# A Method for Detecting Amplitude-Phase Joint Characteristic Parameters of Wireless Channel for Generating Key Parameters

Authors: Qiongying Tan; Shuanglin Huang; Sanjun Liu
Journal: Complexity
 (2021)

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## Abstract

Aiming at the problems of poor adaptability and low secret key generation rate of secret key generation scheme based on single characteristic parameter of wireless channel, a secret key generation method based on joint characteristic parameters of amplitude and phase of wireless channel is proposed. This method is based on the single-eavesdropping wireless fading channel model; the joint characteristic model of amplitude and phase of wireless fading channel is established; it detects and extracts the joint characteristic parameters of amplitude and phase of channel, and then the proposed characteristics are quantified by using the equal probability joint quantization strategy of amplitude and phase to generate the secret key random parameters. In this paper, the amplitude and phase joint feature parameter detection method of wireless channel can not only improve the generation rate of random secret key parameters but also make the eavesdropping party’s eavesdropping error rate closer to 0.5. The test results show that the proposed scheme can significantly improve the rate, reliability, and security of generating key random parameters.


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## Body

## 1. Introduction

With the rapid promotion of Internet of Things technology and the explosive growth of the number of wireless users, various wireless communication security issues have aroused widespread concern in the academic community [1, 2]. Due to the complexity of the wireless communication network topology, the openness of wireless channel transmission, and the mobility of wireless communication terminals, it is difficult for traditional communication security mechanisms based on modern cryptography to guarantee the security of wireless communication systems [3, 4]. With the development of artificial intelligence and the improvement of computing power, wireless information transmission is facing great challenges [5, 6]. Different from the traditional secrecy mechanisms that protect data security through encryption technology, the physical layer security technology through the clever use of the physical characteristics of the communication channel ensures the security of wireless network communication [7, 8]. In order to provide a feasible way of thinking for the problem of wireless communication security, the combination of the two technologies will form a complete communication security solution, which also provides the possibility to realize the ideal secure communication of one time padding [9, 10]. In recent years, people’s research on higher communication frequency band [11, 12] and communication detection technology and control scheme [13] makes the physical layer security technology of extracting key through wireless channel more mature and has a broader application range [14, 15].The research of physical layer secret key generation technology can be traced back to the early 1990s. Ahlswede et al. [16] proposed the idea of using characteristic parameters of wireless channel as a shared random source. Subsequently, Hershey proved the feasibility of generating the secret key based on the characteristic parameters of the wireless channel [17] and opened a precedent in the research of physical layer secret key generation technology. Since then, relevant research studies on the secret extraction technology based on wireless channel characteristics have emerged one after another, among which the most common is the secret key generation based on the channel amplitude (RSS) [18, 19] characteristic. However, because RSS is greatly affected by the location of both parties in communication, moreover, the amount of information provided is small, resulting in a low the secret key generation rate and poor solution adaptability. In response to this problem, the secret key extraction method based on channel phase characteristics [20, 21], channel multipath delay [22], channel state information (CSI) [23, 24], and channel frequency response (CFR) [25, 26] has been proposed one after another, which improved the secret key generation rate to a certain extent. However, because the channel phase is more sensitive to interference and noise, the consistency of the generated secret key is reduced. Owing to the multipath, delay is closely related to factors such as signal propagation path and propagation angle, resulting in the high complexity of extracting secrets with multipath delay. Because the CSI obtained by the legitimate parties in the coherence time is less correlated, this makes the matching rate of the secret keys generated by the legitimate parties relatively low. Since CFR has fine-grained channel characteristics, higher hardware facilities are required.Through the analysis of the above schemes, it can be seen that these secret key generation schemes generally have the problems of poor secret key adaptability, low rate of the secret key generation, or high inconsistency rate. In order to overcome these shortcomings, some studies have used multiantenna [27, 28], multibit quantization [29, 30], or adaptive quantization methods [31, 32] to improve the secret key generation rate, but these methods still lose a lot of useful information, which is not conducive to the secret key generation. There are also some studies which start from the wireless channel characteristic information and improve the key generation ability by sending artificial noise [33, 34] and introducing wireless cooperative nodes [35, 36], but these ways add extra cost and are also more complex to implement. In addition, some research studies start from the dimension of wireless channel characteristics, improving the one-dimensional characteristics of the channel and extracting the secret key through the channel multicharacteristic information to overcome the problems of the abovementioned secret key generation scheme. For example, Wang et al. introduced multiple random wireless channel impulse response (CIR) information to generate the key [37] in order to improve the key bit generation rate and the extensibility of the key generation method. In order to obtain higher key generation rate and randomness, Prof. Zhang et al. used CIR containing the amplitude information and phase information of the wireless channel to extract the key [38]. However, the accuracy of acquiring CIR has high requirements on communication hardware and node resources.In view of the above problems, considering the actual communication scenario, a secret key generation scheme based on the joint characteristic parameters of the wireless channel amplitude and phase is proposed. Based on the eavesdropping channel model, the amplitude and phase characteristics of the channel are analyzed, and the equal probability joint quantization strategy of the amplitude and phase is adopted to quantize the extracted joint characteristics of channel amplitude and phase, which ensures the efficiency, reliability, and security of generating the secret key random parameters and improves the adaptability of quantization threshold.


