# Reconstruction of a Synchronous Scrambler Based on Average Check Conformity

Authors: Jiyuan Tan; Limin Zhang; Zhaogen Zhong
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Channel coding technology is indispensable in digital communication systems. In noncooperative contexts, the identification of the channel codes of the synchronous scrambler is essential. In this paper, a new algorithm that directly uses a soft decision sequence for blind reconstruction of the synchronous scrambler is proposed. First, considering imbalanced signal sources and the principle of scrambling and descrambling the synchronous scrambler, the error-containing equation of the synchronous scrambler is established. Second, the average check conformity is introduced to complete the check relationship detection. Then, based on the statistical characteristics of the average check conformity, the corresponding discrimination threshold is established, and the reconstruction of the feedback polynomial of the synchronous scrambler is completed by traversing the possible primitive polynomials. Finally, the verification equation is determined by the method of subsection optimization, which greatly reduces the number of initial states that need to be traversed; this is critical for scramblers with high-order feedback polynomials (i.e., when the order of the feedback polynomial is greater than 15). Simulations show that the algorithm can effectively reconstruct the synchronous scrambler under imbalanced signal sources. Moreover, the proposed algorithm offers improved performance with about 1–2 dB gain at low signal-to-noise ratios compared with existing methods, and its computational complexity is reasonable.


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## Body

## 1. Introduction

In communication systems, there are often many sequences with continuous zeroes or continuous ones in the transmitted data. To enhance the randomness of the transmitted signal and improve the reliability of signal transmission, the signal transmitter scrambles the transmitted signal, so synchronous scramblers are widely used in satellite communications, spread spectrum communications, and cryptographic systems [1, 2]. In noncooperative fields, a third party needs to rapidly identify and analyze the channel codes in an environment with a low signal-to-noise ratio (SNR). The recognition of channel codes using an intercepted sequence of noise-affected code words is commonly known as reverse analysis [3].Techniques for blind code identification have gained increasingly more attention from researchers, and several types of channel codes such as low-density parity-check (LDPC) [4, 5] and turbo [6, 7] codes have been studied. In this paper, our research goal is to reconstruct the synchronous scrambler, and we only use truncated soft decision sequences to accomplish the identification of feedback polynomials and initial states. We assume that the acquired sequence has been demodulated and frame synchronized successfully, which is achievable in practical engineering. In contrast to our solution, the existing algorithms for reconstructing the synchronous scrambler mainly use hard decision sequences (0 and 1).The blind recognition of the synchronous scrambler includes the reconstruction of the feedback polynomial and the identification of the initial state of the scrambler [8–11]. At present, the reconstruction methods of synchronous scrambler polynomials are mainly divided into two categories: reconstruction of polynomial based on source bias [12–15] and reconstruction of polynomial based on channel coding [13, 16–20]. In actual communication, the zeroes and ones in the source sequence are not balanced. The Walsh–Hadamard method based on the bias of the source uses the maximum established quasi-test to find the global optimal solution, which has become a classic and practical channel coding parameter recognition algorithm. In reference [21], the Walsh–Hadamard method was proposed to solve the two-element domain equations. The result obtained after Walsh transformation is the difference between the number of established and unsatisfied equations, and the solution of the equations is obtained based on this. In reference [14], the error-containing equations of the scrambler sequence were reconstructed on the basis of reference [21], and the feedback polynomial of the scrambler was reconstructed based on the Walsh–Hadamard method. However, the order of the scrambler needs to be estimated first, and the complexity of the algorithm increases exponentially with the order of the feedback polynomial. In reference [15], a scrambler reconstruction algorithm based on real-time detection was proposed, which continuously determines the feedback relationship during the Walsh transformation. This method reduces the computational complexity. In reference [16], a Walsh transform based method is used to solve the feedback polynomial recovery and initial state identification. In reference [17], a method based on Gaussian elimination and double detection is proposed, which solves the problem of blind identification of the synchronous scrambler in directly spread spectrum systems.For the estimation of the initial state of the synchronous scrambler, a method of fast correlation attack on convolutional codes was proposed to estimate the initial state [10]. With this method, the identification of the initial state is converted into the decoding of the convolutional code, and then, the Viterbi hard decision decoding algorithm is used for decoding to estimate the initial state of the synchronous scrambler. When the original sequence is recovered from the error, the convolutional code needs to be reconstructed, so the computational complexity of the algorithm is increased. In reference [14], an initial state estimation algorithm was proposed to improve the fast correlation attack. The performance of this algorithm is not limited by the number of taps of the feedback polynomial, but the influence of channel noise and modulation information is not considered. In reference [22], the dual code was used to complete the reconstruction of the feedback polynomial, but the estimation of the initial state in the case of noise was not analyzed in detail. References [23, 24] used the reconstructed feedback polynomial to randomly generate a synchronous scrambler sequence and compared it with the estimation. The cross-correlation operation was performed on the synchronous scrambler sequence, and the initial state estimation was completed according to the peak value of the correlation function and the correlation spectrum.Based on the above analysis, almost all current algorithms use hard decision sequences to reconstruct the synchronous scrambler. However, in practice, intercepted communication is usually a soft sequence containing the modulation method and channel noise [7, 25–28]. Because hard decision sequences result in a loss of channel reliability, their performance is not ideal at low SNR. Hence, there remains a need for a new methodology for synchronous scrambler reconstruction to improve the performance under low SNR.In this paper, we blindly recognize the binary synchronous scrambler for binary phase-shift keying (BPSK) signals over an additive white Gaussian noise (AWGN) channel. The proposed method is different from the hard decision algorithm because it uses a soft decision sequence instead of a hard decision sequence. The contributions of this paper are summarized as follows:(1)
We define the concept of the average check conformity [28], which measures the probability that the error-containing equation is established correctly. The average check conformity can be blindly computed by an intercepted soft decision sequence.(2)
We analyze the statistical characteristics of the average check conformity. Moreover, a threshold based on the minimum error criterion is obtained, which can quickly detect the irreducible factors of the feedback polynomial.(3)
We propose a method for finding the check equation and solving the error-containing equation when the order of the feedback polynomial is greater than 15, which greatly reduces the number of initial states that need to be traversed.The remainder of this paper is organized as follows. Section2 summarizes the principle and mathematical model of the synchronous scrambler. Section 3 establishes the average check conformity degree, solves the discriminant threshold, and analyzes the computational complexity of the algorithm for the recovery of the feedback polynomial. In Section 4, the verification equation is determined by the method of subsection optimization when the order of the feedback polynomial is greater than 15. In Section 5, our proposed method is evaluated by Monte Carlo simulations, and its performance is compared with that of the hard decision method. Finally, the conclusions are provided in Section 6.


## 2. Principle of the Synchronous Scrambler and the Walsh–Hadamard Transform Method

### 2.1. Principle of the Synchronous Scrambler

The scrambling process of the synchronous scrambler is based on a linear feedback shift register (LFSR), and the scrambling process is shown in Figure1.Figure 1
Scrambling process of the synchronous scrambler.An L-stage LFSR is composed of a weighted feedback network andL registers (Figure 1), which are called the first-stage register, second-stage register,..., and L-th stage register from left to right. Each register has two possible states, 0 and 1, and the operator ⊕ represents modulo-2 addition. When a shift pulse is added, the output content at the previous moment is fed back to the first-stage register, the content of each register is shifted to the next stage, and the current output st is generated by operation. Therefore, given initial values s0,s1,…,sL−1, under the action of the shift pulse, the L-stage LFSR will output a sequence st, which satisfies the linear recurrence relation (or feedback logic)(1)st=c1st−1⊕c2st−2⊕⋯⊕cLst−L,where ci∈GF2,i=1,2,…,L. This sequence is called the L-stage LFSR sequence, and its linearity is mainly reflected in its feedback logic being linear. Any consecutive L term in a sequence of L-stage LFSR is called a state of the sequence. The state s0,s1,…,sL−1 is called the initial state.The scrambling process can be expressed as(2)yk=xk⊕∑i=1L⊕cisk−i,where ∑⊕ means modulo-2 summation, and GF2 represents a binary domain. The value of ci is 0 or 1, and c0=cL=1. The feedback polynomial of the scrambler can be expressed as(3)fx=1+c1x+c2x2+⋯+cLxL.The mathematical model of the synchronous scrambler studied is shown in Figure2.Figure 2
Synchronous scrambler model. LFSR, linear feedback shift register.The model expression can be expressed as(4)rk=zk+nk,where nk is the channel noise, zk is the modulated scrambling sequence, and k represents the k-th moment. The biased sources are Pxk=0=1/2+ε, 0≤ε≤0.5. This article assumes that the received scrambler sequence adopts BPSK modulation, the signal amplitude is A, and the noise is Gaussian white noise with the mean value of zero and the variance of σ2. The SNR is defined as(5)SNR=10⋅log10A22σ2.For the receiver, the intercepted information sequence isrk, and the information sequence that needs to be acquired is xk. At this time, the feedback polynomial coefficient ci needs to be known. In actual communication, most of the LFSR orders used by the synchronous scrambler are 3–60 orders, generally three-term or five-term [9], and are the original polynomials [19]. The reconstruction of polynomials in this study is also based on this foundation.


