# Underlying Security Transmission Design for Orthogonal Time Frequency Space (OTFS) Modulation
## 3 2D Permutation-Aided OTFS Secure Transmission
### Design Of 2D Permutation-Aided OTFS Secure Transmission  
As shown in Figure 1, the proposed OTFS secure transmission is mainly divided into four stages: (1) security sequence generation, (2) interleaving replacement, (3) signal modulation, and (4) signal demodulation. * security sequence generation  
[PERSON] and [PERSON] detect the channel information through channel measurement and use the channel information as the input of the GOST method (detailed in Section 4)-based  
Figure 1: Flowchart for 2D permutation-aided OTFS secure transmission. secret key generation algorithm to generate the index sequence (secret key) that controls the permutation, denoted by \(D_{x}\), \(D_{y}\), \(D_{z}\), \(D_{w}\). * interleaving replacement According to the index sequence, row permutation and column permutation are performed on the IFFT matrix and the FFT matrix, respectively, as follows: \[ \begin{split} F_{M\ast M}&=\frac{1}{M}e^{-2\pi( \pi-1)(\beta-1)/M}\\ &=\left[w_{0}^{T},w_{1}^{T},\cdots,w_{M-1}^{T}\right],\end{split} \] (5) where \(\pi\) and \(\beta\) are the row index and column index of the standard FFT matrix, and \(w_{0}^{T},w_{1}^{T},\cdots,w_{M-1}^{T}\) are the row vectors. The index vectors \(D_{x}\) and \(D_{y}\) are used to replace the rows and columns of the standard FFT matrix. The new FFT matrix can be obtained by the following two steps. During the first step, the rows of the FFT matrix are permutated by the index vectors \(D_{x}\). The new FFT matrix can be expressed as \[ \begin{split} F_{M\ast M}^{\prime}&=\left[w_{D_{x} (0)}^{T},w_{D_{x}(1)}^{T},\cdots,w_{D_{x}(M-1)}^{T}\right]\\ &=[r_{0},r_{1},\cdots,r_{M-1}].\end{split} \] (6) During the second step, the columns of the FFT matrix are permutated by the index vectors \(D_{y}\). Then, the modified FFT matrix can be expressed as \[ \begin{split} F_{M\ast M}^{\prime\prime}&=\left[r _{D_{y}(0)}^{T},r_{D_{y}(1)}^{T},\cdots,r_{D_{y}(M-1)}^{T}\right]\\ &=\left[c_{0}^{T},c_{1}^{T},\cdots,c_{M-1}^{T}\right].\end{split} \] (7) To describe row permutation and column permutation more clearly, an example for a \(3\times 3\) matrix is given in Figure 2. Assuming that \(D_{x}=(3,1,2)\) and \(D_{y}=(3,1,2)\), the entire permutation process is described as follows. Firstly, perform row permutation by using the row index sequence \(D_{x}\). Secondly, perform column permutation by using the column index sequence \(D_{y}\). The modified IFFT matrix is obtained by repeating the above steps using the index vector \(D_{z}\), \(D_{w}\). * OTFS modulation The signal \(x[k,l]\) in the DD domain is converted into the signal \(X[n,m]\) in the TF domain, as follows: \[X[n,m]=P_{ISFFT}\left(\frac{1}{\sqrt{NM}}\sum_{k=0}^{N-1}\sum_{l=0}^{M-1}x[k,l ]e^{2\pi\left(\frac{n}{N}-\frac{m}{M}\right)}\right),\] (8) where \(n\in\{0,\ldots,N-1\}\), \(m\in\{0,\ldots,M-1\}\), \(P_{ISFFT}(\cdot)\) denotes the ISFFT transform after the FFT matrix and IFFT matrix permutate. Then, let us multiply the TF domain signal \(X[n,m]\) by a multiple \(U\) group of conversion matrices \(T\in\{T_{1},T_{2}\cdots,T_{U}\}\), as follows: \[X_{i}[n,m]=T_{i}X[n,m],\]  
Figure 2: Flowchart of matrix permutation. where \(i=1,2,\cdots U\), \(T_{i}\) is the \(i\)-th conversion matrix. Note that \(T_{i}=\left[t_{i},t_{i}^{(1)},\cdots,t_{i}^{(N-1)}\right]\). \(t_{i}^{(1)}\) is obtained by cyclically shifting \(t_{i}\), in which \(t_{i}=[p_{1},0,\cdots,0,p_{2},0,\cdots,0,p_{3},0,\cdots,0,p_{4},0,\)\(\cdots,0]\)[25], and then generating a multiple \(U\) group of time-domain candidate signals through the Heisenberg transform. After this, it selects the group with the smallest PAPR for transmission, as follows:  
\[s_{i}(t)=\sum_{n=0}^{N-1}\sum_{m=0}^{M-1}X_{i}[n,m]g_{tx}(t-nT)e^{jz\pi\triangle \beta(t-nT)}, \tag{10}\]  
\[s(t)=min(s_{1}(t),s_{2}(t),\cdots,s_{U}(t)), \tag{11}\]  
where \(g_{tx}(t)\) is the transmission pulse shape. This can be seen as a two-dimensional extension of the OFDM modulation transform.