## 2. System Model Design

The wireless eavesdropping channel system model studied is shown in Figure1. It includes legitimate communication parties Alice and Bob and a third-party eavesdropper Eve. The channels when the legitimate parties exchange information are AB and BA, respectively, and their channel characteristics are represented by hAB and hBA, respectively, the eavesdropping channels between Alice and Eve and Bob and Eve are AE and BE, respectively, and the channel characteristics are represented by hAE and hBE, respectively. The system works in time division duplex (TDD) mode and realizes the detection of wireless channel characteristics in two stages. Firstly, in the time slot TA, Alice first sends a wireless channel detection signal to Bob and Bob and Eve receive the signal; then, Bob sends a channel detection signal to Alice in the TB time slot and Alice and Eve receive the signal. Assume that the internode channels in this model are all Jakes Rayleigh multipath fading channels, and Eve is a passive eavesdropper, who can receive the signals transmitted between Alice and Bob and know the secret key generation algorithm between the legitimate parties.Figure 1
Wireless eavesdropping channel system model.


### 2.1. TA Time Slot

In the TA time slot, Alice sends the channel detection signal agreed by both parties to Bob:(1)sAt=Aej2πfct+θAt,where A, fc, and θAt are the signal amplitude, carrier frequency, and initial phase of sAt, respectively. The signals received by Bob and Eve are, respectively, as follows:(2)yBt=hABsAt+nABt,(3)yAEt=hAEsAt+nAEt,where nABt and nAEt are the noise received by Bob and Eve, respectively, satisfying nABt∼CN0,σnAB2 and nAEt∼CN0,σnAE2.


### 2.2. TB Time Slot

Bob sends the detection signalsBt to Alice in the time slot TB, and sBt=sAt. The signals received by Alice and Eve are(4)yAt=hBAsBt+nBAt,(5)yBEt=hBEsBt+nBEt,where nBAt and nBEt are the noise received by Alice and Eve in the current slot, respectively, which satisfy nBAt∼CN0,σnBA2 and nBEt∼CN0,σnBE2.Because the signal is affected by various obstacles during the propagation of the wireless channel, the transmitted signal reaches the receiving end along different propagation paths. Due to the fast time-varying characteristics and complexity of the wireless channel characteristics, the channel characteristics between nodes are dynamic random variables, resulting in the fading characteristics of the received signals. In order to analyze the multipath fading characteristics of the wireless channel, the channel characteristics between nodes are decomposed into the superposition of the channel characteristics of several wireless propagation paths. Take the channel characteristichAB between Alice and Bob as an example and express it as follows:(6)hAB=∑k=1NABaABkte−jφABktδt−τABk,where NAB is the number of wireless channel propagation paths between Alice and Bob, aABkt, τABk, and φABkt are the fading factors of the kth path from Alice to Bob, signal delay, and phase shift, respectively, ∗ is the convolution symbol, and δ⋅ is the Dirac function. Similarly, hBA=∑k=1NBAaBAkte−jφBAktδt−τABk. Since the channel characteristics between Alice and Bob satisfy short-term reciprocity, hAB≈hBA. However, The locations of Eve, Alice, and Bob are not the same; that is, the channel transmission characteristics hAE and hBE between Eve, Alice, and Bob are completely different from the main channel characteristics hAB and hBA, making Eve unable to obtain any information about hAB and hBA. The above guarantees the security of the random parameter extraction of the wireless channel characteristic secret key.


## 2.1. TA Time Slot

In the TA time slot, Alice sends the channel detection signal agreed by both parties to Bob:(1)sAt=Aej2πfct+θAt,where A, fc, and θAt are the signal amplitude, carrier frequency, and initial phase of sAt, respectively. The signals received by Bob and Eve are, respectively, as follows:(2)yBt=hABsAt+nABt,(3)yAEt=hAEsAt+nAEt,where nABt and nAEt are the noise received by Bob and Eve, respectively, satisfying nABt∼CN0,σnAB2 and nAEt∼CN0,σnAE2.


## 2.2. TB Time Slot

Bob sends the detection signalsBt to Alice in the time slot TB, and sBt=sAt. The signals received by Alice and Eve are(4)yAt=hBAsBt+nBAt,(5)yBEt=hBEsBt+nBEt,where nBAt and nBEt are the noise received by Alice and Eve in the current slot, respectively, which satisfy nBAt∼CN0,σnBA2 and nBEt∼CN0,σnBE2.Because the signal is affected by various obstacles during the propagation of the wireless channel, the transmitted signal reaches the receiving end along different propagation paths. Due to the fast time-varying characteristics and complexity of the wireless channel characteristics, the channel characteristics between nodes are dynamic random variables, resulting in the fading characteristics of the received signals. In order to analyze the multipath fading characteristics of the wireless channel, the channel characteristics between nodes are decomposed into the superposition of the channel characteristics of several wireless propagation paths. Take the channel characteristichAB between Alice and Bob as an example and express it as follows:(6)hAB=∑k=1NABaABkte−jφABktδt−τABk,where NAB is the number of wireless channel propagation paths between Alice and Bob, aABkt, τABk, and φABkt are the fading factors of the kth path from Alice to Bob, signal delay, and phase shift, respectively, ∗ is the convolution symbol, and δ⋅ is the Dirac function. Similarly, hBA=∑k=1NBAaBAkte−jφBAktδt−τABk. Since the channel characteristics between Alice and Bob satisfy short-term reciprocity, hAB≈hBA. However, The locations of Eve, Alice, and Bob are not the same; that is, the channel transmission characteristics hAE and hBE between Eve, Alice, and Bob are completely different from the main channel characteristics hAB and hBA, making Eve unable to obtain any information about hAB and hBA. The above guarantees the security of the random parameter extraction of the wireless channel characteristic secret key.


## 3. The Secret Key Generation Scheme Design

Based on the above wireless eavesdropping channel system model, this section will propose a secret key extraction method based on the joint amplitude and phase characteristics of the wireless channel. Firstly, Alice and Bob analyze the channel characteristics through the wireless channel detection process and extract the wireless channel amplitude characteristics and phase characteristics. Then, based on the extracted channel characteristics, the statistical equations of the probability characteristics of the amplitude fading and phase distribution of the wireless channel are established. Finally, in order to ensure the reliability of the random parameters of the generated secret key, the equal probability joint quantization strategy of the amplitude and phase is used to quantify the extracted wireless channel characteristics. The received channel detection signals by each node have been given in the previous chapter. Next, we will discuss the probability statistical characteristic equation of the amplitude fading and phase distribution of the wireless channel and the equal probability joint quantization strategy of the amplitude and phase.


