CN111246463A - Covert communication method based on skew parameter of alpha stable distribution random process - Google Patents

Covert communication method based on skew parameter of alpha stable distribution random process Download PDF

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CN111246463A
CN111246463A CN202010083698.3A CN202010083698A CN111246463A CN 111246463 A CN111246463 A CN 111246463A CN 202010083698 A CN202010083698 A CN 202010083698A CN 111246463 A CN111246463 A CN 111246463A
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徐志江
金文兵
程文锋
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Zhejiang Institute of Mechanical and Electrical Engineering Co Ltd
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04W—WIRELESS COMMUNICATION NETWORKS
    • H04W12/00—Security arrangements; Authentication; Protecting privacy or anonymity
    • H04W12/03—Protecting confidentiality, e.g. by encryption
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04W—WIRELESS COMMUNICATION NETWORKS
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Abstract

A hidden communication method based on skew parameters in an alpha stable distribution random process is characterized in that a skewed alpha stable distribution random sequence is used as a carrier, and hidden bits transmitted by legal users modulate the skew polarity of the carrier sequence in the front half bit period and the back half bit period to make the absolute values of the carrier sequences identical but the polarities opposite, so that hidden signals in the whole bit period have the characteristic of symmetrical alpha stable distribution; the covert signal transmitted by this method is very similar to the symmetric alpha-stationary distributed interference present in many wireless communication networks, so that the presence of covert communication is not easily perceived by an eavesdropper. The invention provides an estimation formula of skewness with very small calculation amount and deduces a corresponding probability density function to obtain a bit error rate formula, and indicates the relationship between the bit error rate and the characteristic index, the skewness and the sample number. The invention has the advantages of easy realization, low requirement on the computing performance of equipment and better concealment.

Description

Covert communication method based on skew parameter of alpha stable distribution random process
Technical Field
The invention relates to the technical field of wireless communication and information security, in particular to a covert communication method based on skewness parameters of an alpha stable distribution random process.
Background
In daily life, communication technologies such as LTE and 5G, WiFi are becoming increasingly unknown. Currently and in the future, more and more attention is paid to communication security issues related to protection of personal privacy, property data and the like. One approach is to encrypt the network connection between the transport layer and the application layer through SSL and TLS. However, another problem is introduced during encryption, and encryption technology encrypts plaintext into a bunch of messy codes and reminds an eavesdropper that important information exists in transmitted data. Another method is to use a covert communication method, so that the transmitted covert signal has a random characteristic similar to that of environmental interference, and an eavesdropper cannot distinguish the source of noise (from the covert signal transmitted by a legitimate user or the environmental interference under multiple users). Therefore, the existence of hidden transmission of a legal user cannot be detected under the strict monitoring of an eavesdropper, and the transmitted information is not cracked, so that the key that the important information of the user is not eavesdropped is realized.
Many documents state that in wireless networks such as ad hoc, Wi-Fi, cellular and LET, multiple access interference follows a symmetrical alpha-stationary distribution rather than the generally assumed gaussian (normal) distribution. The invention provides a hidden communication method hidden in environmental interference by utilizing skewness parameters in alpha stable distribution.
Disclosure of Invention
In order to overcome the defects of the prior art and enable a legal user to realize the safe transmission of the covert data, the invention provides a covert communication method based on an alpha stable distribution random process skew parameter, which is easy to realize, has low requirement on the calculation performance of equipment and has better concealment.
