CN111525976A - A covert communication method based on normal random process mean parameter modulation - Google Patents

Abstract

A covert communication method based on normal random process mean parameter modulation. The normal random sequence with non-zero mean is used as the carrier, and the hidden bits transmitted by legitimate users are used to modulate the mean polarity of the normal random sequence, and then the carrier and bipolar Multiplying the pseudo-random codes of , so that the covert signal of the whole bit period has the characteristics of zero mean; the covert signal transmitted by this method is very similar to the Gaussian noise that must exist in the receiver, so that the eavesdropper is not easy to detect the covert communication. The presence. The invention provides the structure diagram of the covert communication system, the probability density function of the covert signal, the mean value estimator, and deduces the bit error rate formula under the additive white Gaussian noise channel. The relationship between the number of samples, the signal-to-noise ratio, and the average value of the carrier. The invention is easy to implement, has low requirements on the computing performance of the equipment, and has good concealment.

Description

A covert communication method based on normal random process mean parameter modulation

technical field

The invention relates to the technical field of wireless communication and information security, in particular to a covert communication method based on normal random process mean value parameter modulation.

Background technique

In current and future wireless communication devices, in the process of data interaction between legitimate users, the issue of data security has received more and more attention. One solution is to encrypt the network connection between the transport layer and the application layer through standard protocols such as SSL/TLS or a proprietary protocol. But encryption also introduces another problem. The encryption technology encrypts the plaintext into a bunch of garbled characters, but instead reminds the eavesdropper that there is important information in the transmitted data. Eavesdroppers may not be able to decipher the content of the communication between legitimate users, but they can transmit interference signals, making the communication between legitimate users unable to proceed normally.

The present invention adopts the method of covert communication, and the sent covert signal has random statistical characteristics similar to environmental interference, so that eavesdroppers cannot distinguish the source of the received signal (the covert signal sent by legitimate users, or the environmental interference under multi-user conditions) . With this kind of covert communication method with low probability of interception, eavesdroppers are likely to fail to detect the existence of covert transmissions between legitimate users, let alone crack the transmitted information, so that important information of users is not eavesdropped.

The additive white Gaussian noise channel is a classic channel in communication. The invention uses a non-zero mean normal distribution random sequence as a carrier, combined with a bipolar pseudo-random code, and proposes a covert communication method hidden in environmental interference.

SUMMARY OF THE INVENTION

In order to enable legal users to realize the safe transmission of covert data, the present invention proposes a covert communication method based on normal distribution random process mean parameter modulation. The Gaussian noise that must exist in the network has similar statistical characteristics, so that the eavesdropper cannot distinguish whether the received signal comes from environmental interference or a concealed signal, which has a strong concealment. In addition, the structure of the covert communication system and the synchronization of the pseudo-random code are unknown to the eavesdropper, which increases the difficulty for the eavesdropper to decipher.

The present invention provides the following technical solutions:

A covert communication method based on normal random process mean value parameter modulation, the covert communication method includes information hiding processing of a covert communication system transmitter and information recovery processing of a covert communication system receiver;

The information concealment processing of the described covert communication system transmitter includes the following steps:

1-1: The random sequence generated by the normal distribution noise generator, the mean value of which is modulated by the hidden bit b, that is, when b is '0', within a bit period, as the normal distribution noise generator of the carrier, Generate a normally distributed random sequence of length n, mean μ, and variance σ ² , denoted as {x _k , k=1,...,n}～N(μ,σ ² ); for b it is '1 ', the normal distribution noise generator generates a normal distribution random sequence with mean -μ and variance σ ² , denoted as {x _k , k=1,...,n}～N(-μ,σ ² ) ;

1-2: Pseudo-random code generator, in one bit period, generates an orthogonal or quasi-orthogonal pseudo-random code sequence of length n, unipolar and bipolar conversion, and converts unipolar '0', '1 ', respectively converted to bipolar '+1', '-1', the pseudo-random sequence after single-bipolar transformation, denoted as {m _k , k=1,...,n};

1-3: The bipolar pseudo-random code sequence is multiplied by the normal distribution sequence modulated by the concealed bit to obtain the concealed signal sequence {s _k =m _k x _k , k=1,...,n}, The covert signal exhibits the characteristic of zero mean normal distribution, that is, the mean of each bit covert signal sequence is 0;

