# Analysis of Low Probability of Detection Capability for Chaotic Standard Map-Based FH-OFDMA System
## 2 FH Patterns for FH-OFDMA System
In addition, receivers of the FH-OFDMA systems using the CSM and cat map-based FH patterns can start parsing modulated data after receiving \(N_{L}\) symbol buffers. Now, we show the SBS-based FH pattern and its FH pattern indexes of \(x\)-axis and \(y\)-axis. In this study, we choose a formula of the Latin squares-based FH pattern as one of the SBS-based FH patterns [13]. We can obtain the SBS-based FH pattern of the \(j\)-th data for the \(k\)-th user using Latin squares as  
\[ \begin{cases}L_{\alpha,x}^{k}(j)=j\\ L_{\alpha,y}^{k}(j)=\big{(}(k-j)\alpha^{-1}\big{)}\text{mod}\,N_{L}\end{cases} , \tag{3}\]  
where \(\alpha\) is the seed number of the FH pattern for \(\alpha=2,\ldots,N_{L}-1\). We introduce the cat map-based FH pattern using block symmetric ciphers [13]. In the cat map-based FH pattern, the generated symbols appear to be hopped in both time and frequency domains, since the transmitted symbol of each user appears at each period once, but the transmission order is mixed. The cat map-based FH pattern of the \(j\)-th data for the \(k\)-th user can be obtained by  
\[ \begin{bmatrix}T_{\alpha,x}^{k}(j)\\ T_{\alpha,y}^{k}(j)\end{bmatrix} = \begin{bmatrix}b_{11}&b_{12}\\ b_{21}&b_{22}\end{bmatrix}\begin{bmatrix}c_{\alpha}^{k}(j)\\ j\end{bmatrix} (\text{mod}\,\,N_{L}), \tag{4}\]  
where \(b_{11}\), \(b_{12}\), \(b_{21}\), and \(b_{22}\) are positive integers satisfying relations \(b_{11}\times b_{22}-b_{12}\times b_{21}=1\) and \(\alpha b_{12}>b_{11}\). The cluster index of \(c_{\alpha}^{k}(j)\) of the \(j\)-th data for the \(k\)-th user using a seed number of \(\alpha\) is given by  
\[c_{\alpha}^{k}(j)=\Big{(}(k-j)\alpha^{-1}\Big{)}\text{mod}\,N_{L},j=0,\ldots,N _{L}-1, \tag{5}\]  
where \(j\) denotes the data index of consecutive points \((i,j)\) for which \(L_{\alpha}(i,j)=k\) and \(L_{\alpha}(i,j)=(\alpha i+j)\) mod \(N_{L}\), \(i,j=0,\ldots,N_{L}-1\)[13]. The CSM is widely used to design block symmetric ciphers for LPD capability. Beside the cat map-based FH pattern, the symbols generated by the CSM-based FH pattern are hopped in both time and frequency domains, and the transmitted sequences are mixed. Based on the complexity of detection probability, the CSM-based FH pattern provides greater LPD capability than the cat map and SBS-based FH patterns. The CSM-based FH pattern of the \(j\)-th data for the \(k\)-th user can be obtained by  
Figure 1: Contiguous frequency hopping orthogonal frequency division multiplexing (FH-OFDM) symbol structure for spectrum sensing. \[ \begin{cases}C^{k}_{\alpha,x}(j)=\Big{(}c^{k}_{\alpha}(j)+j+r_{x}+r_{y}\Big{)} \text{mod}\ N_{L}\\ C^{k}_{\alpha,y}(j)=\Big{(}j+r_{y}+\Big{[}K_{\text{C}}\sin\frac{C^{k}_{\alpha,x} (j)N_{L}}{2\pi}\Big{]}\Big{)}\text{mod}\ N_{L}\end{cases} , \tag{6}\]  
where the cluster index of \(c^{k}_{\alpha}(j)\) of the \(j\)-th data for the \(k\)-th user using a seed number \(\alpha\) can be obtained by (5). \(r_{x}\) and \(r_{y}\) are constant integers from 0 to \(N_{L}-1\), and \(K_{\text{C}}\) is a positive integer. \([x]\) is the nearest integer to \(x\). Figure 2 shows an example of how to assign the three FH patterns in time and subcarrier domains, and we compare the three FH patterns, i.e., the SBS, cat map, and CSM-based patterns. To increase the detection probability from the point of view of eavesdropping detectors, we assume that the detectors receive a redundant number, \(N_{s}\), of contiguous FH-OFDMA frames consisting of \(N_{L}\) symbols. For the examples in Figure 2, we set \(N_{L}=5\), \(N_{\text{FT}}=20\), \(N_{s}=4\), number of transmitting users as 3, and indexes of the users as \(k=1\), 2, and 5. In Figure 2a, we show an example of an SBS-based FH pattern using Latin squares-based strategy for seed number, \(\alpha=4\).