# Anti-Jamming Low-Latency Channel Hopping Protocol for Cognitive Radio Networks
## 3 Preliminary and System Model  
We proposed a symmetric and without pre-shared secrets solution in this paper. In the considered CRN, there are \(N_{c}\) channels with the channel set \(C\), \(C=\{1,2,3,\ldots,N_{c}\}\). The SUs are randomly deployed across the network, and their traffic loads vary. SU \(a\) is identified by an \((n_{e}+1)\)-ary ID and \(k_{a}\) available channels. Let \(C_{a}\) be the available channel set of SU \(a\). There is at least one commonly available channel between any two SUs. Three kinds of jammers are randomly deployed in the CRN [17; 18; 20; 25; 27]:  
* Static jammers: They will consistently jam a fixed channel.
* Arbitrary jammers: They will jam a channel chosen randomly.
* Clever jammers: Each intelligent jammer will have a cognitive radio, and it will jam the channel most frequently used by a particular SU.  
The key variables used in this paper are presented in Table 2.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Variable** & **Definition** \\ \hline \(n_{e}\) & order of an ELP \\ \(N_{u}\) & number of SUs \\ \(N_{c}\) & number of channels \\ \(C\) & channel set \\ \(k_{i}\) & number of available channels of SU \(i\) \\ \(C_{i}\) & available channel set of SU \(i\) \\ \(1D_{i}\) & SU \(i\)’s \((n_{e}+1)\)-ary ID \\ \(ID_{i}^{n}\) & \(n\)th digit of SU \(i\)’s \((n_{e}+1)\)-ary ID \\ \(R_{i}\) & receiver set of SU \(i\) \\ \(R_{i}^{n}\) & \(n\)th receivers of SU \(i\) \\ \(M_{i}\) & SU \(i\)’s hopping matrix \\ \(M_{i}^{n}\) & \(n\)th column of \(M_{i}\) \\ \(FR_{i}^{n}\) & fixed \(R\) frame symbol of \(M_{i}^{n}\) \\ \(uFR_{i}^{n}\) & unfixed \(R\) frame symbol set in each column of \(M_{i}\) \\ \(uFR_{i,x}^{n}\) & unfixed \(R\) frame symbol set of \(M_{i}^{n}\) \\ \(uFR_{i,x}^{n}\) & \(x\)th unfixed \(R\) frame symbol of the \(n\)th column of \(M_{i}\) \\ \(e_{i}^{n}\) & ELP pattern of \(M_{i}^{n}\) \\ \(e_{i}^{n}\) & \(x\)th symbol in the ELP pattern of \(M_{i}^{n}\) \\ \(CHS_{i}\) & SU \(i\)’s channel hopping sequence \\ \(NT_{i}\) & number of \(T\) frames of SU \(i\) \\ \(NR_{i}\) & number of \(R\) frames of SU \(i\) \\ \(L_{c}\) & length of a column in a channel hopping matrix \\ \(\lambda_{i}^{m}\) & ID digit corresponding to \(M_{i}^{m}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Variable listing.  
In the considered CRN, time is divided into several equal-sized slots. An SU switches channels based on its channel hopping sequence. The proposed protocol uses the ELP sequence to generate the channel hopping sequence of each SU.  
* Extended Langford pairing  
A Langford pairing (LP) is a permutation of the sequence of \(2n\) symbols, where \(n\) is the order of the Langford pairing [34; 35]. In a LP of order \(n\), there are \(n\) distinct symbols, each of which appears twice. The distance between two identical symbols is equal to the symbol's value. An example LP of order 3 is \((2,3,1,2,1,3)\). Appending two zeros at the beginning of an LP produces an extended LP (ELP) [33]. The ELP of order 3 built from the above LP is \((0,0,2,3,1,2,1,3)\). Let \(n_{e}\) be the order of an ELP sequence. The rotated ELP sequences are considered to be different patterns. The different patterns produced by rotating the ELP sequence \((0,0,2,3,1,2,1,3)\) to the right by \(\sigma\) times with \(n_{e}=3\) is shown in Figure 1.