The information recovery process of the covert communication system receiver includes the following steps:

2-1: In the case of synchronization, the receiver generates a pseudo-random sequence consistent with the transmitter; the pseudo-random sequence undergoes unipolar and bipolar transformation, and the unipolar '0' and '1' are respectively transformed into For bipolar '+1', '-1', denote the synchronous bipolar pseudo-random code as {m _k , k=1,...,n};

然后{r_k，k＝1，...，n}与同步双极性的伪随机序列{m_k，k＝1，...，n}与相乘，得到{u_k＝r_km_k，k＝1，...，n}序列，送入到均值估计器中，从长度为n的{y_k}随机序列中，得到均值这个参数的估计值

估计器为：2-2: n random sequences received by the receiver in one bit period {r _k =s _k wk,k=1,...,n, where wk,k=1,...,n are additive White Gaussian noise sequence with mean 0 and variance

Then {r _k , k=1,...,n} is multiplied by the synchronous bipolar pseudo-random sequence {m _k , k=1,...,n} to obtain {u _k =r _k m _k , k=1,...,n} sequence is sent to the mean estimator, and the estimated value of the mean parameter is obtained from the {y _k } random sequence of length n

The estimator is:

硬判决的规则是：2-3: Use hard decision to get an estimate of hidden bits

The rules for hard judgment are:

In a covert communication system, a pair of synchronous pseudo-random code generators are needed. The functions of the pseudo-random code here are: first, to make the transmitted covert signal exhibit zero-average characteristics; second, to distinguish different users, the receiver only has Consistent with the pseudo-random code of the transmitter, the non-zero mean normal random sequence modulated by the concealed bits can be recovered from the received signal; third, the eavesdropper must master the structure of the concealed communication system and the synchronous pseudo-random code in order to decipher the concealment communication.

Further, the pseudo-random code selects an approximately orthogonal Gold code.

If the pseudo-random code sequence is used to distinguish users, the pseudo-random code sequence is the same for each hidden bit.

具体地说，随机变量X～N(μ，σ²)，离散等概率随机变量M，其分布列P(M＝-1)＝P(M＝1)＝1/2，M与X相互独立，令S＝MX，则S的概率密度函数为Further, as the non-zero mean normal distribution random noise of the carrier, when the ratio of the mean μ to the standard deviation σ is less than 0.4, that is, μ≤kσ (0<k≤0.4), it is different from the bipolar pseudo-random code. After the sequences are multiplied, the hidden signal obtained is approximately a normally distributed random noise with zero mean, and its variance is

Specifically, random variables X～N(μ, σ ² ), discrete equal-probability random variable M, its distribution sequence P(M=-1)=P(M=1)=1/2, M and X are independent of each other , let S=MX, then the probability density function of S is

When μ≤kσ(0<k≤0.4), f _S (s) is approximated by g _S (s)

in the formula

为Going a step further, the mean estimator

for

Correspondingly, when the pseudo-random code sequence of the receiver is synchronized with the pseudo-random code sequence of the transmitter, the mean of the estimator is

The variance of the estimator is

为加性白高斯噪声的方差。in the formula

is the variance of additive white Gaussian noise.

When the probability of hidden bits '0', '1' is equal, and the pseudo-random code sequences of the sender and receiver are synchronized, the bit error rate ρ under the additive white Gaussian channel is

该公式指出了误比特率与采样个数n、隐蔽信号的均值μ与标准差之比r、以及信噪比r之间的关系，当收发双方的伪随机码序列不同步时，误比特率

这说明窃听者如果不能与发射机的伪随机码同步，无法获取任何有效信息。where the complementary error function is defined as

This formula points out the relationship between the bit error rate and the number of samples n, the ratio of the mean value μ to the standard deviation r of the covert signal, and the signal-to-noise ratio r. When the pseudo-random code sequences of the sender and receiver are not synchronized, the bit error rate

This means that an eavesdropper cannot obtain any valid information if he cannot synchronize with the pseudo-random code of the transmitter.