### 3.1. Channel Characteristic Analysis Stage

Because the amplitude of the transmitted signal over the Rayleigh fading channel obeys the Rayleigh distribution and the phase obeys the uniform distribution [39] of 0,2π, the original signal sent by the legitimate parties is known to Eve. In order to ensure the security of the generated secret key random parameters, the amplitude difference value and phase difference value between the received signal and the known detection signal are used as the wireless channel characteristics to jointly generate the secret key random parameters. By reducing the correlation between the amplitude characteristics and phase characteristics of the wireless channel extracted by the eavesdropper and the legitimate party, the confidentiality of the random parameters of the secret key generation between Alice and Bob is improved.The amplitude difference between the received signal and the known detection signal is defined as the channel amplitude characteristic, and the phase difference is defined as the channel phase characteristic. Assume that Bob’s the channel amplitude characteristic and the channel phase characteristic are, respectively, represented byrABt and θABt as follows:(7)rABt=yBt−A,(8)θABt=arctanimagyBtrealyBt−θAt.The channel amplitude characteristic and channel phase characteristic extracted by Bob are combined into the channel amplitude-phase joint two-dimensional characteristics, which is denoted asrABt,θABt. Similarly, the channel amplitude characteristic extracted by Alice is rBAt=yAt−A, and the phase characteristic is θBAt=arctanimagyAt/realyAt−θBt. The joint characteristic of amplitude and phase is rBAt,θBAt. The channel amplitude characteristics of Eve eavesdropping are rAEt=yAEt−A and rBEt=yBEt−A, respectively, and the phase characteristics are θAEt=arctanimagyAEt/realyAEt−θAt and θBEt=arctanimagyBEt/realyAEt−θBt, respectively. The amplitude and phase joint two-dimensional characteristics of the channel are rAEt,θAEt and rBEt,θBEt, respectively.Through the channel characteristic analysis stage, the joint two-dimensional characteristics information of the channel amplitude and phase of each node is obtained. However, using the channel amplitude and phase joint two-dimensional characteristics to generate the required random parameters of the secret key, it is also necessary to quantify the extracted joint two-dimensional characteristics information through the channel characteristic quantization stage.


### 3.2. Channel Characteristic Quantization Stage

Taking Bob node as an example, the quantization process ofrABt,θABt is analyzed. Firstly, the quantization process of channel amplitude fading characteristics is analyzed. Because yB obeys Rayleigh distribution, its probability density function is(9)fyB=yBσAB2exp−yB22σAB2,in which σAB2 is the average power of Bob’s received signal yB, which satisfies σAB2=EyB2. According to formulae (7) and (9), the probability density function of rABt can be derived in two stages. When the first stage is rAB>sAt, the probability density function of rABt satisfies frAB=rAB+sAt/σAB2exp−rAB+sAt2/2σAB2. When the second stage is 0<rAB<sAt, frAB=rAB+sAt/σAB2exp−rAB+sAt2/2σAB2−sAt−rAB/σAB2exp−sAt−rAB2/2σAB2. To facilitate the calculation below, let f1=rAB+sAt/σAB2exp−rAB+sAt2/2σAB2 and f2=sAt−rAB/σAB2exp−sAt−rAB2/2σAB2, and you can change rABt. The probability density function of is expressed as(10)frAB=f1,rAB>sAt,f1−f2,0<rAB<sAt,0,Other.According to equation (10), the distribution of rAB is not uniform, and the distribution characteristics are affected by the original signal amplitude. In order to ensure the adaptability of quantization threshold and the security of random parameters of the secret key generation, the rAB interval adopts equal probability quantization algorithm, so that the measured value falls on each quantization interval with equal probability, so as to realize the automatic adjustment of quantization threshold of channel amplitude characteristics and improve the randomness of generating the secret key random parameters.Secondly, the quantization process of channel phase characteristic is discussed. Since the phasearctanimagyBt/realyBt of the received signal obeys uniform distribution within 0,2π, let ϕABt=arctanimagyBt/realyBt, then its probability density function is(11)fϕAB=12π,0≤ϕAB≤2π,0,other.Combining with equations (8) and (11), the probability density function of θAB can be deduced as(12)fθAB=12π,−θA≤θAB<2π−θA,0,Other.It can be seen from equation (12) that θABt is a random variable uniformly distributed on [−θA, 2π−θA], which can directly divide the θABt region by using the equal probability quantization algorithm to ensure the randomness of the generated secret key random parameters. This method of joint quantization of channel amplitude and phase characteristics using the equal probability quantization algorithm is called the equal probability joint quantization strategy of the amplitude and phase.Finally, assuming that the quantization bit length of channel amplitude characteristic and channel phase characteristic is bothn/2, the whole quantization space is divided into q=2n parts. The equal probability joint quantization strategy of the amplitude and phase is used to quantify the amplitude and phase characteristics of wireless channel. Firstly, the channel amplitude characteristic region is divided into 2n/2 concentric circles so that the probability of each channel amplitude characteristic measurement value falling into the region between each two concentric circles is equal. Next, the interval −θA,2π−θA where the channel phase characteristics are located is evenly divided into 2n/2 parts, and quantization interval was 2π/2n/2. Taking the quantization bit number n=4 as an example, the region division diagram corresponding to the joint quantization of 2 bit channel amplitude characteristic and 2 bit channel phase characteristic is as shown in Figure 2.Figure 2
Joint quantization interval of amplitude and phase.The quantization space in the figure takes 0 as the center of the circle, and the amplitude space is divided into four parts of equal probability:0,Q1, Q1,Q2, Q2,Q3 and Q3,Q4. For the quantization thresholds Q1, Q2, Q3, and Q4 representing the channel amplitude characteristic of the system, let j∈1,2,…,2n/2, then Qj=p−1j/2n/2, pQj=∫0QjfrABdrAB, where Q4≈+∞. Taking 0 as the starting point and 2π/2n/2 as the phase interval, the space is further evenly divided into four equally spaced sections: −θA,π/2−θA, π/2−θA,π−θA, π−θA,3π/2−θA, and 3π/2−θA,2π−θA. The joint quantization scheme of channel amplitude and phase is shown in Table 1.Table 1
Joint quantization scheme of 2 bit channel amplitude and 2 bit channel phase.
IntervalQuantitative valuesQuantization interval10000θtε−θA,π/2−θA∩rtε0,Q120001θtε−θA,π/2−θA∩rtεQ1,Q230010θtε−θA,π/2−θA∩rtεQ2,Q340011θtε−θA,π/2−θA∩rtεQ3,Q450100θtεπ/2−θA,π−θA∩rtε0,Q160101θtεπ/2−θA,π−θA∩rtεQ1,Q270110θtεπ/2−θA,π−θA∩rtεQ2,Q380111θtεπ/2−θA,π−θA∩rtεQ3,Q491000θtεπ−θA,3π/2−θA∩rtε0,Q1101001θtεπ−θA,3π/2−θA∩rtεQ1,Q2111010θtεπ−θA,3π/2−θA∩rtεQ2,Q3121011θtεπ−θA,3π/2−θA∩rtεQ3,Q4131100θtε3π/2−θA,2π−θA∩rtε0,Q1141101θtε3π/2−θA,2π−θA∩rtεQ1,Q2151110θtε3π/2−θA,2π−θA∩rtεQ2,Q3161111θtε3π/2−θA,2π−θA∩rtεQ3,Q4The region of channel amplitude-phase joint characteristics is divided by the equal probability joint quantization strategy of the amplitude and phase, and then each quantization region is labeled with corresponding quantization bit sequence. Finally, according to the extracted channel amplitude and phase joint characteristics value fallen in the quantization interval, its corresponding quantization bit sequence is determined, and the corresponding secret key random parameters are formed.