### 2.2. Walsh–Hadamard Transform Method

From the scrambling of the synchronous scrambler in Figure1, it can be seen that the relationship between the input and output of each level of LFSR is as follows(6)sk=c1sk−1⊕c2sk−2⊕⋯⊕cL−1sk−L+1⊕cLsk−L,According to equation (5), the following equations are established(7)s1s2⋯sLsL+1s2s3⋯sL+1sL+2⋮⋮⋯⋮⋮sN−LsN−L+1⋯sN−1sNcLcL−1⋮c11=00⋮0,where N is the length of the intercepted scrambler sequence and L is the order of the feedback polynomial.Next,yN is substituted into equation (6) to obtain the error-containing equations of the synchronous scrambler(8)y1y2⋯yLyL+1y2y3⋯yL+1yL+2⋮⋮⋯⋮⋮yN−LyN−L+1⋯yN−1yNcLcL−1⋮c11=00⋮0,where yN represents the N-th symbol of the received scrambling code sequence.The Walsh–Hadamard transform method is an effective algorithm for solving the feedback polynomial of the maximum length sequence (m-sequence). The physical meaning of the transformed spectral coefficients can be expressed as the difference between the number of statistics that make the equation true and the number that does not hold [13]. Therefore, the algorithm will map the solution with the largest number of equations to the feedback relationship as the estimated value of the LFSR feedback polynomial. The following is a brief review of the feedback relationship estimation algorithm based on the Walsh–Hadamard transform.Hadamard matrixCm, m≥2 can be expressed as follows:(9)Cm=Cm−1Cm−1Cm−1−Cm−1.The initial matrix is given by(10)C1=111−1.The2m-dimensional row vector a, which is transformed into a 2m-dimensional row vector b by the Walsh–Hadamard transform, can be expressed as follows:(11)b=aCm.The algorithm for estimating the feedback relationship using the Walsh–Hadamard transform is as follows:(1)
Group the received sequenceyN according to the length L+1 and equation (7) with overlapping and continuous grouping and perform decimal conversion. The received data will be N−L decimal numbers.(2)
Construct a2L+1-dimensional vector a with these N−L decimal numbers, that is, mark the position of the i-th element in the vector as i−1, and the value of the i-th element is equal to i−1 in the N−L decimal numbers. The number of occurrences is 1≤i≤2L+1.(3)
The2L+1-dimensional vector a is generated by the Walsh–Hadamard transform to generate the 2L+1-dimensional vector b.(4)
Convert the binary representation of the position of the largest element except the first element in vectorb into a feedback relationship. Assuming that the largest element appears at the n-th position, the binary representation of n is n0n1…nL, where 2≤n≤2L+1 is a positive integer, n0n1…nL∈0,1. Then the estimated feedback relationship is(12)fx=nL+nL−1x+⋯+n0xL.The first element is excluded in Step4 because the first element corresponds to the all-zero feedback coefficient, which will cause the value of the first element to be equal to N−L, and the all-zero feedback coefficient is meaningless, so the all-zero solution is excluded. According to the physical meaning of the Walsh–Hadamard transform, the criterion for finding the largest element except the first element after transformation is called the criterion of maximum established number.The computational complexity of the algorithm increases exponentially with the order of the feedback polynomial. Furthermore, the performance of the algorithm under low SNR needs to be improved.


## 2.1. Principle of the Synchronous Scrambler

The scrambling process of the synchronous scrambler is based on a linear feedback shift register (LFSR), and the scrambling process is shown in Figure1.Figure 1
Scrambling process of the synchronous scrambler.An L-stage LFSR is composed of a weighted feedback network andL registers (Figure 1), which are called the first-stage register, second-stage register,..., and L-th stage register from left to right. Each register has two possible states, 0 and 1, and the operator ⊕ represents modulo-2 addition. When a shift pulse is added, the output content at the previous moment is fed back to the first-stage register, the content of each register is shifted to the next stage, and the current output st is generated by operation. Therefore, given initial values s0,s1,…,sL−1, under the action of the shift pulse, the L-stage LFSR will output a sequence st, which satisfies the linear recurrence relation (or feedback logic)(1)st=c1st−1⊕c2st−2⊕⋯⊕cLst−L,where ci∈GF2,i=1,2,…,L. This sequence is called the L-stage LFSR sequence, and its linearity is mainly reflected in its feedback logic being linear. Any consecutive L term in a sequence of L-stage LFSR is called a state of the sequence. The state s0,s1,…,sL−1 is called the initial state.The scrambling process can be expressed as(2)yk=xk⊕∑i=1L⊕cisk−i,where ∑⊕ means modulo-2 summation, and GF2 represents a binary domain. The value of ci is 0 or 1, and c0=cL=1. The feedback polynomial of the scrambler can be expressed as(3)fx=1+c1x+c2x2+⋯+cLxL.The mathematical model of the synchronous scrambler studied is shown in Figure2.Figure 2
Synchronous scrambler model. LFSR, linear feedback shift register.The model expression can be expressed as(4)rk=zk+nk,where nk is the channel noise, zk is the modulated scrambling sequence, and k represents the k-th moment. The biased sources are Pxk=0=1/2+ε, 0≤ε≤0.5. This article assumes that the received scrambler sequence adopts BPSK modulation, the signal amplitude is A, and the noise is Gaussian white noise with the mean value of zero and the variance of σ2. The SNR is defined as(5)SNR=10⋅log10A22σ2.For the receiver, the intercepted information sequence isrk, and the information sequence that needs to be acquired is xk. At this time, the feedback polynomial coefficient ci needs to be known. In actual communication, most of the LFSR orders used by the synchronous scrambler are 3–60 orders, generally three-term or five-term [9], and are the original polynomials [19]. The reconstruction of polynomials in this study is also based on this foundation.


## 2.2. Walsh–Hadamard Transform Method

From the scrambling of the synchronous scrambler in Figure1, it can be seen that the relationship between the input and output of each level of LFSR is as follows(6)sk=c1sk−1⊕c2sk−2⊕⋯⊕cL−1sk−L+1⊕cLsk−L,According to equation (5), the following equations are established(7)s1s2⋯sLsL+1s2s3⋯sL+1sL+2⋮⋮⋯⋮⋮sN−LsN−L+1⋯sN−1sNcLcL−1⋮c11=00⋮0,where N is the length of the intercepted scrambler sequence and L is the order of the feedback polynomial.Next,yN is substituted into equation (6) to obtain the error-containing equations of the synchronous scrambler(8)y1y2⋯yLyL+1y2y3⋯yL+1yL+2⋮⋮⋯⋮⋮yN−LyN−L+1⋯yN−1yNcLcL−1⋮c11=00⋮0,where yN represents the N-th symbol of the received scrambling code sequence.The Walsh–Hadamard transform method is an effective algorithm for solving the feedback polynomial of the maximum length sequence (m-sequence). The physical meaning of the transformed spectral coefficients can be expressed as the difference between the number of statistics that make the equation true and the number that does not hold [13]. Therefore, the algorithm will map the solution with the largest number of equations to the feedback relationship as the estimated value of the LFSR feedback polynomial. The following is a brief review of the feedback relationship estimation algorithm based on the Walsh–Hadamard transform.Hadamard matrixCm, m≥2 can be expressed as follows:(9)Cm=Cm−1Cm−1Cm−1−Cm−1.The initial matrix is given by(10)C1=111−1.The2m-dimensional row vector a, which is transformed into a 2m-dimensional row vector b by the Walsh–Hadamard transform, can be expressed as follows:(11)b=aCm.The algorithm for estimating the feedback relationship using the Walsh–Hadamard transform is as follows:(1)
Group the received sequenceyN according to the length L+1 and equation (7) with overlapping and continuous grouping and perform decimal conversion. The received data will be N−L decimal numbers.(2)
Construct a2L+1-dimensional vector a with these N−L decimal numbers, that is, mark the position of the i-th element in the vector as i−1, and the value of the i-th element is equal to i−1 in the N−L decimal numbers. The number of occurrences is 1≤i≤2L+1.(3)
The2L+1-dimensional vector a is generated by the Walsh–Hadamard transform to generate the 2L+1-dimensional vector b.(4)
Convert the binary representation of the position of the largest element except the first element in vectorb into a feedback relationship. Assuming that the largest element appears at the n-th position, the binary representation of n is n0n1…nL, where 2≤n≤2L+1 is a positive integer, n0n1…nL∈0,1. Then the estimated feedback relationship is(12)fx=nL+nL−1x+⋯+n0xL.The first element is excluded in Step4 because the first element corresponds to the all-zero feedback coefficient, which will cause the value of the first element to be equal to N−L, and the all-zero feedback coefficient is meaningless, so the all-zero solution is excluded. According to the physical meaning of the Walsh–Hadamard transform, the criterion for finding the largest element except the first element after transformation is called the criterion of maximum established number.The computational complexity of the algorithm increases exponentially with the order of the feedback polynomial. Furthermore, the performance of the algorithm under low SNR needs to be improved.


