Is this permutation secure? Let vector ${\bf d} \in \{ \pm 1 \}^n$ be the message we want to send. In my system, ${\bf d}$ is multiplied by an $n \times n$ Fourier matrix ${\bf F}$, as follows
$$ {\bf x} = {\bf F} {\bf d}  $$
where
$$ {\bf F} = 
 \begin{pmatrix}
  1 &amp; 1 &amp; 1 &amp; \cdots &amp; 1 \\
  1 &amp; e^{jw} &amp; e^{j2w}&amp;\cdots &amp; e^{j(n-1)w} \\
  \vdots  &amp; \vdots  &amp; \vdots &amp; \ddots &amp; \vdots \\
  1 &amp; e^{j(n-1)w} &amp;e^{j2(n-1)w}&amp; \cdots &amp; e^{j(n-1)(n-1)w} 
 \end{pmatrix}$$
We perform secret permutation $P$ for ${\bf x}$ provided that only the legitimate parties know the permutation and $P$ changes for every transmission.

Does multiplying by ${\bf F}$ help to diffuse?

Is this actually breakable?

If so, what kind of cryptanalysis can be used?
This is problematic as stated. You need to specify a probability distribution for that complex matrix, but the complex field is infinite. This then implies that you need to also carefully define some detection/quantization mechanism.
So, why complex numbers?