## 3.1. Channel Characteristic Analysis Stage

Because the amplitude of the transmitted signal over the Rayleigh fading channel obeys the Rayleigh distribution and the phase obeys the uniform distribution [39] of 0,2π, the original signal sent by the legitimate parties is known to Eve. In order to ensure the security of the generated secret key random parameters, the amplitude difference value and phase difference value between the received signal and the known detection signal are used as the wireless channel characteristics to jointly generate the secret key random parameters. By reducing the correlation between the amplitude characteristics and phase characteristics of the wireless channel extracted by the eavesdropper and the legitimate party, the confidentiality of the random parameters of the secret key generation between Alice and Bob is improved.The amplitude difference between the received signal and the known detection signal is defined as the channel amplitude characteristic, and the phase difference is defined as the channel phase characteristic. Assume that Bob’s the channel amplitude characteristic and the channel phase characteristic are, respectively, represented byrABt and θABt as follows:(7)rABt=yBt−A,(8)θABt=arctanimagyBtrealyBt−θAt.The channel amplitude characteristic and channel phase characteristic extracted by Bob are combined into the channel amplitude-phase joint two-dimensional characteristics, which is denoted asrABt,θABt. Similarly, the channel amplitude characteristic extracted by Alice is rBAt=yAt−A, and the phase characteristic is θBAt=arctanimagyAt/realyAt−θBt. The joint characteristic of amplitude and phase is rBAt,θBAt. The channel amplitude characteristics of Eve eavesdropping are rAEt=yAEt−A and rBEt=yBEt−A, respectively, and the phase characteristics are θAEt=arctanimagyAEt/realyAEt−θAt and θBEt=arctanimagyBEt/realyAEt−θBt, respectively. The amplitude and phase joint two-dimensional characteristics of the channel are rAEt,θAEt and rBEt,θBEt, respectively.Through the channel characteristic analysis stage, the joint two-dimensional characteristics information of the channel amplitude and phase of each node is obtained. However, using the channel amplitude and phase joint two-dimensional characteristics to generate the required random parameters of the secret key, it is also necessary to quantify the extracted joint two-dimensional characteristics information through the channel characteristic quantization stage.