The invention provides the following technical scheme:
a hidden communication method based on alpha stable distribution random process skewness parameter comprises information hiding processing of a sending end of a hidden communication system and information recovery processing of a receiving end of the hidden communication system;
the information hiding processing of the sending end of the covert communication system comprises the following steps:
1-1 generating a randomly distributed sequence, denoted S (α), following an alpha-stationary distribution with a characteristic index of α and a skewness of β with a stationary distribution noise generator over a bit period TbHas N random sequences, marked as { Xk，k＝1，...，N}；
1-2: the masked bit b, encoded by Manchester, at an intermediate time of each bit period, i.e. TbAt/2, a jump occurs; if b is 1, jumping from low level to high level; if b is equal to 0, jumping from high level to low level, single-polarity and double-polarity conversion is carried out, and unipolar '0' and unipolar '1' are respectively converted into bipolar '-1' and bipolar '-1';
1-3, after multiplying the random sequence output by the stable distribution noise generator with '-1' of single and double polarity transformation, the polarity of the skewness is changed, namely the random sequence distributed by S (α, - β) is obtained, and after multiplying with '+ 1', the random sequence still distributed by S (α) is obtained;
thus, if the concealment bit b is 0, the random sequence s of the first half bit period0＝{-XkN/2 is a random sequence S following the distribution of S (α, - β), the random sequence S of the second half of the bit period1＝{XkN is a random sequence that follows the distribution of S (α), whereas if the concealment bit b is 1, the random sequence S of the first half bit period0＝{XkN/2 is a random sequence S following the distribution of S (α), the second half of the bit period1＝{XkN is a random sequence that follows the distribution of S (α, - β);
the information recovery processing of the covert communication system receiving end comprises the following steps:
2-1: in the case of synchronization, the receiving end is in one bit period TbAfter demultiplexing the received N random sequences, the N random sequences are divided into 2 sequences of length N/2, i.e. r ═ r0，r1](ii) a Then make a pair of r1X (-1) processing such that the skewness sign is the same for the first and second half bit periods; and multiplexing the 2 random sequences with the length of N/2 to combine the random sequences into a random sequence with the length of N, and recording the random sequence as the random sequence with the length of N
Figure BDA0002381241190000031
Under ideal channel, i.e. without attenuation, delay and noise, it is obvious that
Figure BDA0002381241190000032
2-2: from length N
Figure BDA0002381241190000033
In random sequences, the estimated value of the parameter skewness is estimated
Figure BDA0002381241190000034
The estimated value of skewness is
Figure BDA0002381241190000035
2-3: obtaining estimates of concealed bits using hard decisions
Figure BDA0002381241190000036
The hard decision rule is:
Figure BDA0002381241190000037
further, the estimated value of skewness
Figure BDA0002381241190000038
Has a probability density function of
Figure BDA0002381241190000039
In the formula (I), the compound is shown in the specification,
Figure BDA00023812411900000310
on the basis, the estimator is an approximate unbiased estimation obtained by numerical integration through mathematic mathematical tool software, and when the characteristic index α tends to be 1 or 2, the variance becomes large, and the performance of the estimator deteriorates.
Still further, when the hidden bits '0', '1', etc. are at equal probability, the bit error rate ρ under the ideal channel is
Figure BDA0002381241190000041
Wherein Pr (e) represents the probability of event Pr (e); sign (t) is a sign function defined as
Figure BDA0002381241190000042
The error function erf (z) is defined as
Figure BDA0002381241190000043
The equation indicates the relationship between the bit error rate and the characteristic index α, the skew β, and the number of samples N.
Further, in the case of an additive white Gaussian noise channel, the divergence ratio is defined as
Figure BDA0002381241190000044
Where gamma is the divergence of the alpha plateau,
Figure BDA0002381241190000045
the technical idea of the invention is that the covert communication system comprises a covert transmitter and a receiver, wherein at a transmitting end, the covert bit modulates a skewness β parameter of alpha stable distribution random noise, and at a receiving end, the skewness β value is estimated, so that the covert bit is recovered.
The transmitter in the covert communication system has '0' for the covert bit b, and transmits the random distribution of S (α, - β) distribution in the first half bit periodThe noise, the second half bit period emits S (α) distributed random distributed noise, the first half bit period emits S (α) distributed random distributed noise and the second half bit period emits S (α, - β) distributed random distributed noise, the hidden bit b is '1', the first half bit period emits S (α) distributed random distributed noise, each bit period has { X }kThe skewness of the N/2 samples before and after the N samples is different, but noise with symmetrical alpha stable distribution is presented in the whole bit period, namely, the skewness β of the whole sample is 0, which is similar to the environmental noise with symmetrical alpha stable distribution in the wireless network.
A receiver in the covert communication system obtains a synchronous position through the fraction low-order covariance; obtaining the estimated value of skewness from the received N samples in one bit period through the symbol fraction low-order moment
Figure BDA0002381241190000051
And obtaining the estimation of the hidden bit through hard decision.
The hidden signal transmitted by the invention is similar to the environmental interference in the wireless network, namely, an eavesdropper cannot easily distinguish whether the received signal is the environmental interference or the hidden signal, and the hidden signal has strong concealment. In addition, the structure of the covert communication system is unknown to the eavesdropper, which increases the difficulty of hacking by the eavesdropper.