With the increase of SNR, the bit error rate decreases, but there is an error floor (error floor). When the SNR tends to infinity, the value of the floor of the bit error rate is the bit error rate of the ideal channel, which is equal to

In the present invention, the covert communication system includes a covert transmitter and a receiver. At the transmitter, the covert bits modulate the mean value parameter of the normally distributed random carrier; at the receiver, the mean value is estimated, thereby recovering the covert bits;

The transmitter in the covert communication system, for the covert bit b is '0', in a bit period, as a normal distribution noise generator of the carrier, generates a positive signal with a length of n, a mean value of μ, and a variance of σ ² . Normally distributed random sequence, denoted as {x _k , k= ¹ , . , a normal distribution random sequence with variance σ ² , denoted as {x _k , k=1,...,n}～N(-μ,σ ² ). Then, the normal distribution random sequence with non-zero mean value of n is multiplied by the bipolar, equal probability pseudo-random code sequence {m _k , k=1,...,n} to obtain the covert signal sequence { _sk = m _k x _k , k=1, . . . , n}. The covert signal exhibits the characteristic of zero mean normal distribution, that is, the mean value of each bit covert signal sequence is 0, which is similar to the additive white Gaussian noise in wireless communication.

经过硬判决，得到隐蔽比特的估计。The receiver in the covert communication system multiplies the received sequence with the synchronized pseudo-random code sequence to recover a normal random sequence with a non-zero mean value; obtains an estimated value of the mean value through the mean value estimator

After a hard decision, an estimate of the hidden bits is obtained.

In covert communication systems, a pair of synchronized pseudo-random code generators are required. The approximately orthogonal Gold code is selected as the pseudo-random code. The functions of the pseudo-random code: firstly, it makes the transmitted covert signal show the characteristic of zero mean value; secondly, it is used to distinguish different users; thirdly, it increases the difficulty for eavesdroppers to crack.

The idea of the present invention is as follows: a non-zero mean normal distribution random sequence is used as a carrier, and the transmitted concealed bits modulate the mean polarity of the carrier, and then multiplied with a bipolar pseudo-random code, so that the concealed signal presents a zero mean positive This is very similar to the Gaussian noise that must exist in the receiver, and the eavesdropper is not sensitive to it, thus achieving the purpose of covert communication.

The beneficial effects of the invention are as follows: the given covert communication system has very low computational complexity and is suitable for devices with weak computing power such as the Internet of Things; the formula of the probability density function of covert signals and the performance analysis of the estimator are given, And the bit error rate formula of covert communication system under additive white Gaussian noise channel is deduced.

Description of drawings

Figure 1 is a structural diagram of a covert communication system.

FIG. 2 is a graph of relative error when the probability density function of the mixed random process when μ=0.4σ and σ=1 is approximated to a normal distribution.

Figure 3 is a normal distribution probability diagram of the estimator when n=1000 and the channel signal-to-noise ratio is -10dB.

Figure 4 shows the bit error rate of the covert system under the additive white Gaussian noise channel when μ=0.4σ and n=100.

Detailed ways

The present invention will be described in detail below with reference to the accompanying drawings.

1 to 4, a covert communication method based on normal distribution random process mean parameter modulation, information hiding processing of a covert communication system transmitter and information recovery processing of a covert communication system receiver;

The information concealment processing of the described covert communication system transmitter includes the following steps:

1-1: Use a normal distribution noise generator to generate a random distribution sequence that obeys a distribution with mean (-1) ^b μ and variance σ ² , denoted as N((-1) ^b μ, σ ² ), where b is a binary hidden bit, b∈{0,1}; μ and σ are real numbers greater than 0, set the bit period as T _b , and generate n independent and identically distributed random sequences within a T _b , denoted as {x _k , k= ¹ , . N(-μ,σ ² ) distributed random sequence;

1-2: The pseudo-random code generator generates pseudo-random codes with probabilities of '0' and '1', and after unipolar and bipolar conversion, the unipolar '0' and '1' are converted to bipolar respectively. '+1', '-1'. Denote the bipolar pseudo-random code sequence as {m _k , k=1,...,n};