## 3.2. Channel Characteristic Quantization Stage

Taking Bob node as an example, the quantization process ofrABt,θABt is analyzed. Firstly, the quantization process of channel amplitude fading characteristics is analyzed. Because yB obeys Rayleigh distribution, its probability density function is(9)fyB=yBσAB2exp−yB22σAB2,in which σAB2 is the average power of Bob’s received signal yB, which satisfies σAB2=EyB2. According to formulae (7) and (9), the probability density function of rABt can be derived in two stages. When the first stage is rAB>sAt, the probability density function of rABt satisfies frAB=rAB+sAt/σAB2exp−rAB+sAt2/2σAB2. When the second stage is 0<rAB<sAt, frAB=rAB+sAt/σAB2exp−rAB+sAt2/2σAB2−sAt−rAB/σAB2exp−sAt−rAB2/2σAB2. To facilitate the calculation below, let f1=rAB+sAt/σAB2exp−rAB+sAt2/2σAB2 and f2=sAt−rAB/σAB2exp−sAt−rAB2/2σAB2, and you can change rABt. The probability density function of is expressed as(10)frAB=f1,rAB>sAt,f1−f2,0<rAB<sAt,0,Other.According to equation (10), the distribution of rAB is not uniform, and the distribution characteristics are affected by the original signal amplitude. In order to ensure the adaptability of quantization threshold and the security of random parameters of the secret key generation, the rAB interval adopts equal probability quantization algorithm, so that the measured value falls on each quantization interval with equal probability, so as to realize the automatic adjustment of quantization threshold of channel amplitude characteristics and improve the randomness of generating the secret key random parameters.Secondly, the quantization process of channel phase characteristic is discussed. Since the phasearctanimagyBt/realyBt of the received signal obeys uniform distribution within 0,2π, let ϕABt=arctanimagyBt/realyBt, then its probability density function is(11)fϕAB=12π,0≤ϕAB≤2π,0,other.Combining with equations (8) and (11), the probability density function of θAB can be deduced as(12)fθAB=12π,−θA≤θAB<2π−θA,0,Other.It can be seen from equation (12) that θABt is a random variable uniformly distributed on [−θA, 2π−θA], which can directly divide the θABt region by using the equal probability quantization algorithm to ensure the randomness of the generated secret key random parameters. This method of joint quantization of channel amplitude and phase characteristics using the equal probability quantization algorithm is called the equal probability joint quantization strategy of the amplitude and phase.Finally, assuming that the quantization bit length of channel amplitude characteristic and channel phase characteristic is bothn/2, the whole quantization space is divided into q=2n parts. The equal probability joint quantization strategy of the amplitude and phase is used to quantify the amplitude and phase characteristics of wireless channel. Firstly, the channel amplitude characteristic region is divided into 2n/2 concentric circles so that the probability of each channel amplitude characteristic measurement value falling into the region between each two concentric circles is equal. Next, the interval −θA,2π−θA where the channel phase characteristics are located is evenly divided into 2n/2 parts, and quantization interval was 2π/2n/2. Taking the quantization bit number n=4 as an example, the region division diagram corresponding to the joint quantization of 2 bit channel amplitude characteristic and 2 bit channel phase characteristic is as shown in Figure 2.Figure 2
Joint quantization interval of amplitude and phase.The quantization space in the figure takes 0 as the center of the circle, and the amplitude space is divided into four parts of equal probability:0,Q1, Q1,Q2, Q2,Q3 and Q3,Q4. For the quantization thresholds Q1, Q2, Q3, and Q4 representing the channel amplitude characteristic of the system, let j∈1,2,…,2n/2, then Qj=p−1j/2n/2, pQj=∫0QjfrABdrAB, where Q4≈+∞. Taking 0 as the starting point and 2π/2n/2 as the phase interval, the space is further evenly divided into four equally spaced sections: −θA,π/2−θA, π/2−θA,π−θA, π−θA,3π/2−θA, and 3π/2−θA,2π−θA. The joint quantization scheme of channel amplitude and phase is shown in Table 1.Table 1
Joint quantization scheme of 2 bit channel amplitude and 2 bit channel phase.
IntervalQuantitative valuesQuantization interval10000θtε−θA,π/2−θA∩rtε0,Q120001θtε−θA,π/2−θA∩rtεQ1,Q230010θtε−θA,π/2−θA∩rtεQ2,Q340011θtε−θA,π/2−θA∩rtεQ3,Q450100θtεπ/2−θA,π−θA∩rtε0,Q160101θtεπ/2−θA,π−θA∩rtεQ1,Q270110θtεπ/2−θA,π−θA∩rtεQ2,Q380111θtεπ/2−θA,π−θA∩rtεQ3,Q491000θtεπ−θA,3π/2−θA∩rtε0,Q1101001θtεπ−θA,3π/2−θA∩rtεQ1,Q2111010θtεπ−θA,3π/2−θA∩rtεQ2,Q3121011θtεπ−θA,3π/2−θA∩rtεQ3,Q4131100θtε3π/2−θA,2π−θA∩rtε0,Q1141101θtε3π/2−θA,2π−θA∩rtεQ1,Q2151110θtε3π/2−θA,2π−θA∩rtεQ2,Q3161111θtε3π/2−θA,2π−θA∩rtεQ3,Q4The region of channel amplitude-phase joint characteristics is divided by the equal probability joint quantization strategy of the amplitude and phase, and then each quantization region is labeled with corresponding quantization bit sequence. Finally, according to the extracted channel amplitude and phase joint characteristics value fallen in the quantization interval, its corresponding quantization bit sequence is determined, and the corresponding secret key random parameters are formed.


## 4. Performance Analysis

In order to measure the system performance of the proposed scheme, this section mainly analyzes the reliability, efficiency, and safety of the scheme.