## 3. Reconstruction of Feedback Polynomial of the Synchronous Scrambler

### 3.1. Establishment of Average Check Conformity

For the convenience of explanation, thek line of the error-containing equation is listed and discussed separately, and the following error-containing check equation is obtained(13)∑l=1L+1yk,l⋅cL−l=0,where 1≤k≤N−L, and yk,l represents the k-th row and l-th column of matrix A in equation (7), where A is y1y2…yLyL+1y2y3…yL+1yL+2⋮⋮…⋮⋮yN−LyN−L+1…yN−1yN.To measure the possibility of the establishment of the equations, the establishment of the check conformity [28] is as follows:(14)Fk=∏l=0L1−2Pyk,l=1|rk,l⋅cL−l.In equation (14), yk,0,yk,1,…,yk,l is the hard decision sequence, the corresponding soft decision sequence is rk,0,rk,1,…,rk,l, and Pyk,l=1|rk,l is the probability that the hard decision sequence yk,l takes the value of one under the condition that the intercepted scrambler soft decision sequence is rk,l.When∑l=1L+1yk,l⋅cL−1=0,Pyk,l=|rk,l⋅cL−1<0.5, and the check conformity is mapped to 1; otherwise, it is mapped to −1. The posterior probability Pyk,l=1|rk,l in equation (14) is derived as follows:(15)Prk,l|yk,l=0=12πσe−rk,l+A2/2σ2,(16)Prk,l|yk,l=1=12πσe−rk,l−A2/2σ2.According to the Bayesian formula,(17)Pyk,l=1|rk,l=Prk,l|yk,l=1⋅Pyk,l=1Prk,l,and Prk,l can be expressed as(18)Prk,l=Prk,l|yk,l=1⋅Pyk,l=1+Prk,l|yk,l=0⋅Pyk,l=0.Since there is no prior information, letPyk,l=0=Pyk,l=1=0.5. Using equations (15)–(18), the following expressions can be obtained:(19)Pyk,l=1|rk,l=e2A⋅rk,l/σ2e2A⋅rk,l/σ2+1.By substituting equations (19) into (14), the expression of the check conformity can be obtained as(20)Fk=∏l=0L1−2⋅e2A⋅rk,l/σ2e2A⋅rk,l/σ2+1⋅cL−l.Considering all error-containing equations in equation (8), the average check conformity is obtained as follows:(21)F¯=1N−L∑k=1N−LFk.Equation (21) can be transformed into(22)F¯=1N−L∑k=1N−L∏l∈Ω1−e2A⋅rk,l/σ2e2A⋅rk,l/σ2+1,where Ω=L|cL=1.When the polynomial to be traversed is a feedback polynomial, the number of valid and nonconforming error-checking equations (8) is different due to the imbalanced ratio of zeroes to ones in the information source. At this time, the ratio of positive to negative values in Fk,k=0,1,…,N−L is not balanced. The larger the SNR, the closer the F¯ is to 1 or −1. Otherwise, the probability of the equation is 0.5. The value of F¯ hovers around 0.


### 3.2. Solution of the Discriminant Threshold

For the statistical characteristics of the average check conformity, the corresponding discriminant threshold needs to be introduced to complete the identification of the feedback polynomial.RememberG=cLcL−1⋯c11, and suppose the number of taps equal to one is m and the corresponding set of tap positions is G1=cLcn⋯c11. Currently, the scrambling hard decision sequence participating in the check equation is yk,1yk,L−n+1⋯yk,Lyk,L+1.When the check equation is established, the symbol 1 modulo-2 plus zero inyk,1yk,L−n+1⋯yk,Lyk,L+1 participating in the check equation is equal to zero. At this time, the number of corresponding symbol 1 is an even number, and all possible composition conditions can be expressed as(23)Vk,0=∑j=0m/2Cm2j,where  means round down, and Cmn=n⋅n−1Ln−m+1/m⋅m−1L1.The mean and variance of each possible situation are statistically averaged, and the results are as follows:(24)μk,0=∑j=0m/2Cm2jVk,0∫−∞∞1−e2A⋅rk,l/σ2e2A⋅rk,l/σ2+1⋅pr|y=1dx2j⋅∫−∞∞1−e2A⋅rk,l/σ2e2A⋅rk,l/σ2+1⋅pr|y=0dxm−2j,(25)σk,02=∑j=0m/2Cm2jVk,0⋅∫−∞∞1−e2A⋅rk,l/σ2e2A⋅rk,l/σ2+12⋅pr|y=1dx2j⋅∫−∞∞1−e2A⋅rk,l/σ2e2A⋅rk,l/σ2+12⋅pr|y=0dxm−2j−μk,02.Otherwise, the code element 1 modulo-2 plus one inyk,1yk,L−n+1⋯yk,Lyk,L+1 participating in the check equation is equal to one. At this time, the number of corresponding code element 1 is an odd number, and all possible composition conditions can be expressed as(26)Vk,1=∑j=0m/2Cm2j−1.The mean and variance of each possible situation are statistically averaged, and the results are as follows:(27)μk,1=∑j=0m/2Cm2j−1Vk,1∫−∞∞1−e2A⋅rk,l/σ2e2A⋅rk,l/σ2+1⋅pr|y=1dx2j−1⋅∫−∞∞1−e2A⋅rk,l/σ2e2A⋅rk,l/σ2+1⋅pr|y=0dxm−2j+1,(28)σk,12=∑j=0m/2Cm2j−1Vk,1⋅∫−∞∞1−e2A⋅rk,l/σ2e2A⋅rk,l/σ2+12⋅pr|y=1dx2j−1⋅∫−∞∞1−e2A⋅rk,l/σ2e2A⋅rk,l/σ2+12⋅pr|y=0dxm−2j+1−μk,12.When the traversed polynomial is scrambled to generate the polynomial, the number of validl0 and the number of invalid l1 of the check equation are consistent with the source imbalance, namely,(29)l0l1=0.5+ε0.5−ε.Otherwise, the number of symbol 1 in the sequenceyk,1yk,L−n+1⋯yk,Lyk,L+1 participating in the check equation is random, and all compositions can be expressed as(30)V1=∑j=0mCmj.The mean and variance of each possible situation are statistically averaged, and the results are as follows:(31)μ1=∑j=0mCmjV∫−∞∞1−e2A⋅rk,l/σ2e2A⋅rk,l/σ2+1⋅pr|y=1dxj⋅∫−∞∞1−e2A⋅rk,l/σ2e2A⋅rk,l/σ2+1⋅pr|y=0dxm−j,(32)σ12=∑j=0mCmjV1⋅∫−∞∞1−e2A⋅rk,l/σ2e2A⋅rk,l/σ2+12⋅pr|y=1dxj⋅∫−∞∞1−e2A⋅rk,l/σ2e2A⋅rk,l/σ2+12⋅pr|y=0dxm−j−μ12.There is no analytical solution for the integral expressions in equations (23)–(32). To complete fast calculations and achieve higher calculation accuracy, this paper uses numerical integration to solve them.First, the following two hypothesis tests are given:H0: The traversed polynomial is not the synchronous scrambler feedback polynomial.H1: The traversed polynomial is the synchronous scrambler feedback polynomial.According to the law of large numbers, when the intercepted scrambler sequence lengthN is large and the traversed polynomial is not the feedback polynomial, F¯ obeys the Gaussian distribution with mean μ1 and variance σ22=σ12/N. H0 can be expressed as(33)H0:F¯∼Νμ1,σ22.Otherwise,F¯ obeys the Gaussian distribution with mean μ3 and variance σ32=σ2/N, that is,(34)H1:F¯∼Νμ3,σ32,where μ3=0.5+εμk,0+0.5−εμk,1 and σ2=0.5+ε2⋅σk,02+0.5−ε2⋅σk,12.Supposing the discrimination threshold isΛ, then the false alarm probability Pf and the missed alarm probability Pn are, respectively,(35)Pf=∫Λ∞12πσ2e−y−μ12/σ22dx=21−ΦΛ−μ1σ2,(36)Pn=∫−∞Λ12πσ3e−y−μ32/σ32dx=ΦΛ−μ3σ3.In practical application, using the Neyman Pearson criterion, it is first necessary to determinePf, and then, according to equation (33), obtain the discrimination threshold(37)Λ=σ2Φ−11−Pf/2.WhenF¯>Λ, H1 is established; otherwise, H0 is established.


### 3.3. Steps for Reconstructing the Feedback Polynomial

The steps for reconstructing the feedback polynomial of the synchronous scrambler based on the average check conformity are as follows.Step 1.
Convert the intercepted scrambler soft decision sequence into a posterior probability sequence according to equation (19).Step 2.
Construct and store 3-tap and 5-tap polynomials of order 3–60.Step 3.
Determine the false alarm probabilityPf.Step 4.
Traverse the polynomial in step2, and construct an error-containing check equation according to equation (8).Step 5.
Calculate the average check conformity according to equations (21) and (22).Step 6.
Calculate the discriminant threshold according to equation (37). If F¯≥Λ, the feedback polynomial is identified; otherwise, return to step 4 until F¯≥Λ.
The flowchart of the algorithm is shown in Figure3.Figure 3
Flowchart of the algorithm.


### 3.4. Analysis of the Computational Complexity of Reconstructing the Feedback Polynomial

Assume that the length of the intercepted scrambler sequence isN, the number of terms of the scrambler polynomial is d, the order of the scrambling polynomial is L, and the number of traversed polynomials is S.The computational complexity of the algorithm in this study is mainly focused on the calculation of the check conformity. In the process of traversing the polynomial for the first time,d−1 multiplication operations and N−L addition operations are required. Considering the worst case, the feedback polynomial is not recognized until the last polynomial is traversed. The maximum calculation amount is Sd−1 multiplications and SN−L additions. The computational complexity of the Walsh–Hadamard method mainly focuses on the Hadamard transformation. The computational complexity is 2L+1⋅2L+1 addition operations. In the case of fast algorithms, the computational complexity still requires 2L+1L+1⋅S addition operations. At the same time, to obtain the statistical vector, N−L row vectors are converted from binary to decimal. The complexity of the algorithm increases exponentially with the order of the scrambling polynomial. As the order of the polynomial increases, the computational complexity of the algorithm proposed in this paper becomes much smaller than that of the Walsh–Hadamard method.


