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.

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 $A\in GL_4$. $\square$