### 4.1. Reliability

The matching rate of the random secret key parameters generated by both legitimate parties is used as the reliability measurement index. Due to the existence of objective interference factors such as the random noise of wireless channel and the difference of wireless devices, there are deviations in the joint characteristics of amplitude and phase extracted by the legitimate parties. Under the premise of a large number of measurements of channel amplitude characteristics and channel phase characteristics, the deviations of channel amplitude characteristics and phase characteristics [40, 41] of each node are independent of each other and follow a Gaussian distribution with an average of 0. Suppose rBAt,θBAt and rABt,θABt all fall in the same quantization space and the deviations of the channel phase characteristic values extracted by Alice and Bob are ΔθA and ΔθB, respectively, the deviation of channel amplitude characteristic value is ΔrA and ΔrB, respectively. According to the measured values of big data channel characteristics, the variances of unbiased estimation of Alice and Bob channel amplitude and phase characteristics are σΔrA2, σΔθA2 and σΔrB2, σΔθB2, respectively. At this point, the probability that Alice extracts the channel amplitude characteristic value rBA0=rBA+ΔrA falls in the intervals Qj−1,Qj and the channel phase characteristic value θBA0=θBA+ΔθA falls in the interval 2πi−1/2n/2−θA,2πi/2n/2−θA is(13)PAij=∫Qj−1Qj12πσΔrAexp−x−rBA22σΔrA2dx∫2πi−1/2n/2−θA2πi/2n/2−θA12πσΔθAexp−y−θBA22σΔθA2dy,where i∈1,2,…,2n/2. Similarly, the probability that the channel amplitude characteristic value rAB0=rAB+ΔrB extracted by Bob falls in the amplitude interval Qj−1,Qj and the channel phase characteristic value θAB0=θAB+ΔθB extracted by Bob falls in the phase interval 2πi−1/2n/2−θA,2πi/2n/2−θA is(14)PBij=∫Qj−1Qj12πσΔrBexp−x−rAB22σΔrB2dx∫2πi−1/2n/2−θA2πi/2n/2−θA12πσΔθBexp−y−θAB22σΔθB2dy.Then, the probability that the quantization results of Alice and Bob’s channel amplitude and phase joint two-dimensional characteristics fall within the same quantization interval is as follows:(15)PAB=∑i∑jPAijPBij.


### 4.2. Security

The bit inconsistency rate of the secret key random parameter between Eve and the legitimate node is used to characterize the security of communication between legitimate nodes. Taking the legitimate node Bob as an example, the bit inconsistency rate between the random parameters of the secret key generated by the eavesdropper and the legitimate party is calculated.The probability of Eve eavesdropping the channel amplitude characteristic value falls in the areaQj−1,Qj and the channel phase characteristic value falls in 2πi−1/2n/2−θA,2πi/2n/2−θA is as follows:(16)PEij=∫Qj−1Qj12πσΔrEexp−x−rE22σΔrE2dx∫2πi−1/2n/2−θA2πi/2n/2−θA12πσΔθEexp−y−θE22σΔθE2dy,where rE=rAEt,rBEt, θE=θAEt,θBEt,σΔrE2, and σΔθE2 are the variances of unbiased estimation of channel amplitude and phase characteristics extracted by Eve according to eavesdropping signals. If the quantization result of channel amplitude-phase joint characteristics extracted by Eve and Bob falls in the same quantization interval, the probability is PBE, meeting PBE=∑i∑jPBijPEij; if gray code with minimum bit error is used for binary coding of quantization interval, the coding of adjacent quantization interval has only one bit difference, and then the bit inconsistency rate of Eve generated secret key random parameters and Bob’s can be approximately expressed as(17)PIBE≈1−PBElog2q.


### 4.3. Efficiency

The maximum secret key generation rate is a secret key parameter to reflect the efficiency of the secret key generation system, and it is also an important index of the system security performance. The maximum secret key generation rate is defined as the mutual information between the legitimate parties to extract channel characteristics [42]. LetrBAt,θBAt=X, rABt,θABt=Y, and rAEt,θAEt,rBEt,θBEt=Z, then the maximum secret key generation rate of the scheme can be expressed as follows:(18)CK=minIX;Y,IX;Y|Z.If the eavesdropping channel and the legitimate channel are independent of each other, then(19)CK=HX+HY−HX,Y.Assuming that the amplitude characteristics and phase characteristics of the wireless channel obtained by the legitimate parties are independent of each other and obey the complex Gaussian distribution with the mean value of 0, thenprBA=1/2πσΔrAexp−rBA2/2σΔrA2, pθBA=1/2πσΔθAexp−θBA2/2σΔθA2, prAB=1/2πσΔrBexp−rAB2/2σΔrB2, pθAB=1/2πσΔθBexp−θAB2/2σΔθB2, prBA,θBA=prBA∗pθBA, prAB,θAB=prAB∗pθAB, and pX,Y=prBA,θBA,rAB,θAB.Thus, the above equation (19) can be reformulated as(20)CK=−∫0+∞∫−θA2π−θAprBA,θBAlog2prBA,θBAdθBAdrBA−∫0+∞∫−θA2π−θAprAB,θABlog2prAB,θABdθABdrAB+∫0+∞∫0+∞∫−θA2π−θA∫−θA2π−θApX,Ylog2pX,YdθABdθBAdrABdrBA.


## 4.1. Reliability

The matching rate of the random secret key parameters generated by both legitimate parties is used as the reliability measurement index. Due to the existence of objective interference factors such as the random noise of wireless channel and the difference of wireless devices, there are deviations in the joint characteristics of amplitude and phase extracted by the legitimate parties. Under the premise of a large number of measurements of channel amplitude characteristics and channel phase characteristics, the deviations of channel amplitude characteristics and phase characteristics [40, 41] of each node are independent of each other and follow a Gaussian distribution with an average of 0. Suppose rBAt,θBAt and rABt,θABt all fall in the same quantization space and the deviations of the channel phase characteristic values extracted by Alice and Bob are ΔθA and ΔθB, respectively, the deviation of channel amplitude characteristic value is ΔrA and ΔrB, respectively. According to the measured values of big data channel characteristics, the variances of unbiased estimation of Alice and Bob channel amplitude and phase characteristics are σΔrA2, σΔθA2 and σΔrB2, σΔθB2, respectively. At this point, the probability that Alice extracts the channel amplitude characteristic value rBA0=rBA+ΔrA falls in the intervals Qj−1,Qj and the channel phase characteristic value θBA0=θBA+ΔθA falls in the interval 2πi−1/2n/2−θA,2πi/2n/2−θA is(13)PAij=∫Qj−1Qj12πσΔrAexp−x−rBA22σΔrA2dx∫2πi−1/2n/2−θA2πi/2n/2−θA12πσΔθAexp−y−θBA22σΔθA2dy,where i∈1,2,…,2n/2. Similarly, the probability that the channel amplitude characteristic value rAB0=rAB+ΔrB extracted by Bob falls in the amplitude interval Qj−1,Qj and the channel phase characteristic value θAB0=θAB+ΔθB extracted by Bob falls in the phase interval 2πi−1/2n/2−θA,2πi/2n/2−θA is(14)PBij=∫Qj−1Qj12πσΔrBexp−x−rAB22σΔrB2dx∫2πi−1/2n/2−θA2πi/2n/2−θA12πσΔθBexp−y−θAB22σΔθB2dy.Then, the probability that the quantization results of Alice and Bob’s channel amplitude and phase joint two-dimensional characteristics fall within the same quantization interval is as follows:(15)PAB=∑i∑jPAijPBij.