1-3: The random sequence output by the normally distributed noise generator is multiplied by the bipolar pseudo-random sequence, denoted as {s _k =x _k m _k , k=1,...,n}; single and double After multiplying the '-1' of the polarity transformation, the polarity of the mean value of the random sequence is changed; multiplying with '+1' does not change the polarity of the mean value of the random sequence; the sequence _sk containing the hidden information is transmitted to the channel go;

Let M be a discrete random variable, and the distribution is listed as P(M=-1)=P(M=1)=1/2; let X be a continuous random variable, and the probability density function is f _X (x); M and X is independent of each other; when M=m, the conditional probability density function of X is f _X|M (x|m); let S=MX, it can be proved that S is a continuous random variable, and the probability density function is f _S (s) for

Subsequent accompanying drawings 2 and 3 verify that under certain conditions, the random variable S can be approximated to a normal distribution, and further, from formula (1), f _S (-s)=f _S (s), also That is, the transmitted {s _k , k=1, . However, after the multiplication of bipolar pseudo-random codes, it is approximately a normal distribution with a mean of 0. In this way, the transmitted signal containing hidden information, like the noise of a Gaussian channel, obeys the normal value of 0. distribution, it is difficult for eavesdroppers to distinguish, therefore, it has good concealment.

The information recovery process of the covert communication system receiver includes the following steps:

2-1: In the case of synchronization, the receiver generates a pseudo-random sequence consistent with the transmitter; the pseudo-random sequence undergoes unipolar and bipolar conversion, and the unipolar '0' and '1' are converted to bipolar respectively. Polarity '+1', '-1', denote the synchronous bipolar pseudo-random code as {m _k , k=1,...,n};

2-2: The sequence after the additive white Gaussian noise channel is {r _k =s _k +n _k , k1,...,n; then with the synchronous bipolar pseudo-random sequence mk,,k=1,... . and n are multiplied to obtain {r _k m _k , k=1, ..., n} sequence, which is sent to the mean estimator, and the mean is estimated as:

硬判决的规则是：2-3: Use hard decision to get an estimate of hidden bits

The rules for hard judgment are:

Referring to FIG. 2, a covert communication method based on normal distribution random process mean parameter modulation, the covert signal including covert bits is approximately normal distribution with zero mean value, and the details are as follows:

Formula (1) gives the probability density function of the covert signal, where the probability density function of the random variable X is:

Therefore, the probability density function of the covert signal is:

When μ≤kσ(0＜k≤0.4), f _S (s) can be approximated by g _S (s),

图2给出了σ＝1，k＝0.4时，公式(5)近似为公式(6)时的相对误差，相对误差定义为in

Figure 2 shows the relative error when σ=1, k=0.4, formula (5) is approximated to formula (6), and the relative error is defined as

The value range in Figure 2 is [-3σ _S , 3σ _S ], and according to the "3σ" criterion of normal distribution, 99.7% probability falls within this range. In addition, taking different σ values, the curve shape of the graph is exactly the same. As can be seen from the figure, the maximum relative error is within 6%, and this approximation is acceptable. In particular, for smaller values of k, the approximation works well. For example, k=0.3, the maximum relative error is within 2%; k=0.2, the maximum relative error is within 0.4%.

的正态分布随机过程。经过加性白高斯信道后的输出为R＝S+W＝MX+W (8)Referring to FIG. 3 , the probability map of the normal distribution of the estimator when n=1000 and the channel signal-to-noise ratio is -10dB. The transmitted covert signal S=MX, and its probability density function is formula (1). When μ≤kσ (0<k≤0.4) is selected, it can be approximated as follows: the given mean value of formula (6) is 0 and the variance is

A normally distributed random process. The output after the additive white Gaussian channel is R=S+W=MX+W (8)

的白高斯噪声，定义信噪比为隐蔽信号的功率与加性白高斯噪声的功率之比，也即where W is the mean value of 0 and the variance is

The white Gaussian noise is defined as the ratio of the power of the covert signal to the power of the additive white Gaussian noise, that is,

If the receiver's pseudo-random code generator M' is synchronized with the transmitter, M'=M, then the input mean estimator is