Before the structure of a covert communication system is given, an alpha stable distribution is introduced. The random process has no closed probability density function expression and is defined by its characteristic function of
ΦX(t)＝exp[-iδt-|σt|α(1-iβsign(t)ξ(t，α))]
Here exp [ ·]An exponential function is defined as the function of the exponent,
Figure BDA0002381241190000061
Figure BDA0002381241190000062
constant α ∈ (0, 2)]Called characteristic index (characteristic index), β e [ -1, 1]Called skewness (shewness), σ ∈ R+Called scale, γ ═ σαCalled divergence (dispersion), δ ∈ R is called position (location) — take the alpha stable distribution as S (α, σ, δ), in the present invention, unless otherwise specified, δ is assumed to be 0, σ is 1, and simply S (α) — if a random variable X obeys the alpha stable distribution, it is taken as X to S (α).
The hidden bit to be transmitted is converted into an alpha stable distribution random sequence with opposite skewness symbols in the front half bit period and the rear half bit period, but the whole sequence presents the characteristic of symmetrical alpha stable distribution, is similar to the environmental interference in a wireless network, and is insensitive to an eavesdropper, so that the aim of hidden communication is fulfilled.
The invention has the beneficial effects that: the skewness estimation method based on the symbol fraction low-order moment has low calculation complexity and is suitable for equipment with weak calculation power, such as the Internet of things and the like; gives the skewness
Figure BDA0002381241190000063
The probability density function formula and the bit error rate formula of the covert communication system obtain the performance of the covert communication system.
Drawings
Fig. 1 is a block diagram of a covert communication system.
Fig. 2 shows the fractional low-order covariance when α is 1.43, β is 0.5, and N is 400.
Fig. 3 is a graph of probability density functions of the estimator when α is 1.43, β is 0.8, and N is 1000.
Fig. 4 shows the mean value of the estimator when α < 1 and N is 1000.
Fig. 5 shows the variance of α < 1, N1000 estimator.
Fig. 6 shows that α is 0.5, and the bit error rate is under the ideal channel for different numbers of samples.
Fig. 7 shows the bit error rate of α ═ 0.5, for different numbers of samples in the additive white gaussian noise channel.
Detailed Description
The present invention will now be described in detail with reference to the accompanying drawings.
Referring to fig. 1-7, a covert communication method based on alpha stable distribution random process skewness parameter includes information hiding processing at the transmitting end of covert communication system and information recovery processing at the receiving end of covert communication system,
the information hiding processing of the sending end of the covert communication system comprises the following steps:
1-1 generating a randomly distributed sequence following an S (α) distribution with a steady distribution noise generator, with a bit period of TbAt a time TbHas N random sequences, marked as { Xk，k＝1，...，N}；
1-2: the masked bit b, encoded by Manchester, at an intermediate time of each bit period, i.e. TbAt/2, jumping, if b is equal to 1, jumping from low level to high level; if b is equal to 0, jumping from high level to low level, single-polarity and double-polarity conversion is carried out, and unipolar '0' and unipolar '1' are respectively converted into bipolar '-1' and bipolar '-1';
1-3, after multiplying the random sequence output by the stable distribution noise generator with '-1' of single and double polarity transformation, the polarity of the skewness is changed, namely the random sequence distributed by S (α, - β) is obtained, and after multiplying with '+ 1', the random sequence still distributed by S (α) is obtained;
thus, if the concealment bit b is 0, the random sequence s of the first half bit period0＝{-XkN/2 is a random sequence S following the distribution of S (α, - β), the random sequence S of the second half of the bit period1＝{XkN is a random sequence that follows the distribution of S (α), whereas if the concealment bit b is 1, the random sequence S of the first half bit period0＝{XkN/2 is a random sequence S following the distribution of S (α), the second half of the bit period1＝{XkN is a random sequence that follows the distribution of S (α, - β);
the information recovery processing of the covert communication system receiving end comprises the following steps:
2-1: in the case of synchronization, the receiving end is in one bit period TbThe received N random sequences are r ═ r0，r1]From the description of the transmitting end, it can be seen that, no matter whether the hidden bit is '0' or '1', the skewness sign of the half bit period before and after transmission is opposite, therefore, r is done at the receiving end1Processing the x (-1) to make the skewness signs of the front and back half bit periods the same, wherein the demultiplexing divides the N random sequences into 2 random sequences with the length of N/2, and the multiplexing is just opposite, and combines the 2 random sequences with the length of N/2 into one random sequence with the length of N;
