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On the q-bentness of Boolean functions

Abstract

For each non-constant q in the set of n-variable Boolean functions, the q-transform of a Boolean function f is related to the Hamming distances from f to the functions obtainable from q by nonsingular linear change of basis. Klapper conjectured that no Boolean function exists with its q-transform coefficients equal to \(\pm \, 2^{n/2}\) (such function is called q-bent) when q is non-affine balanced. In our early work, we only gave partial results to confirm this conjecture for small n. Here we prove thoroughly that the conjecture is true for all n by investigating the nonexistence of the partial difference sets in abelian groups with special parameters. We also introduce a new family of functions called \((\delta ,q)\)-bent functions, which give a measurement of q-bentness.

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Notes

  1. 1.

    If an \(h\in {\mathcal {B}}_n\), we say that the rank of h is the least positive integer r such that for some \(B\in GL_n\) the function \(h_B\) depends only on r variables.

  2. 2.

    In some papers G is a group with addition [3, 4], while in others G is a group with multiplication [9,10,11].

  3. 3.

    In this case, we use \(4\mu + 2^{n/2+1}\) in place of \(4\mu - 2^{n/2+1}\).

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Acknowledgements

The authors wish to thank Prof. Cunsheng Ding for some suggestions on the theory of difference sets. Thanks also go to the anonymous referees and the editor for their time and useful comments. The work was partially supported by the National Natural Science Foundation of China under grant No. 61772292 and by the Provincial Natural Science Foundation of Fujian under Grant No.  2018J01425. A. Klapper was partially supported by the National Science Foundation under Grant No. CNS-1420227. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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Correspondence to Zhixiong Chen.

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Communicated by C. Carlet.

Appendix: Example 1 re-visited

Appendix: Example 1 re-visited

In this appendix we give a proof for Example 1.

We write \(\mathrm {supp}(f)=\{\alpha _i : 1\le i\le 6\}\) with

$$\begin{aligned} \alpha _1=1100, ~ \alpha _2=1011, ~ \alpha _3=0111, ~ \alpha _4=0011,~ \alpha _5=1101, ~ \alpha _6=1110. \end{aligned}$$

We check that \(\alpha _1+\alpha _2+\alpha _3=0000\) and \(\alpha _4+\alpha _5+\alpha _6=0000.\) However, in the set \(\mathrm {supp}(q)\), we only have \(1100+1010+0110=0000\) and among vectors left there are no three vectors such that they are linearly dependent. Similar result holds in the set \(V_4\setminus (\mathrm {supp}(q)\cup \{0000\})\). This means that there are no \(A\in GL_4\) such that \(\mathrm {supp}(f_A)\subseteq \mathrm {supp}(q)\) or \(\mathrm {supp}(f_A)\subseteq V_4\setminus (\mathrm {supp}(q)\cup \{0000\})\). In other words, we always have

$$\begin{aligned} \mathrm {supp}(f_A)\cap \mathrm {supp}(q)\ne \emptyset \end{aligned}$$

and

$$\begin{aligned} \mathrm {supp}(f_A)\cap (V_4\setminus (\mathrm {supp}(q)\cup \{0000\}))\ne \emptyset \end{aligned}$$

for all \(A\in GL_4\). Hence we get

$$\begin{aligned} wt(f_A+q)\in \{4,6,8,10,12\}, \end{aligned}$$

from which we derive \(W_q(f)(A)\in \{0,\pm \, 4, \pm \,8\}\) for all \(A\in GL_4\). \(\square \)

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Chen, Z., Gu, T. & Klapper, A. On the q-bentness of Boolean functions. Des. Codes Cryptogr. 87, 163–171 (2019). https://doi.org/10.1007/s10623-018-0494-1

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Keywords

  • Boolean function
  • Walsh–Hadamard transform
  • Bent function
  • q-transform
  • q-bent function
  • Partial difference set

Mathematics Subject Classification

  • 94A60
  • 06E30
  • 05B10