## 3.1. Establishment of Average Check Conformity

For the convenience of explanation, thek line of the error-containing equation is listed and discussed separately, and the following error-containing check equation is obtained(13)∑l=1L+1yk,l⋅cL−l=0,where 1≤k≤N−L, and yk,l represents the k-th row and l-th column of matrix A in equation (7), where A is y1y2…yLyL+1y2y3…yL+1yL+2⋮⋮…⋮⋮yN−LyN−L+1…yN−1yN.To measure the possibility of the establishment of the equations, the establishment of the check conformity [28] is as follows:(14)Fk=∏l=0L1−2Pyk,l=1|rk,l⋅cL−l.In equation (14), yk,0,yk,1,…,yk,l is the hard decision sequence, the corresponding soft decision sequence is rk,0,rk,1,…,rk,l, and Pyk,l=1|rk,l is the probability that the hard decision sequence yk,l takes the value of one under the condition that the intercepted scrambler soft decision sequence is rk,l.When∑l=1L+1yk,l⋅cL−1=0,Pyk,l=|rk,l⋅cL−1<0.5, and the check conformity is mapped to 1; otherwise, it is mapped to −1. The posterior probability Pyk,l=1|rk,l in equation (14) is derived as follows:(15)Prk,l|yk,l=0=12πσe−rk,l+A2/2σ2,(16)Prk,l|yk,l=1=12πσe−rk,l−A2/2σ2.According to the Bayesian formula,(17)Pyk,l=1|rk,l=Prk,l|yk,l=1⋅Pyk,l=1Prk,l,and Prk,l can be expressed as(18)Prk,l=Prk,l|yk,l=1⋅Pyk,l=1+Prk,l|yk,l=0⋅Pyk,l=0.Since there is no prior information, letPyk,l=0=Pyk,l=1=0.5. Using equations (15)–(18), the following expressions can be obtained:(19)Pyk,l=1|rk,l=e2A⋅rk,l/σ2e2A⋅rk,l/σ2+1.By substituting equations (19) into (14), the expression of the check conformity can be obtained as(20)Fk=∏l=0L1−2⋅e2A⋅rk,l/σ2e2A⋅rk,l/σ2+1⋅cL−l.Considering all error-containing equations in equation (8), the average check conformity is obtained as follows:(21)F¯=1N−L∑k=1N−LFk.Equation (21) can be transformed into(22)F¯=1N−L∑k=1N−L∏l∈Ω1−e2A⋅rk,l/σ2e2A⋅rk,l/σ2+1,where Ω=L|cL=1.When the polynomial to be traversed is a feedback polynomial, the number of valid and nonconforming error-checking equations (8) is different due to the imbalanced ratio of zeroes to ones in the information source. At this time, the ratio of positive to negative values in Fk,k=0,1,…,N−L is not balanced. The larger the SNR, the closer the F¯ is to 1 or −1. Otherwise, the probability of the equation is 0.5. The value of F¯ hovers around 0.


## 3.2. Solution of the Discriminant Threshold

For the statistical characteristics of the average check conformity, the corresponding discriminant threshold needs to be introduced to complete the identification of the feedback polynomial.RememberG=cLcL−1⋯c11, and suppose the number of taps equal to one is m and the corresponding set of tap positions is G1=cLcn⋯c11. Currently, the scrambling hard decision sequence participating in the check equation is yk,1yk,L−n+1⋯yk,Lyk,L+1.When the check equation is established, the symbol 1 modulo-2 plus zero inyk,1yk,L−n+1⋯yk,Lyk,L+1 participating in the check equation is equal to zero. At this time, the number of corresponding symbol 1 is an even number, and all possible composition conditions can be expressed as(23)Vk,0=∑j=0m/2Cm2j,where  means round down, and Cmn=n⋅n−1Ln−m+1/m⋅m−1L1.The mean and variance of each possible situation are statistically averaged, and the results are as follows:(24)μk,0=∑j=0m/2Cm2jVk,0∫−∞∞1−e2A⋅rk,l/σ2e2A⋅rk,l/σ2+1⋅pr|y=1dx2j⋅∫−∞∞1−e2A⋅rk,l/σ2e2A⋅rk,l/σ2+1⋅pr|y=0dxm−2j,(25)σk,02=∑j=0m/2Cm2jVk,0⋅∫−∞∞1−e2A⋅rk,l/σ2e2A⋅rk,l/σ2+12⋅pr|y=1dx2j⋅∫−∞∞1−e2A⋅rk,l/σ2e2A⋅rk,l/σ2+12⋅pr|y=0dxm−2j−μk,02.Otherwise, the code element 1 modulo-2 plus one inyk,1yk,L−n+1⋯yk,Lyk,L+1 participating in the check equation is equal to one. At this time, the number of corresponding code element 1 is an odd number, and all possible composition conditions can be expressed as(26)Vk,1=∑j=0m/2Cm2j−1.The mean and variance of each possible situation are statistically averaged, and the results are as follows:(27)μk,1=∑j=0m/2Cm2j−1Vk,1∫−∞∞1−e2A⋅rk,l/σ2e2A⋅rk,l/σ2+1⋅pr|y=1dx2j−1⋅∫−∞∞1−e2A⋅rk,l/σ2e2A⋅rk,l/σ2+1⋅pr|y=0dxm−2j+1,(28)σk,12=∑j=0m/2Cm2j−1Vk,1⋅∫−∞∞1−e2A⋅rk,l/σ2e2A⋅rk,l/σ2+12⋅pr|y=1dx2j−1⋅∫−∞∞1−e2A⋅rk,l/σ2e2A⋅rk,l/σ2+12⋅pr|y=0dxm−2j+1−μk,12.When the traversed polynomial is scrambled to generate the polynomial, the number of validl0 and the number of invalid l1 of the check equation are consistent with the source imbalance, namely,(29)l0l1=0.5+ε0.5−ε.Otherwise, the number of symbol 1 in the sequenceyk,1yk,L−n+1⋯yk,Lyk,L+1 participating in the check equation is random, and all compositions can be expressed as(30)V1=∑j=0mCmj.The mean and variance of each possible situation are statistically averaged, and the results are as follows:(31)μ1=∑j=0mCmjV∫−∞∞1−e2A⋅rk,l/σ2e2A⋅rk,l/σ2+1⋅pr|y=1dxj⋅∫−∞∞1−e2A⋅rk,l/σ2e2A⋅rk,l/σ2+1⋅pr|y=0dxm−j,(32)σ12=∑j=0mCmjV1⋅∫−∞∞1−e2A⋅rk,l/σ2e2A⋅rk,l/σ2+12⋅pr|y=1dxj⋅∫−∞∞1−e2A⋅rk,l/σ2e2A⋅rk,l/σ2+12⋅pr|y=0dxm−j−μ12.There is no analytical solution for the integral expressions in equations (23)–(32). To complete fast calculations and achieve higher calculation accuracy, this paper uses numerical integration to solve them.First, the following two hypothesis tests are given:H0: The traversed polynomial is not the synchronous scrambler feedback polynomial.H1: The traversed polynomial is the synchronous scrambler feedback polynomial.According to the law of large numbers, when the intercepted scrambler sequence lengthN is large and the traversed polynomial is not the feedback polynomial, F¯ obeys the Gaussian distribution with mean μ1 and variance σ22=σ12/N. H0 can be expressed as(33)H0:F¯∼Νμ1,σ22.Otherwise,F¯ obeys the Gaussian distribution with mean μ3 and variance σ32=σ2/N, that is,(34)H1:F¯∼Νμ3,σ32,where μ3=0.5+εμk,0+0.5−εμk,1 and σ2=0.5+ε2⋅σk,02+0.5−ε2⋅σk,12.Supposing the discrimination threshold isΛ, then the false alarm probability Pf and the missed alarm probability Pn are, respectively,(35)Pf=∫Λ∞12πσ2e−y−μ12/σ22dx=21−ΦΛ−μ1σ2,(36)Pn=∫−∞Λ12πσ3e−y−μ32/σ32dx=ΦΛ−μ3σ3.In practical application, using the Neyman Pearson criterion, it is first necessary to determinePf, and then, according to equation (33), obtain the discrimination threshold(37)Λ=σ2Φ−11−Pf/2.WhenF¯>Λ, H1 is established; otherwise, H0 is established.


## 3.3. Steps for Reconstructing the Feedback Polynomial

The steps for reconstructing the feedback polynomial of the synchronous scrambler based on the average check conformity are as follows.Step 1.
Convert the intercepted scrambler soft decision sequence into a posterior probability sequence according to equation (19).Step 2.
Construct and store 3-tap and 5-tap polynomials of order 3–60.Step 3.
Determine the false alarm probabilityPf.Step 4.
Traverse the polynomial in step2, and construct an error-containing check equation according to equation (8).Step 5.
Calculate the average check conformity according to equations (21) and (22).Step 6.
Calculate the discriminant threshold according to equation (37). If F¯≥Λ, the feedback polynomial is identified; otherwise, return to step 4 until F¯≥Λ.
The flowchart of the algorithm is shown in Figure3.Figure 3
Flowchart of the algorithm.


## 3.4. Analysis of the Computational Complexity of Reconstructing the Feedback Polynomial

Assume that the length of the intercepted scrambler sequence isN, the number of terms of the scrambler polynomial is d, the order of the scrambling polynomial is L, and the number of traversed polynomials is S.The computational complexity of the algorithm in this study is mainly focused on the calculation of the check conformity. In the process of traversing the polynomial for the first time,d−1 multiplication operations and N−L addition operations are required. Considering the worst case, the feedback polynomial is not recognized until the last polynomial is traversed. The maximum calculation amount is Sd−1 multiplications and SN−L additions. The computational complexity of the Walsh–Hadamard method mainly focuses on the Hadamard transformation. The computational complexity is 2L+1⋅2L+1 addition operations. In the case of fast algorithms, the computational complexity still requires 2L+1L+1⋅S addition operations. At the same time, to obtain the statistical vector, N−L row vectors are converted from binary to decimal. The complexity of the algorithm increases exponentially with the order of the scrambling polynomial. As the order of the polynomial increases, the computational complexity of the algorithm proposed in this paper becomes much smaller than that of the Walsh–Hadamard method.


