Is this permutation secure?

Question

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

$$ {\bf F} = \begin{pmatrix} 1 & 1 & 1 & \cdots & 1 \\ 1 & e^{jw} & e^{j2w}&\cdots & e^{j(n-1)w} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & e^{j(n-1)w} &e^{j2(n-1)w}& \cdots & e^{j(n-1)(n-1)w} \end{pmatrix}$$ We perform secret permutation $P$ for ${\bf x}$ provided that only the legitimate parties know the permutation and $P$ changes for every transmission.

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

2. Is this actually breakable?

3. If so, what kind of cryptanalysis can be used?

Answer 1

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?

Answer 2

Multiplying by $F$ cannot help. It is publicly known, and easily invertible. Therefore an adversary can easily undo it, leaving them with simply the permuted inputs $\mathbf{Px}$.

Moreover, permuting the input cannot be IND-CPA secure. This is because permutation matrices leave norms invariant, meaning:

$$\lVert \mathbf{Px}\rVert_p = \lVert \mathbf{x}\rVert_p$$ For any $p$-norm (including the "$\ell_0$-norm", meaning the Hamming weight). This means that frequency analysis can be used to attack enciphering via solely permuting the input. In general these ciphers are known as transposition ciphers.