after multiplexing, it is recorded as
Figure BDA0002381241190000081
Under ideal channel, i.e. without attenuation, delay and noise, it is obvious that
Figure BDA0002381241190000082
2-2: from length N
Figure BDA0002381241190000083
In random sequences, the estimated value of the parameter skewness is estimated
Figure BDA0002381241190000084
The estimated value of skewness is
Figure BDA0002381241190000085
2-3: obtaining estimates of concealed bits using hard decisions
Figure BDA0002381241190000086
The hard decision rule is:
Figure BDA0002381241190000091
referring to FIG. 2, a covert communication method based on alpha stable distribution random process skewness parameter adopts a method of fractional low order covariance synchronously, and a random sequence { X with the length of N and complying with S (α) distributionkK 1., N }, the fractional low-order covariance is defined as:
Figure BDA0002381241190000092
wherein N is1＝max(0，-k)，N2Min (N-k, N), a + b < α, notation p of random variable X, defined as X<p>＝sign(X)|X|p
Fig. 2 shows a graph of the fractional low-order covariance of the signal transmitted by the concealment system to the channel in step 1-3, where α is 1.43, β is 0.5, N is 400, and a is b is α/4, it can be seen that the fractional low-order covariance is maximum only when k is 0, i.e. full synchronization, and besides, the fractional low-order covariance is small, and in particular, k is ± 400, ± 800, i.e. no periodicity occurs at ± 2 bit periods.
Referring to fig. 3, a probability density function curve of the estimator when α is 1.25, β is 0.8, and n is 1000 is given.
If the random variables X-S (α), then their signed fractional lower order moments are
Figure BDA0002381241190000093
Wherein E [ X ]]Is the expectation of a random variable X; gamma function
Figure BDA0002381241190000094
Figure BDA0002381241190000095
p∈(-2, -1) ∪ (-1, α.) when p is 0
Figure BDA0002381241190000101
Using ensemble averaging
Figure BDA0002381241190000102
Alternative statistical average E [ sign (X)]Obtaining an estimate of skewness
Figure BDA0002381241190000103
Is composed of
Figure BDA0002381241190000104
From the central limit theorem, S0Subject to a normal distribution with a mean value of
Figure BDA0002381241190000105
With a second order origin moment of
Figure BDA0002381241190000106
Thus, its variance is
Figure BDA0002381241190000107
That is to say S0Has a probability density function of
Figure BDA0002381241190000108
Known as S0Probability density function of
Figure BDA0002381241190000109
Then according to the correlation theorem in the random process, get
Figure BDA00023812411900001010
Function of probability densityNumber of
Figure BDA00023812411900001011
Is composed of
Figure BDA0002381241190000111
Referring to fig. 4 and 5, the mean and variance of the estimator when the sample number N is 1000 and the characteristic index α < 1 are given
Figure BDA0002381241190000112
The variance is expressed as
Figure BDA0002381241190000113
These two expressions have no closed solution, but can be numerically calculated using mathematic mathematical software. As can be seen from the view in figure 4,
Figure BDA0002381241190000114
it can be seen from fig. 5 that as the characteristic index α goes towards 1, the variance becomes larger and the performance of the estimator deteriorates, the same conclusion holds for α > 2.
Referring to fig. 6, the bit error rate p of the concealed bits '0', '1', etc. can be directly obtained by combining the hard decision rule based on the probability density function of the known estimator and the probability of the concealed bits '0', '1', etc. the bit error rate p of different sample numbers N in the ideal channel is shown when α is 0.5
Figure BDA0002381241190000121
Simulation results show that the skew degree and the number of samples are increased, and the bit error rate can be reduced. Meanwhile, the simulation result is very consistent with the theoretical derivation.
Referring to fig. 7, we show the bit error rate for additive white gaussian noise channel for different sample numbers N and skewness when α is 0.5. since the alpha-stationary random process has no variance (power) in the conventional sense, we use the divergence to define the signal-to-noise ratio, called the divergence ratio, defined as the ratio of the divergence
Figure BDA0002381241190000122
Where γ is the divergence of the alpha plateau, γGIs the divergence of the gaussian noise and is,
Figure BDA0002381241190000123
Figure BDA0002381241190000124
is the gaussian noise power. It can be seen from the graph that under the condition of low divergence ratio, the bit error rate is higher, that is, the influence of gaussian noise on the estimator is larger; on the other hand, when the divergence is relatively large, there is an error bit rate platform (error floor), that is, no matter how high the divergence ratio is, the bit error rate can not be reduced any more. This is because the limit at this time is the ideal channel, and therefore the bit error rate plateau value is the bit error rate at the ideal channel.