## 4. Estimation of the Initial State

### 4.1. Establishment of Average Check Conformity

From the scrambling of the synchronous scrambler in Figure1, the initial state of the synchronous scrambler is S0∧=s1,s2,…,sL.The state at each moment satisfies the following relation:(38)sk+n=∑i=1ncisk+n−i.The recurrence relation on GF(2) of the LFSR state can be obtained as(39)S0∧=s1,s2,…,sL,S1∧=s2,s3,…,s1+L=S0∧T,S2∧=s3,s4,…,s2+L=S1∧T=S0∧T2,⋮Sk−1∧=sk,sk+1,…,sk+L−1=Sk−2∧T=S0∧Tk−1,Sk∧=sk+1,sk+2,…,sk+L=Sk−1∧T=S0∧Tk.Among them,T is the state transition matrix of LFSR, and the matrix is expressed as follows:(40)T=00⋯0cL10⋯0cL−101⋯0cL−2⋮⋮⋱⋮⋮00⋯1c1L×L.According to the definition of linear block code, the LFSR sequenceSN=s1,s2,…,sN of length N can be regarded as an N,L linear block code whose information vector is S0∧=s1,s2,…,sL. The generator matrix of the linear block code can be expressed as follows:(41)GLFSR=G1,G2,…GN=10⋯0cLg1i⋯g1N01⋯0cL−1g2i⋯g2N⋮⋮⋱⋮⋮⋮⋮⋮00⋯1c1gLi⋯gLNL×N,where Gi=TLi−L,L+1<i≤N, and TLi−L represents the L column of the state transition matrix T raised to the power of i−L on GF(2).Therefore, the LFSR sequenceSN and the information vector S0∧ satisfy the following equation:(42)SN=S0∧⋅GLFSR.Transposing equation (42), we obtain(43)10⋯001⋯0⋮⋮⋱⋮00⋯1cLcL−1⋯c1g1ig2i⋯gLi⋮⋮⋱⋮g1Ng2N⋯gLN⋅s1s2⋮sL=s1s2⋮sN.For the received scrambling sequencey1,y2,…,yN, the error-containing equations for the initial state of the synchronous scrambler can be established as follows:(44)10⋯0y101⋯0y2⋮⋮⋱⋮⋮00⋯1yLcLcL−1⋯c1yL+1g1ig2i⋯gLiyL+2⋮⋮⋱⋮⋮g1Ng2N⋯gLNyN⋅s1s2⋮sL1=00⋮0.According to Section3.1, the check conformity is established to verify whether the possibility of equation (44) is correct:(45)F1k=∏l=1L+11−2hk,l⋅Sl∧=∏l=1L1−2hk,l⋅Sl∧⋅1−2Pyk,L+1=1|rk,L+1⋅SL+1∧=∏l=1L1−2hk,l⋅Sl∧⋅1−2⋅e2A⋅rk,l/σ2e2A⋅rk,l/σ2+1The statistical average is taken for the check conformity, and the average check conformity is obtained as follows(46)F¯1=1N∑k=1N∏l=1L1−2hk,l⋅Sl∧⋅1−2⋅e2A⋅rk,l/σ2e2A⋅rk,l/σ2+1.


### 4.2. Subsection Optimization at High Order

The number of initial states of the synchronous scrambler that need to be traversed is2L. When the order of the feedback polynomial is high, the number of polynomials that need to be traversed increases exponentially. The idea of solving the error-containing equations in segments is introduced to reduce the number of initial states that need to be traversed, thereby reducing the computational complexity of the algorithm.First, estimate the initial state of the firstL10<L1≤L bits of the initial state, find the GLFSR column in the L+1∼N column of the generator matrix t of the linear block code, and then establish the following check equation(47)Gi1⊕Gi2⊕⋯⊕GitT=qi1,qi2,…,qiL1,0,0,…,0=Qi,i=1,2,…,φ1.In equation (47), φ1 represents the number of established verification equations, and qiL1=1 in binary vector Qi.According to the verification equation, equation (47), the following equation can be obtained(48)∑j=1t⊕sij=qi1,qi2,…,qiL1⋅s1,s2,…,sL1T.From equation (48), ∑j=1t⊕sij is a linear combination ofs1,s2,…sL1. So, the following expression holds:(49)q11q12⋯q1L1q21q22⋯q2L1⋮⋮⋱⋮qφ11qφ21⋯qφ1L1⋅s1s2⋮sL1=∑j=1t⊕s1j∑j=1t⊕s2j⋮∑j=1t⊕sφ1j.According to equation (49), the following error-containing equations can be established(50)q11q12⋯q1L1∑j=1t⊕s1jq21q22⋯q2L1∑j=1t⊕s2j⋮⋮⋱⋮⋮qφ1qφ2⋯qφL1∑j=1t⊕sφ1j⋅s1s2⋮sL11=00⋮0.After the error-containing equations are established, the optimal solution ofs1,s2,…,sL1 can be obtained by solving the error-containing equation according to the method of establishing the degree of agreement in Section 4.1 so that the first L1 position of the initial state can be estimated.The second step estimatessL1+1,sL1+2,…,sL1+L2 and finds the t column in the L+1∼N column of the generator matrix GLFSR, and establishes the following check equation(51)Gi1⊕Gi2⊕⋯⊕GitT=qi1′,…,qiL1′,qiL1+1,…,qiL1+L2,0,…,0,i=1,2,…,φ2.In equation (51), qiL1+L2=1.According to the verification equation, equation (51), the following equation can be obtained(52)∑j=1t⊕sij=qi1′,…,qiL1′,qiL1+1,…,qiL1+L2⋅s1,s2,…,sL1+L2.TFrom equation (52), the following equation can be established(53)q11′q12′⋯q1L1+L2q21′q22′⋯q2L1+L2⋮⋮⋱⋮qφ21′qφ21′⋯qφ2L1+L2⋅s1s2⋮sL1+L2=∑j=1t⊕s1j∑j=1t⊕s2j⋮∑j=1t⊕sφ2j.Then, a system of equations containing errors can be established(54)q11′q12′⋯q1L1+L2∑j=1t⊕s1jq21′q22′⋯q2L1+L2∑j=1t⊕s2j⋮⋮⋱⋮⋮qφ21′qφ22′⋯qφ2L1+L2∑j=1t⊕sφ2j⋅s1s2⋮sL1+L21=00⋮0.The equation can be solved by checking the degree of conformity to complete the estimation ofsL1+1,sL1+2,…,sL1+L2. If the number of digits is large, we can continue to estimate according to the method of the second step until the entire initial state is estimated, and the estimated number of steps is determined according to the series L of the LFSR. When L≤15, it is not necessary to solve the equation piecewise. When 15<L≤30, it is enough to solve the system of equations containing errors in two steps. Assuming that the identification of the initial states is completed in two steps, the number of initial states that need to be traversed to solve the equation without using subsections is 2L1⋅2L−L1, whereas the number of initial states to be traversed to solve the error-containing equation in subsections is only 2L1+2L−L1. Hence, the number of states to be traversed is greatly reduced. Since it is necessary to find and solve the check equation in sections, the required amount of intercepted scrambled data will increase when solving in subsections. Therefore, when using subsections, the amount of intercepted data is increased in exchange for a reduction in computational complexity.


### 4.3. Analysis of the Computational Complexity of Estimating the Initial State

WhenL≤15, the algorithm in this paper does not need to solve the equation in sections. It is assumed that the length of the intercepted scrambling code sequence is N, the number of taps of the scrambling code polynomial is m, and the number of initial states to be traversed is 2L. The computational complexity of the algorithm in this paper is mainly focused on the calculation of the check compliance. In the process of traversing the initial state, addition and multiplication of the magnitude of ON⋅2L need to be performed. When L>15, the number of initial states traversed is 2L1+2L−L1. Then ON⋅2L1+2L−L1-scale additions and multiplications need to be performed.The computational complexity of Viterbi decoding is aboutOd⋅2B, where B is the fixed constraint length of the constructed convolutional code, the code rate is R=1/d+1, and the complexity of finding the check equation is approximately ∑j=1tCNt. When L≤15, since the algorithm in this paper does not need to find the check equation, the computational complexity is lower than that Viterbi decoding. When L>15, due to the need to solve the equation in sections, this paper also looks for the check equation. The algorithm complexity is mainly reflected in the calculation of the Viterbi decoding and check compliance. The computational complexity in this paper is slightly higher than that Viterbi decoding, but the algorithm complexity in this paper is within the affordable range.


