# Distinction of Scrambled Linear Block Codes Based on Extraction of Correlation Features
## 2 Scrambled Linear Block Code Model  
The basis of the scrambling code is the linear-feedback shift register (LFSR), in which the self-synchronous scrambling code introduces an exclusive OR logic between the feedback logic output and the first-stage register, and the obtained result is used as the input of the register. The independent m-sequence generated by the LFSR is added to the information sequence, thereby generating the scrambling sequence. The self-synchronous scrambler and synchronous scrambler are shown in Figure 1a,b, respectively.  
An \(L\)-stage LFSR's feedback network is made up of L registers, such as the first-stage register and the second-stage register, and an \(L\)-th stage register from left to right. Each register can have two states, 0 and 1, and the operation \(\oplus\) means the addition of modulo-2. When a shift pulse is added, the output content of the previous moment is fed back to the first-level register, the content of each level register is shifted to the next level, and the current output \(s_{t}\) is generated by operation. Therefore, after giving the initial values, \(s_{0},s_{1},\ldots,s_{L-1}\), under the action of the shift pulse, the \(L\)-stage LFSR outputs a sequence \(\{\textbf{s}_{t}\}\), which satisfies the feedback logic  
\[s_{t}=c_{1}s_{t-1}\oplus c_{2}s_{t-2}\oplus\cdots\oplus c_{L}s_{t-L}, \tag{1}\]  
where \(c_{i}\in GF(2),1\leq i\leq L\). This sequence is called the LFSR sequence, and its linearity is mainly reflected in its feedback logic being linear. Any consecutive \(L\) term in a sequence of the \(L\)-stage LFSR is called the state of the sequence. \((s_{0},s_{1},\ldots,s_{L})\) is called the initial state [16].  
The self-synchronous scrambling process can be expressed as  
\[y_{k}=x_{k}\oplus\sum_{i=1}^{L}\oplus c_{i}y_{k-i}. \tag{2}\]  
Additionally, the synchronous scrambling process can be expressed as  
\[y_{k}=x_{k}\oplus\sum_{i=1}^{L}c_{i}s_{k-i}, \tag{3}\]  
where \(\sum\oplus\) means modulo-2 accumulation. \(c_{i}\in GF(2)\), \(i=1,2,\ldots,L\) is the feedback coefficient of the LFSR, \(GF(2)\) represents the 2-element domain, and the value of \(c_{i}\) is 0 or 1, \(c_{0}=c_{L}=1\). The generator polynomial of the scrambling code can be expressed as  
\[f(x)=1+c_{1}x+c_{2}x^{2}+\cdots+c_{L}x^{L}. \tag{4}\]  
Figure 1: Scrambler schematic. (**a**) Self-synchronous scrambler; (**b**) Synchronous scrambler.  
In actual communication system information transmission, most of the sources become unbiased sources through channel coding [17], so the signal model studied in this paper is shown in Figure 2[17; 18].  
The noise is Gaussian white noise with mean 0, variance \(\sigma^{2}\), and signal amplitude \(A\). The signal-to-noise ratio (SNR) is defined as  
\[\text{SNR}=10\cdot\log\Bigl{(}A^{2}/2\sigma^{2}\Bigr{)}. \tag{5}\]  
For the receiver, first, we intercept a sequence of code elements of a linear block code, and we only need to determine whether the sequence of the code elements is scrambled using LFSR. When scrambling is performed, it is further determined whether the code element sequence employs a self-synchronous scrambler as shown in Figure 1a or a synchronous scrambler as shown in Figure 1b. Further identification of the scrambling parameters is only possible when the above three types of code element sequences are distinguished. In the next section, we focus on how to effectively identify whether the intercepted information sequence is scrambled and what kind of scrambling method is used.