Claims (4)

1. A covert communication method based on skew parameters in an alpha stable distribution random process is characterized in that: the method comprises the steps of information hiding processing of a sending end of the covert communication system and information recovery processing of a receiving end of the covert communication system;
the information hiding processing of the sending end of the covert communication system comprises the following steps:
1-1 generating a randomly distributed sequence, denoted S (α), following an alpha-stationary distribution with a characteristic index of α and a skewness of β with a stationary distribution noise generator over a bit period TbHas N random sequences, marked as { Xk，k＝1，...，N}；
1-2: the masked bit b, encoded by Manchester, at an intermediate time of each bit period, i.e. TbAt/2, a jump occurs; jump from low level if b is 1Changing to a high level; if b is equal to 0, jumping from high level to low level, single-polarity and double-polarity conversion is carried out, and unipolar '0' and unipolar '1' are respectively converted into bipolar '-1' and bipolar '-1';
1-3, after multiplying the random sequence output by the stable distribution noise generator with '-1' of single and double polarity transformation, the polarity of the skewness is changed, namely the random sequence distributed by S (α, - β) is obtained, and after multiplying with '+ 1', the random sequence still distributed by S (α) is obtained;
thus, if the concealment bit b is 0, the random sequence s of the first half bit period0＝{-XkN/2 is a random sequence S following the distribution of S (α, - β), the random sequence S of the second half of the bit period1＝{XkK is N/2, …, N is a random sequence that follows the distribution of S (α), whereas if the concealment bit b is 1, the random sequence S of the first half bit period0＝{XkN/2 is a random sequence S following the distribution of S (α), the second half of the bit period1＝{XkN is a random sequence that follows the distribution of S (α, - β);
the information recovery processing of the covert communication system receiving end comprises the following steps:
2-1: in the case of synchronization, the receiving end is in one bit period TbAfter demultiplexing the received N random sequences, the N random sequences are divided into 2 random sequences of length N/2, i.e. r ═ r0，r1](ii) a Then make a pair of r1X (-1) processing such that the skewness sign is the same for the first and second half bit periods; and multiplexing the 2 random sequences with the length of N/2 to combine the random sequences into a random sequence with the length of N, and recording the random sequence as the random sequence with the length of N
Figure FDA0002381241180000021
Under ideal channel, i.e. without attenuation, delay and noise, it is obvious that
Figure FDA0002381241180000022
2-2: from length N
Figure FDA0002381241180000023
In random sequences, the estimated value of the parameter skewness is estimated
Figure FDA0002381241180000024
The estimated value of skewness is
Figure FDA0002381241180000025
2-3: obtaining estimates of concealed bits using hard decisions
Figure FDA0002381241180000026
The hard decision rule is:
Figure FDA0002381241180000027
2. the covert communication method based on alpha-steadily distributed random process skewness parameters of claim 1, wherein: skewness estimation
Figure FDA0002381241180000028
Has a probability density function of
Figure FDA0002381241180000029
In the formula (I), the compound is shown in the specification,
Figure FDA00023812411800000210
on the basis, the estimator is an approximate unbiased estimation obtained by numerical integration through mathematic mathematical tool software, and when the characteristic index α tends to be 1 or 2, the variance becomes large, and the performance of the estimator deteriorates.
3. The covert communication method based on alpha-steadily distributed random process skewness parameters of claim 1 or 2, wherein: when the hidden bits are '0', '1', etc., the bit error rate ρ under the ideal channel is
Figure FDA0002381241180000031
The equation indicates the relationship between the bit error rate and the characteristic index α, the skew β, and the number of samples N.
4. The covert communication method based on alpha-steadily distributed random process skewness parameters of claim 3, wherein: under the additive white Gaussian noise channel, the divergence ratio is defined as
Figure FDA0002381241180000032
Where gamma is the divergence of the alpha plateau,
Figure FDA0002381241180000033
the bit error rate is reduced with the increase of the divergence ratio, but a bit error rate platform exists, and when the divergence ratio tends to be infinite, the value of the bit error rate platform is the bit error rate of an ideal channel.
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