## 4.1. Establishment of Average Check Conformity

From the scrambling of the synchronous scrambler in Figure1, the initial state of the synchronous scrambler is S0∧=s1,s2,…,sL.The state at each moment satisfies the following relation:(38)sk+n=∑i=1ncisk+n−i.The recurrence relation on GF(2) of the LFSR state can be obtained as(39)S0∧=s1,s2,…,sL,S1∧=s2,s3,…,s1+L=S0∧T,S2∧=s3,s4,…,s2+L=S1∧T=S0∧T2,⋮Sk−1∧=sk,sk+1,…,sk+L−1=Sk−2∧T=S0∧Tk−1,Sk∧=sk+1,sk+2,…,sk+L=Sk−1∧T=S0∧Tk.Among them,T is the state transition matrix of LFSR, and the matrix is expressed as follows:(40)T=00⋯0cL10⋯0cL−101⋯0cL−2⋮⋮⋱⋮⋮00⋯1c1L×L.According to the definition of linear block code, the LFSR sequenceSN=s1,s2,…,sN of length N can be regarded as an N,L linear block code whose information vector is S0∧=s1,s2,…,sL. The generator matrix of the linear block code can be expressed as follows:(41)GLFSR=G1,G2,…GN=10⋯0cLg1i⋯g1N01⋯0cL−1g2i⋯g2N⋮⋮⋱⋮⋮⋮⋮⋮00⋯1c1gLi⋯gLNL×N,where Gi=TLi−L,L+1<i≤N, and TLi−L represents the L column of the state transition matrix T raised to the power of i−L on GF(2).Therefore, the LFSR sequenceSN and the information vector S0∧ satisfy the following equation:(42)SN=S0∧⋅GLFSR.Transposing equation (42), we obtain(43)10⋯001⋯0⋮⋮⋱⋮00⋯1cLcL−1⋯c1g1ig2i⋯gLi⋮⋮⋱⋮g1Ng2N⋯gLN⋅s1s2⋮sL=s1s2⋮sN.For the received scrambling sequencey1,y2,…,yN, the error-containing equations for the initial state of the synchronous scrambler can be established as follows:(44)10⋯0y101⋯0y2⋮⋮⋱⋮⋮00⋯1yLcLcL−1⋯c1yL+1g1ig2i⋯gLiyL+2⋮⋮⋱⋮⋮g1Ng2N⋯gLNyN⋅s1s2⋮sL1=00⋮0.According to Section3.1, the check conformity is established to verify whether the possibility of equation (44) is correct:(45)F1k=∏l=1L+11−2hk,l⋅Sl∧=∏l=1L1−2hk,l⋅Sl∧⋅1−2Pyk,L+1=1|rk,L+1⋅SL+1∧=∏l=1L1−2hk,l⋅Sl∧⋅1−2⋅e2A⋅rk,l/σ2e2A⋅rk,l/σ2+1The statistical average is taken for the check conformity, and the average check conformity is obtained as follows(46)F¯1=1N∑k=1N∏l=1L1−2hk,l⋅Sl∧⋅1−2⋅e2A⋅rk,l/σ2e2A⋅rk,l/σ2+1.


## 4.2. Subsection Optimization at High Order

The number of initial states of the synchronous scrambler that need to be traversed is2L. When the order of the feedback polynomial is high, the number of polynomials that need to be traversed increases exponentially. The idea of solving the error-containing equations in segments is introduced to reduce the number of initial states that need to be traversed, thereby reducing the computational complexity of the algorithm.First, estimate the initial state of the firstL10<L1≤L bits of the initial state, find the GLFSR column in the L+1∼N column of the generator matrix t of the linear block code, and then establish the following check equation(47)Gi1⊕Gi2⊕⋯⊕GitT=qi1,qi2,…,qiL1,0,0,…,0=Qi,i=1,2,…,φ1.In equation (47), φ1 represents the number of established verification equations, and qiL1=1 in binary vector Qi.According to the verification equation, equation (47), the following equation can be obtained(48)∑j=1t⊕sij=qi1,qi2,…,qiL1⋅s1,s2,…,sL1T.From equation (48), ∑j=1t⊕sij is a linear combination ofs1,s2,…sL1. So, the following expression holds:(49)q11q12⋯q1L1q21q22⋯q2L1⋮⋮⋱⋮qφ11qφ21⋯qφ1L1⋅s1s2⋮sL1=∑j=1t⊕s1j∑j=1t⊕s2j⋮∑j=1t⊕sφ1j.According to equation (49), the following error-containing equations can be established(50)q11q12⋯q1L1∑j=1t⊕s1jq21q22⋯q2L1∑j=1t⊕s2j⋮⋮⋱⋮⋮qφ1qφ2⋯qφL1∑j=1t⊕sφ1j⋅s1s2⋮sL11=00⋮0.After the error-containing equations are established, the optimal solution ofs1,s2,…,sL1 can be obtained by solving the error-containing equation according to the method of establishing the degree of agreement in Section 4.1 so that the first L1 position of the initial state can be estimated.The second step estimatessL1+1,sL1+2,…,sL1+L2 and finds the t column in the L+1∼N column of the generator matrix GLFSR, and establishes the following check equation(51)Gi1⊕Gi2⊕⋯⊕GitT=qi1′,…,qiL1′,qiL1+1,…,qiL1+L2,0,…,0,i=1,2,…,φ2.In equation (51), qiL1+L2=1.According to the verification equation, equation (51), the following equation can be obtained(52)∑j=1t⊕sij=qi1′,…,qiL1′,qiL1+1,…,qiL1+L2⋅s1,s2,…,sL1+L2.TFrom equation (52), the following equation can be established(53)q11′q12′⋯q1L1+L2q21′q22′⋯q2L1+L2⋮⋮⋱⋮qφ21′qφ21′⋯qφ2L1+L2⋅s1s2⋮sL1+L2=∑j=1t⊕s1j∑j=1t⊕s2j⋮∑j=1t⊕sφ2j.Then, a system of equations containing errors can be established(54)q11′q12′⋯q1L1+L2∑j=1t⊕s1jq21′q22′⋯q2L1+L2∑j=1t⊕s2j⋮⋮⋱⋮⋮qφ21′qφ22′⋯qφ2L1+L2∑j=1t⊕sφ2j⋅s1s2⋮sL1+L21=00⋮0.The equation can be solved by checking the degree of conformity to complete the estimation ofsL1+1,sL1+2,…,sL1+L2. If the number of digits is large, we can continue to estimate according to the method of the second step until the entire initial state is estimated, and the estimated number of steps is determined according to the series L of the LFSR. When L≤15, it is not necessary to solve the equation piecewise. When 15<L≤30, it is enough to solve the system of equations containing errors in two steps. Assuming that the identification of the initial states is completed in two steps, the number of initial states that need to be traversed to solve the equation without using subsections is 2L1⋅2L−L1, whereas the number of initial states to be traversed to solve the error-containing equation in subsections is only 2L1+2L−L1. Hence, the number of states to be traversed is greatly reduced. Since it is necessary to find and solve the check equation in sections, the required amount of intercepted scrambled data will increase when solving in subsections. Therefore, when using subsections, the amount of intercepted data is increased in exchange for a reduction in computational complexity.


## 4.3. Analysis of the Computational Complexity of Estimating the Initial State

WhenL≤15, the algorithm in this paper does not need to solve the equation in sections. It is assumed that the length of the intercepted scrambling code sequence is N, the number of taps of the scrambling code polynomial is m, and the number of initial states to be traversed is 2L. The computational complexity of the algorithm in this paper is mainly focused on the calculation of the check compliance. In the process of traversing the initial state, addition and multiplication of the magnitude of ON⋅2L need to be performed. When L>15, the number of initial states traversed is 2L1+2L−L1. Then ON⋅2L1+2L−L1-scale additions and multiplications need to be performed.The computational complexity of Viterbi decoding is aboutOd⋅2B, where B is the fixed constraint length of the constructed convolutional code, the code rate is R=1/d+1, and the complexity of finding the check equation is approximately ∑j=1tCNt. When L≤15, since the algorithm in this paper does not need to find the check equation, the computational complexity is lower than that Viterbi decoding. When L>15, due to the need to solve the equation in sections, this paper also looks for the check equation. The algorithm complexity is mainly reflected in the calculation of the Viterbi decoding and check compliance. The computational complexity in this paper is slightly higher than that Viterbi decoding, but the algorithm complexity in this paper is within the affordable range.


## 5. Experiments and Discussion

### 5.1. Analysis of Recognition Rate of Feedback Polynomial

Without special instructions, the following simulation check defaults to a given false alarm probabilityPf=10−3, the feedback polynomial is fx=1+x7+x8, the SNR is varied from 0 dB to 14 dB with an interval of 2 dB, and the number of Monte Carlo simulations is 1000.


#### 5.1.1. Verification of the Effectiveness of the Algorithm

When the SNR is 8 dB,ε=0.1,Ns=2×105 and ε=0.15,Ns=5×104. The statistics, ergodic polynomials, and discriminant threshold distribution diagrams after all polynomials are traversed are shown in Figures 4(a) and 4(b).Figure 4
Relationship between statistics and discriminant threshold. (a)ε=0.1. (b)ε=0.15.
(a)(b)It can be seen from Figure4 that when the traversed polynomial is correct, the statistics have an obvious prominent spectrum, and the average verification conformity spectrum is significantly higher than the discrimination threshold; otherwise, the average verification conformity spectrum is lower than the discrimination threshold. Therefore, the algorithm proposed in this paper can effectively identify the feedback polynomial.


#### 5.1.2. Effect of Bit Error Rate on the Algorithm

The source imbalance degree of the intercepted synchronous scrambler set in the simulation is 0.1. The lengths of the intercepted synchronous scrambler are set as2×105, 3×105, and 4×105; the relationship between the bit error rate (BER) and the probability of correct recognition of the feedback polynomial is determined, and the results are shown in Figure 5.Figure 5
Effect of bit error rate on the algorithm.It can be seen from Figure5 that the recognition rate of the algorithm can be significantly improved by increasing the length of the intercepted synchronous scrambler sequence. It can be seen from the simulation results that the recognition rate of the feedback polynomial can reach over 90%.


#### 5.1.3. Effect of the Intercepted Scramble Sequence Length on the Algorithm

The source imbalance is 0.1 and the length of the intercepted synchronous scrambler sequence is set to2×105, 3×105, and 4×105. Figure 5 shows the relationship between N and the probability of correct recognition of the feedback polynomial under different SNRs.It can be seen from Figure6 that the recognition rate of the algorithm can be significantly improved by increasing N. When the source imbalance is low, N can be increased to overcome the decrease in algorithm recognition performance caused by the decrease in ε. The algorithm has better recognition performance under low SNR. When ε=0.1, N=4×105, and the SNR is 3 dB, the recognition rate of the feedback polynomial can reach over 90%.Figure 6
Influence of the length of the intercepted synchronous scrambler sequence on the recognition rate of the algorithm.


