g) \in E\) if the Hamming distance between f and g is equal to \(2^k\). It is shown that the maximum degree of the graph is equal to \(2^k (2^1 + 1) (2^2 + 1) \cdots (2^k + 1)\) and all its vertices of maximum degree are quadratic bent functions. It is obtained that the degree of a vertex from Maiorana—McFarland class is not less than \(2^{2k + 1} - 2^k\). It is proven that the graph is connected for \(2k = 2, 4, 6\), disconnected for \(2k \ge 10\) and its subgraph induced by all functions EA-equivalent to Maiorana—McFarland bent functions is connected.

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Acknowledgements

The work is supported by the Russian Foundation for Basic Research (Projects No. 15-07-01328, 15-31-20635), the Ministry of Education and Science of the Russian Federation and Project No. 0314-2015-0011.

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  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Nikolay Kolomeec

  2. Novosibirsk State University, Novosibirsk, Russia

    Nikolay Kolomeec

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  1. Nikolay Kolomeec
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Correspondence to Nikolay Kolomeec.

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Communicated by D. Jungnickel.

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Kolomeec, N. The graph of minimal distances of bent functions and its properties. Des. Codes Cryptogr. 85, 395–410 (2017). https://doi.org/10.1007/s10623-016-0306-4

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