## 4.2. Security

The bit inconsistency rate of the secret key random parameter between Eve and the legitimate node is used to characterize the security of communication between legitimate nodes. Taking the legitimate node Bob as an example, the bit inconsistency rate between the random parameters of the secret key generated by the eavesdropper and the legitimate party is calculated.The probability of Eve eavesdropping the channel amplitude characteristic value falls in the areaQj−1,Qj and the channel phase characteristic value falls in 2πi−1/2n/2−θA,2πi/2n/2−θA is as follows:(16)PEij=∫Qj−1Qj12πσΔrEexp−x−rE22σΔrE2dx∫2πi−1/2n/2−θA2πi/2n/2−θA12πσΔθEexp−y−θE22σΔθE2dy,where rE=rAEt,rBEt, θE=θAEt,θBEt,σΔrE2, and σΔθE2 are the variances of unbiased estimation of channel amplitude and phase characteristics extracted by Eve according to eavesdropping signals. If the quantization result of channel amplitude-phase joint characteristics extracted by Eve and Bob falls in the same quantization interval, the probability is PBE, meeting PBE=∑i∑jPBijPEij; if gray code with minimum bit error is used for binary coding of quantization interval, the coding of adjacent quantization interval has only one bit difference, and then the bit inconsistency rate of Eve generated secret key random parameters and Bob’s can be approximately expressed as(17)PIBE≈1−PBElog2q.


## 4.3. Efficiency

The maximum secret key generation rate is a secret key parameter to reflect the efficiency of the secret key generation system, and it is also an important index of the system security performance. The maximum secret key generation rate is defined as the mutual information between the legitimate parties to extract channel characteristics [42]. LetrBAt,θBAt=X, rABt,θABt=Y, and rAEt,θAEt,rBEt,θBEt=Z, then the maximum secret key generation rate of the scheme can be expressed as follows:(18)CK=minIX;Y,IX;Y|Z.If the eavesdropping channel and the legitimate channel are independent of each other, then(19)CK=HX+HY−HX,Y.Assuming that the amplitude characteristics and phase characteristics of the wireless channel obtained by the legitimate parties are independent of each other and obey the complex Gaussian distribution with the mean value of 0, thenprBA=1/2πσΔrAexp−rBA2/2σΔrA2, pθBA=1/2πσΔθAexp−θBA2/2σΔθA2, prAB=1/2πσΔrBexp−rAB2/2σΔrB2, pθAB=1/2πσΔθBexp−θAB2/2σΔθB2, prBA,θBA=prBA∗pθBA, prAB,θAB=prAB∗pθAB, and pX,Y=prBA,θBA,rAB,θAB.Thus, the above equation (19) can be reformulated as(20)CK=−∫0+∞∫−θA2π−θAprBA,θBAlog2prBA,θBAdθBAdrBA−∫0+∞∫−θA2π−θAprAB,θABlog2prAB,θABdθABdrAB+∫0+∞∫0+∞∫−θA2π−θA∫−θA2π−θApX,Ylog2pX,YdθABdθBAdrABdrBA.