#### 5.1.4. Effect of Source Imbalance on the Algorithm

The sequence length is set to1×104, 3×104, and 5×104, the SNR is set to 6 dB, and the source imbalance is varied from 0 to 0.2 with an interval of 0.01. The relationship between the source imbalance and the recognition rate of the algorithm is shown in Figure 7.Figure 7
Impact of source imbalance on algorithm recognition rate.It can be seen from Figure7 that under the same SNR, with the increase of the source imbalance, the probability of correct recognition of the algorithm increases. The more imbalanced the source, the higher the recognition rate of the algorithm. With the increase of ε, the number of samples required to reach a certain recognition rate is also reduced.


#### 5.1.5. Effect of the Order of the Feedback Polynomial on the Algorithm

The source imbalance is 0.1 and the sequence length is set to3×105. The feedback polynomials are fx=1+x6+x7, fx=1+x9+x11, and fx=1+x3+x20. The relationship between the order of the feedback polynomial and the recognition rate is determined, and the result is shown in Figure 8.Figure 8
Influence of the order of the feedback polynomial on algorithm performance.With the same intercept sequence length and the same SNR, the lower the order of the feedback polynomial, the higher the recognition rate of the algorithm. Additionally, the recognition rates of the three algorithms increase with the increase of the SNR. The lower the order, the earlier the recognition rate reaches 100%. When the SNR is 4 dB, the probability of correct recognition of the algorithm can reach over 90%, and it has good polynomial order fault tolerance.


#### 5.1.6. Comparison with Other Algorithms

The algorithm in this study is compared with the algorithm based on bit imbalance [13], the real-time detection algorithm based on Walsh–Hadamard transform [15], the Clzeau algorithm [12], the Walsh–Hadamard transform [16], and the greatest common divisor [17]. The intercepted sequence length is set as 3×105 and the source imbalance is 0.1. Through a Monte Carlo simulation experiment, the probability of correct recognition is determined for each of the four algorithms under different SNRs, and the results are shown in Figure 9.Figure 9
Comparison of fault tolerance performance of four algorithms.It can be seen from the figure that the algorithm proposed in this study has an SNR gain of about 1–2 dB at a low SNR. The main reason is that the proposed algorithm makes better use of the soft decision sequence to construct the conformity degree, and it makes great use of the information reliability in the soft sequence. The performance improvement is more obvious under the condition of low SNR.


### 5.2. Analysis of the Recognition Rate of the Initial State

The following simulations verify that each initial state is randomly generated (excluding all zeroes), the feedback polynomials aref1x=1+x7+x10, f2x=1+x3+x7+x9+x10, f3x=1+x14+x15, and f4x=1+x11+x13+x14+x15, and the number of Monte Carlo simulations is 1000.


#### 5.2.1. Influence of SNR on the Initial State Recognition Rate

Whenε=0.1 and N=1000, the curve of probability of correct recognition corresponding to different orders and tap numbers with SNR is shown in Figure 10.Figure 10
Distribution of probability of correct recognition with SNR.It can be seen from the figure that with the increase of SNR, the probability of correct recognition of the initial state gradually increases, and it has a good ability to adapt to low SNR. AtSNR=0dB, the probability of correct recognition of the algorithm can reach 90%. Under the same SNR, the lower the order, the higher the estimation rate of the algorithm. Under the same order, the number of taps has little effect on the estimation rate.


#### 5.2.2. Influence of Source Imbalance Degree on the Initial State Recognition Rate

WhenSNR=6dB and N=1000, the variation curve of the probability of correct recognition corresponding to different orders and tap numbers with the source imbalance is shown in Figure 11.Figure 11
Distribution of probability of correct recognition with source imbalance.It can be seen from the figure that with the increase ofε, the probability of correct recognition of the initial state gradually increases. Under the same ε, the lower the order, the higher the estimation rate of the algorithm. The number of taps has little effect on the estimation rate. When ε is low, the probability of correct recognition can be improved by increasing the length of the synchronous scrambler sequence. According to Figures 10 and 11, the algorithm is greatly affected by the order of the feedback polynomial, and the number of taps does not affect the performance of the algorithm. The lower the order, the higher the estimation rate of the algorithm.


#### 5.2.3. Influence of BER on the Initial State Recognition Rate

Whenε=0.1 and N=1000, the curves of the probability of correct recognition versus the BER corresponding to different orders and numbers of taps are shown in Figure 12.Figure 12
Distribution of probability of correct recognition versus bit error rate.It can be seen from the figure that with the increase of the BER, the probability of correct recognition of the initial state gradually decreases. When the intercepted scrambled data are only 1000 and the BER is10−2, the probability of correct recognition of the algorithm can reach over 90%.


## 5.1. Analysis of Recognition Rate of Feedback Polynomial

Without special instructions, the following simulation check defaults to a given false alarm probabilityPf=10−3, the feedback polynomial is fx=1+x7+x8, the SNR is varied from 0 dB to 14 dB with an interval of 2 dB, and the number of Monte Carlo simulations is 1000.


### 5.1.1. Verification of the Effectiveness of the Algorithm

When the SNR is 8 dB,ε=0.1,Ns=2×105 and ε=0.15,Ns=5×104. The statistics, ergodic polynomials, and discriminant threshold distribution diagrams after all polynomials are traversed are shown in Figures 4(a) and 4(b).Figure 4
Relationship between statistics and discriminant threshold. (a)ε=0.1. (b)ε=0.15.
(a)(b)It can be seen from Figure4 that when the traversed polynomial is correct, the statistics have an obvious prominent spectrum, and the average verification conformity spectrum is significantly higher than the discrimination threshold; otherwise, the average verification conformity spectrum is lower than the discrimination threshold. Therefore, the algorithm proposed in this paper can effectively identify the feedback polynomial.


### 5.1.2. Effect of Bit Error Rate on the Algorithm

The source imbalance degree of the intercepted synchronous scrambler set in the simulation is 0.1. The lengths of the intercepted synchronous scrambler are set as2×105, 3×105, and 4×105; the relationship between the bit error rate (BER) and the probability of correct recognition of the feedback polynomial is determined, and the results are shown in Figure 5.Figure 5
Effect of bit error rate on the algorithm.It can be seen from Figure5 that the recognition rate of the algorithm can be significantly improved by increasing the length of the intercepted synchronous scrambler sequence. It can be seen from the simulation results that the recognition rate of the feedback polynomial can reach over 90%.


### 5.1.3. Effect of the Intercepted Scramble Sequence Length on the Algorithm

The source imbalance is 0.1 and the length of the intercepted synchronous scrambler sequence is set to2×105, 3×105, and 4×105. Figure 5 shows the relationship between N and the probability of correct recognition of the feedback polynomial under different SNRs.It can be seen from Figure6 that the recognition rate of the algorithm can be significantly improved by increasing N. When the source imbalance is low, N can be increased to overcome the decrease in algorithm recognition performance caused by the decrease in ε. The algorithm has better recognition performance under low SNR. When ε=0.1, N=4×105, and the SNR is 3 dB, the recognition rate of the feedback polynomial can reach over 90%.Figure 6
Influence of the length of the intercepted synchronous scrambler sequence on the recognition rate of the algorithm.


### 5.1.4. Effect of Source Imbalance on the Algorithm

The sequence length is set to1×104, 3×104, and 5×104, the SNR is set to 6 dB, and the source imbalance is varied from 0 to 0.2 with an interval of 0.01. The relationship between the source imbalance and the recognition rate of the algorithm is shown in Figure 7.Figure 7
Impact of source imbalance on algorithm recognition rate.It can be seen from Figure7 that under the same SNR, with the increase of the source imbalance, the probability of correct recognition of the algorithm increases. The more imbalanced the source, the higher the recognition rate of the algorithm. With the increase of ε, the number of samples required to reach a certain recognition rate is also reduced.


### 5.1.5. Effect of the Order of the Feedback Polynomial on the Algorithm

The source imbalance is 0.1 and the sequence length is set to3×105. The feedback polynomials are fx=1+x6+x7, fx=1+x9+x11, and fx=1+x3+x20. The relationship between the order of the feedback polynomial and the recognition rate is determined, and the result is shown in Figure 8.Figure 8
Influence of the order of the feedback polynomial on algorithm performance.With the same intercept sequence length and the same SNR, the lower the order of the feedback polynomial, the higher the recognition rate of the algorithm. Additionally, the recognition rates of the three algorithms increase with the increase of the SNR. The lower the order, the earlier the recognition rate reaches 100%. When the SNR is 4 dB, the probability of correct recognition of the algorithm can reach over 90%, and it has good polynomial order fault tolerance.


### 5.1.6. Comparison with Other Algorithms

The algorithm in this study is compared with the algorithm based on bit imbalance [13], the real-time detection algorithm based on Walsh–Hadamard transform [15], the Clzeau algorithm [12], the Walsh–Hadamard transform [16], and the greatest common divisor [17]. The intercepted sequence length is set as 3×105 and the source imbalance is 0.1. Through a Monte Carlo simulation experiment, the probability of correct recognition is determined for each of the four algorithms under different SNRs, and the results are shown in Figure 9.Figure 9
Comparison of fault tolerance performance of four algorithms.It can be seen from the figure that the algorithm proposed in this study has an SNR gain of about 1–2 dB at a low SNR. The main reason is that the proposed algorithm makes better use of the soft decision sequence to construct the conformity degree, and it makes great use of the information reliability in the soft sequence. The performance improvement is more obvious under the condition of low SNR.