另外，M′W服从均值为0、方差为

的正态分布。收发两端的伪随机码序列同步时，X与M′W由于来源不同，它们相互独立，则Y服从均值为μ、方差为

的正态分布。如公式(2)所示，均值μ的估计器为When the pseudo-random code sequence of the receiver is synchronized with the pseudo-random code sequence of the transmitter, that is, when M'=M, M'M=1;

In addition, M'W obeys the mean of 0 and the variance of

normal distribution. When the pseudo-random code sequences at the sending and receiving ends are synchronized, X and M′W are independent of each other due to different sources, so Y obeys the mean value μ and the variance is

normal distribution. As shown in Equation (2), the estimator for the mean μ is

Correspondingly, the mean of the estimator is

The variance of the estimator is

It can be verified through simulation that the estimator obeys a normal distribution, as shown in Figure 3, and the simulation conditions are n=1000, k=0.4, and r=-10dB.

Obviously, when the pseudo-random code sequences at the sending and receiving ends are not synchronized, Y=M'W does not contain the concealed bit modulation signal X, and the concealed bits cannot be demodulated, that is, the bit error rate is 0.5.

Referring to FIG. 4 , the bit error rate performance curve of covert communication under additive white Gaussian noise channel. The simulation conditions are n=100, k=0.4, r=[-10, 10]dB. It can be seen from the simulation diagram that the simulation is very close to the theoretical derivation. The theoretical bit error rate is derived as follows:

Combining equations (12) and (13), the probability density function of the estimator is

Combined with the hard decision rule given by formula (3), the bit error rate ρ is

从该公式可以看出，增大比特采样个数n、隐蔽信号的均值、以及信噪比，都可以降低误比特率。然而，增大数n，降低了传输速率；增大均值μ，增加了隐蔽信号的发射功率，降低了隐蔽性。从图中可看出，在低信噪比的情况下，误比特率比较高，也即高斯噪声对估计器的影响比较大；另一方面，当信噪比较大时，存在一个误比特率平台(error floor)，也即无论信噪比多高，再也不能降低误比特率了。这是因为此时的极限即为理想信道，所以误比特率平台值即为理想信道时的误比特率，等于

where the complementary error function is defined as

It can be seen from this formula that increasing the number of bit samples n, the mean value of the covert signal, and the signal-to-noise ratio can reduce the bit error rate. However, increasing the number n reduces the transmission rate; increasing the mean value μ increases the transmit power of the concealed signal and reduces the concealment. It can be seen from the figure that in the case of low signal-to-noise ratio, the bit error rate is relatively high, that is, the Gaussian noise has a relatively large influence on the estimator; on the other hand, when the signal-to-noise ratio is large, there is a bit error rate. The error floor, that is, no matter how high the signal-to-noise ratio is, the bit error rate can no longer be reduced. This is because the limit at this time is the ideal channel, so the platform value of the bit error rate is the bit error rate of the ideal channel, which is equal to

Claims (7)

1. A hidden communication method based on normal random process mean parameter modulation is characterized in that: the covert communication method comprises information hiding processing of a covert communication system transmitter and information recovery processing of a covert communication system receiver;

the information hiding process of the covert communication system transmitter comprises the following steps:

1-1: the random sequence generated by the normal distribution noise generator has its mean value modulated by the hidden bits, i.e. when b is '0', the normal distribution noise generator as the carrier generates the random sequence with length n, mean value mu and variance sigma in one bit period²Is normally distributed and randomly ordered as { x_k，k＝1，...，n}～N(μ，σ²) (ii) a For b of '1', a normal distribution noise generator generates a mean of- μ and a variance of σ²Is normally distributed and randomly ordered as { x_k，k＝1，...，n}～N(-μ，σ²)；

1-2: a pseudo-random code generator for generating an orthogonal or quasi-orthogonal pseudo-random code sequence with length n in one bit period, a single-polarity and double-polarity conversion for converting unipolar '0' and '1' into bipolar '+ 1' and '1', respectively, and a pseudo-random sequence after the single-polarity and double-polarity conversion is marked as { m_k，k＝1，...，n}；

1-3: multiplying bipolar pseudo-random code sequence by normal distribution sequence modulated by hidden bit to obtain hidden signal sequence { s_k＝m_kx_kK is 1,.. multidot.n }, the concealment signal exhibits the characteristic of zero-mean normal distribution, i.e. the mean value of each bit concealment signal sequence is 0;

the information recovery process of the covert communication system receiver comprises the following steps:

2-1: in case of synchronization, the receiver generates a pseudo-random sequence that is identical to the transmitter; the pseudo-random sequence is converted into unipolar '0' and unipolar '1' through single-polarity and bipolar '1' and bipolar pseudo-random code { m_k，k＝1，...，n}；

2-2: n random sequences r received by the receiver in one bit period_k＝s_k+ wk, k 1., n, where wk, k 1., n is an additive white gaussian noise sequence with a mean of 0 and a variance of 0

Then r_kK 1.. multidata, n is synchronized with the pseudo-random sequence of the synchronous bipolarity { m }_kMultiplying k by 1.. multidot.n to obtain { y }_k＝r_km_kThe sequence k 1.. multidot.n is fed to the mean estimator, and then the length n is used as the length n of the sequence y_kObtaining the estimated value of the parameter of mean value in random sequence

The estimator is as follows:

2-3: obtaining estimates of concealed bits using hard decisions

The hard decision rule is:

in covert communication systems, a pair of synchronized pseudo random code generators are required, where the pseudo random codes function: firstly, the transmitted concealed signal presents the characteristic of zero mean value; secondly, the method is used for distinguishing different users, and the receiver can recover the non-0-mean normal random sequence modulated by the concealed bit from the received signal only if the receiver is consistent with the pseudo-random code of the transmitter; thirdly, the eavesdropper must master the structure of the covert communication system and the synchronous pseudo-random code to break the covert communication.

2. The hidden communication method based on normal random process mean parameter modulation as claimed in claim 1, characterized in that: the pseudo-random codes pick approximately orthogonal Gold codes.

3. The hidden communication method based on normal random process mean parameter modulation as claimed in claim 2, characterized in that: the pseudo-random code sequence, if used to distinguish users, is the same at each concealed bit.

4. The hidden communication method based on normal random process mean parameter modulation as claimed in any of claims 1 to 3, wherein: non-0-mean normal distribution as a carrierWhen the ratio of the mean value mu to the standard deviation sigma is less than 0.4, namely mu is less than or equal to K sigma (K is more than 0 and less than or equal to 0.4), the hidden signal obtained after multiplication with the bipolar pseudo-random code sequence is approximate to the normal distribution random noise with zero mean value, and the variance is

Specifically, the random variables X to N (μ, σ)²) The distribution column P (M ═ 1) ═ 1/2, M and X are independent from each other, and when S is MX, the probability density function of S is

When mu is not more than k sigma (k is more than 0 and not more than 0.4), f_SG for(s)_S(s) approximation

In the formula

5. The hidden communication method based on normal random process mean parameter modulation as claimed in any of claims 1 to 3, wherein: mean estimator

Is composed of

Accordingly, when the pseudorandom code sequence of the receiver is synchronized with the pseudorandom code sequence of the transmitter, the estimator has an average value of

The variance of the estimator is

In the formula

Is the variance of additive white gaussian noise.

6. The hidden communication method based on normal random process mean parameter modulation as claimed in any of claims 1 to 3, wherein: when the hidden bit '0' and '1' are synchronous and the pseudo-random code sequences of the transmitting and receiving parties are synchronous, the bit error rate rho under the additive white Gaussian channel is

The complementary error function in the formula is defined as

The formula indicates the relationship between the bit error rate and the number of samples n, the ratio r of the mean value mu and the standard deviation of the concealed signal and the signal-to-noise ratio r, when the pseudo-random code sequences of the two parties of the transmitter and the receiver are not synchronous, the bit error rate is different

This means that an eavesdropper cannot acquire any valid information if it cannot synchronize with the pseudorandom code of the transmitter.

7. The hidden communication method based on normal random process mean parameter modulation as claimed in claim 6, wherein: with the increase of the signal-to-noise ratio, the bit error rate is reduced, but there is a bit error rate platform error floor, when the signal-to-noise ratio tends to be infinite, the bit error rate platform value is the bit error rate when the ideal channel is equal to the bit error rate when the signal-to-noise ratio is equal to the ideal channel

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