## 5.1.1. Verification of the Effectiveness of the Algorithm

When the SNR is 8 dB,ε=0.1,Ns=2×105 and ε=0.15,Ns=5×104. The statistics, ergodic polynomials, and discriminant threshold distribution diagrams after all polynomials are traversed are shown in Figures 4(a) and 4(b).Figure 4
Relationship between statistics and discriminant threshold. (a)ε=0.1. (b)ε=0.15.
(a)(b)It can be seen from Figure4 that when the traversed polynomial is correct, the statistics have an obvious prominent spectrum, and the average verification conformity spectrum is significantly higher than the discrimination threshold; otherwise, the average verification conformity spectrum is lower than the discrimination threshold. Therefore, the algorithm proposed in this paper can effectively identify the feedback polynomial.


## 5.1.2. Effect of Bit Error Rate on the Algorithm

The source imbalance degree of the intercepted synchronous scrambler set in the simulation is 0.1. The lengths of the intercepted synchronous scrambler are set as2×105, 3×105, and 4×105; the relationship between the bit error rate (BER) and the probability of correct recognition of the feedback polynomial is determined, and the results are shown in Figure 5.Figure 5
Effect of bit error rate on the algorithm.It can be seen from Figure5 that the recognition rate of the algorithm can be significantly improved by increasing the length of the intercepted synchronous scrambler sequence. It can be seen from the simulation results that the recognition rate of the feedback polynomial can reach over 90%.


## 5.1.3. Effect of the Intercepted Scramble Sequence Length on the Algorithm

The source imbalance is 0.1 and the length of the intercepted synchronous scrambler sequence is set to2×105, 3×105, and 4×105. Figure 5 shows the relationship between N and the probability of correct recognition of the feedback polynomial under different SNRs.It can be seen from Figure6 that the recognition rate of the algorithm can be significantly improved by increasing N. When the source imbalance is low, N can be increased to overcome the decrease in algorithm recognition performance caused by the decrease in ε. The algorithm has better recognition performance under low SNR. When ε=0.1, N=4×105, and the SNR is 3 dB, the recognition rate of the feedback polynomial can reach over 90%.Figure 6
Influence of the length of the intercepted synchronous scrambler sequence on the recognition rate of the algorithm.


## 5.1.4. Effect of Source Imbalance on the Algorithm

The sequence length is set to1×104, 3×104, and 5×104, the SNR is set to 6 dB, and the source imbalance is varied from 0 to 0.2 with an interval of 0.01. The relationship between the source imbalance and the recognition rate of the algorithm is shown in Figure 7.Figure 7
Impact of source imbalance on algorithm recognition rate.It can be seen from Figure7 that under the same SNR, with the increase of the source imbalance, the probability of correct recognition of the algorithm increases. The more imbalanced the source, the higher the recognition rate of the algorithm. With the increase of ε, the number of samples required to reach a certain recognition rate is also reduced.


## 5.1.5. Effect of the Order of the Feedback Polynomial on the Algorithm

The source imbalance is 0.1 and the sequence length is set to3×105. The feedback polynomials are fx=1+x6+x7, fx=1+x9+x11, and fx=1+x3+x20. The relationship between the order of the feedback polynomial and the recognition rate is determined, and the result is shown in Figure 8.Figure 8
Influence of the order of the feedback polynomial on algorithm performance.With the same intercept sequence length and the same SNR, the lower the order of the feedback polynomial, the higher the recognition rate of the algorithm. Additionally, the recognition rates of the three algorithms increase with the increase of the SNR. The lower the order, the earlier the recognition rate reaches 100%. When the SNR is 4 dB, the probability of correct recognition of the algorithm can reach over 90%, and it has good polynomial order fault tolerance.


## 5.1.6. Comparison with Other Algorithms

The algorithm in this study is compared with the algorithm based on bit imbalance [13], the real-time detection algorithm based on Walsh–Hadamard transform [15], the Clzeau algorithm [12], the Walsh–Hadamard transform [16], and the greatest common divisor [17]. The intercepted sequence length is set as 3×105 and the source imbalance is 0.1. Through a Monte Carlo simulation experiment, the probability of correct recognition is determined for each of the four algorithms under different SNRs, and the results are shown in Figure 9.Figure 9
Comparison of fault tolerance performance of four algorithms.It can be seen from the figure that the algorithm proposed in this study has an SNR gain of about 1–2 dB at a low SNR. The main reason is that the proposed algorithm makes better use of the soft decision sequence to construct the conformity degree, and it makes great use of the information reliability in the soft sequence. The performance improvement is more obvious under the condition of low SNR.


## 5.2. Analysis of the Recognition Rate of the Initial State

The following simulations verify that each initial state is randomly generated (excluding all zeroes), the feedback polynomials aref1x=1+x7+x10, f2x=1+x3+x7+x9+x10, f3x=1+x14+x15, and f4x=1+x11+x13+x14+x15, and the number of Monte Carlo simulations is 1000.


### 5.2.1. Influence of SNR on the Initial State Recognition Rate

Whenε=0.1 and N=1000, the curve of probability of correct recognition corresponding to different orders and tap numbers with SNR is shown in Figure 10.Figure 10
Distribution of probability of correct recognition with SNR.It can be seen from the figure that with the increase of SNR, the probability of correct recognition of the initial state gradually increases, and it has a good ability to adapt to low SNR. AtSNR=0dB, the probability of correct recognition of the algorithm can reach 90%. Under the same SNR, the lower the order, the higher the estimation rate of the algorithm. Under the same order, the number of taps has little effect on the estimation rate.


### 5.2.2. Influence of Source Imbalance Degree on the Initial State Recognition Rate

WhenSNR=6dB and N=1000, the variation curve of the probability of correct recognition corresponding to different orders and tap numbers with the source imbalance is shown in Figure 11.Figure 11
Distribution of probability of correct recognition with source imbalance.It can be seen from the figure that with the increase ofε, the probability of correct recognition of the initial state gradually increases. Under the same ε, the lower the order, the higher the estimation rate of the algorithm. The number of taps has little effect on the estimation rate. When ε is low, the probability of correct recognition can be improved by increasing the length of the synchronous scrambler sequence. According to Figures 10 and 11, the algorithm is greatly affected by the order of the feedback polynomial, and the number of taps does not affect the performance of the algorithm. The lower the order, the higher the estimation rate of the algorithm.


### 5.2.3. Influence of BER on the Initial State Recognition Rate

Whenε=0.1 and N=1000, the curves of the probability of correct recognition versus the BER corresponding to different orders and numbers of taps are shown in Figure 12.Figure 12
Distribution of probability of correct recognition versus bit error rate.It can be seen from the figure that with the increase of the BER, the probability of correct recognition of the initial state gradually decreases. When the intercepted scrambled data are only 1000 and the BER is10−2, the probability of correct recognition of the algorithm can reach over 90%.


## 5.2.1. Influence of SNR on the Initial State Recognition Rate

Whenε=0.1 and N=1000, the curve of probability of correct recognition corresponding to different orders and tap numbers with SNR is shown in Figure 10.Figure 10
Distribution of probability of correct recognition with SNR.It can be seen from the figure that with the increase of SNR, the probability of correct recognition of the initial state gradually increases, and it has a good ability to adapt to low SNR. AtSNR=0dB, the probability of correct recognition of the algorithm can reach 90%. Under the same SNR, the lower the order, the higher the estimation rate of the algorithm. Under the same order, the number of taps has little effect on the estimation rate.


## 5.2.2. Influence of Source Imbalance Degree on the Initial State Recognition Rate

WhenSNR=6dB and N=1000, the variation curve of the probability of correct recognition corresponding to different orders and tap numbers with the source imbalance is shown in Figure 11.Figure 11
Distribution of probability of correct recognition with source imbalance.It can be seen from the figure that with the increase ofε, the probability of correct recognition of the initial state gradually increases. Under the same ε, the lower the order, the higher the estimation rate of the algorithm. The number of taps has little effect on the estimation rate. When ε is low, the probability of correct recognition can be improved by increasing the length of the synchronous scrambler sequence. According to Figures 10 and 11, the algorithm is greatly affected by the order of the feedback polynomial, and the number of taps does not affect the performance of the algorithm. The lower the order, the higher the estimation rate of the algorithm.


## 5.2.3. Influence of BER on the Initial State Recognition Rate

Whenε=0.1 and N=1000, the curves of the probability of correct recognition versus the BER corresponding to different orders and numbers of taps are shown in Figure 12.Figure 12
Distribution of probability of correct recognition versus bit error rate.It can be seen from the figure that with the increase of the BER, the probability of correct recognition of the initial state gradually decreases. When the intercepted scrambled data are only 1000 and the BER is10−2, the probability of correct recognition of the algorithm can reach over 90%.


## 6. Conclusion

In this paper, we propose a blind reconstruction method for the synchronous scrambler. We introduce the average verification conformity degree to determine the probability of the verification equation being correctly established. The average verification conformity degree obeys different distributions when the correct polynomial is traversed and when the incorrect polynomial is traversed. Then, the optimal discrimination threshold is established by analyzing the statistical characteristics of the average verification conformity degree to complete the identification of the feedback polynomial. Finally, we propose a method for finding the check equation and solving the error-containing equation when the order of the feedback polynomial is greater than 15, which greatly reduces the number of initial states that need to be traversed. Simulation results show that the proposed algorithm has fairly good performance under the influences of different factors. Moreover, the new method is extremely robust to BER and low SNR. Compared with the existing algorithms, the performance of the proposed algorithm is significantly improved, and its computational complexity is reasonable.



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