## 5. Experiment and Result

In order to verify the performance of the designed scheme and evaluate the advantages and disadvantages of the proposed method, three software radio systems were used to simulate the three-node communication of Alice, Bob, and Eve, respectively, and the system performance of the three secret key generation methods in Table2 was analyzed. The system is mainly composed of AD9361 integrated radio frequency module and Xilinx-Zynq450 control module with adjustable frequency band from 70 MHz to 6 GHz. The system works in TDD half-duplex communication mode, with the sampling rate of 3 GHz, radio frequency point of 1.5 GHz, analog bandwidth of 200 MHz, and channel gain of 60. The channel transmission signals are modulated and demodulated by 16QAM. When the distance between Alice and Bob is 80 m, the following test results are obtained through 1000 system experiments.Table 2
The secret key generation scheme corresponding to simulation number.
Simulation numberThe secret key generation scheme nameScheme 1The secret key generation method based on joint amplitude-phase characteristic parameters of wireless channelScheme 2The secret key generation scheme based on phase characteristics of wireless channelScheme 3The secret key generation scheme based on amplitude characteristics of wireless channelFigure3 describes the change trend of secret key random parameter generation rate with SNR of the three secret key generation schemes in Table 2. It can be seen that with the increase in SNR, the three curves in the figure all show an upward trend. This is because the increase in SNR improves the correlation of wireless channel characteristics between legitimate parties and reduces the difference of quantization results between Alice and Bob, thus improving the generation rate of the secret key random parameters between legitimate parties. It can also be seen from the figure that at the same SNR, the secret key random parameter generation rate of scheme 1 is close to the sum of the secret key random parameter generation rates of scheme 2 and scheme 3; this is because scheme 1 uses the channel phase characteristics of scheme 2 and the channel amplitude characteristics of scheme 3 to generate the secret key random parameters so that the amount of information transmitted by scheme 1 each time includes the channel phase information of scheme 2 and the channel amplitude information of scheme 3. Under the same SNR, the secret key random parameter generation rate of scheme 1 approaches the sum of the secret key random parameter generation rates of schemes 2 and 3. In addition, it can be seen from the figure that the generation rate of the secret key random parameters in scheme 2 is higher than that in scheme 3, which is due to the uniform distribution of channel phase characteristics in scheme 2 in the interval of −θA,2π−θA. While the distribution of channel amplitude characteristics in scheme 3 is not uniform, the high probability is concentrated in a certain amplitude range, which leads to the decrease in the generation rate of the random parameters. From the above curves, it can be seen that the proposed scheme in this paper has better secret key random parameter generation rate.Figure 3
Curve of the secret key random parameter generation rate changing with SNR.Figure4 compares the variation curve of the secret key random parameter matching rate of Bob and Alice (Eve) with SNR of three different secret key generation schemes in Table 2. In the figure, P0BE refers to the secret key random parameter matching rate between Bob and Eve. It can be seen from the figure that no matter how the SNR increases, P0BE still maintains within a certain range, and scheme 1 is closer to 0.5. P0AB represents the matching rate of the secret key random parameters between legitimate communication parties. P0AB increases with the increase in SNR. This reason is that the difference between the amplitude characteristics and the phase characteristics of the channel extracted by Alice and Bob decreases with the increase in SNR, which reduces the difference between the quantization results of the legitimate communication parties and increases the probability of falling in the same quantization interval and improves the matching rate between the secret key random parameters. It can also be seen from Figure 4 that under the same SNR, the secret key random parameter matching rate of scheme 1 is higher than that of schemes 2 and 3, which means that the secret key random parameter consistency generated by scheme 1 is better than that in schemes 2 and 3; that is, scheme 1 has more advantages in reliability.Figure 4
The matching rate of the secret key random parameters among Bob, Alice, and Eve changes with the channel SNR.Figure5 compares the change of the bit error rate of the secret key random parameters of Bob and Alice (Eve) with SNR in different secret key generation schemes. PBE is the bit error rate of the secret key random parameter between Bob and Eve. It can be seen that no matter how the SNR changes, PBE basically remains unchanged. In the figure, PAB represents the bit error rate of the secret key random parameters between legitimate parties. PAB decreases with the increase in SNR. This is because the quantization difference between channel amplitude characteristics and channel phase characteristics of legitimate parties decreases with the increase in SNR, which increases the probability that the quantization results of Alice and Bob fall in the same quantization interval, thus improving the bit error rate of the secret key random parameters between legitimate parties. From Figure 5, curve shows that under the same SNR, the bit error rate of the secret key random parameters of scheme 1 is lower than that of schemes 2 and 3. This is because scheme 1 has the phase characteristics of scheme 2 and the amplitude characteristics of scheme 3; by using the diversity of wireless channel characteristic information to improve the consistency of the secret keys random parameters generated by the legal parties, scheme 2 uses the abundant and evenly distributed channel phase characteristics to make the bit error rate of the secret keys random parameters of the legitimate parties lower than scheme 3, which means that the consistent performance of the secret key random parameters of the legitimate parties in scheme 1 is higher than that of schemes 2 and 3. In addition, it can be seen from Figure 5 that the secret key random parameter error rate between the legitimate parties is much smaller than the secret key random parameter error rate between the eavesdropping node and the legitimate node, and the bit error rate of the secret key random parameters between the legitimate party and the eavesdropping party of scheme 1 is higher than that of schemes 2 and 3 and closer to 0.5, which means that the secret key random parameters generated by the legitimate parties of scheme 1 have better security.Figure 5
Variation of bit error rates between the secret key random parameters of Bob and Alice (EVE) with channel SNR.Figure6 describes the change of bit error rate of the secret key random parameters between Bob, Alice, and Eve with the distance between Eve and Bob in different secret key generation schemes when SNR is 30 dB. As it can be seen from Figure 6, PAB of schemes 1–3 is all close to 0, which means that the consistency of the secret key random parameters generated by the legitimate parties is good. PBE of schemes 1–3 increases with the increase in distance. This is because when Eve is farther away from Bob, the correlation between the eavesdropping channel and the legitimate channel is weaker, and the probability that the quantized result of extracting channel characteristics falls in the same interval is smaller, which makes the bit error rate between the secret key random parameter generated by Eve and the legitimate node higher. It can also be seen from the figure that under the same distance, the bit error rate of the secret key random parameter of eavesdropper in scheme 1 is higher than that of schemes 2 and 3 and closer to 0.5, while the bit error rate of the secret key random parameter of the legitimate parties is lower than that of schemes 2 and 3, which means that scheme 1 is superior to schemes 2 and 3 in terms of security and consistency.Figure 6
Analysis of the variation of the secret key random parameter bit error rate with distance.Through the above experimental results, it can be observed that the scheme proposed in this paper has more advantages in terms of efficiency, consistency, and security, which provides an idea for obtaining better communication performance in wireless communication scenarios.


## 6. Conclusion

This paper proposes a secret key generation scheme based on joint characteristic parameters of amplitude and phase of wireless channel. This scheme uses the mutual detection signal of the legitimate parties to obtain the amplitude and phase joint characteristic information of wireless channel, and the extracted characteristic parameters are quantized by the equal probability joint quantization strategy of the amplitude and phase to generate the secret key random parameters, which improves the rate, reliability, and security of generating the secret key random parameters of the legitimate parties. The matching rate and generating rate of the secret key random parameters generated by legitimate parties and the inconsistency rate of the secret key random parameters generated by the eavesdropper and the legitimate party are analyzed systematically. Experimental results show that compared with the secret key generation schemes based on single amplitude or phase characteristics of wireless channel, the proposed scheme has significant advantages in the generation efficiency, consistency, and security of the secret key random parameters. However, the security in the process of information transmission has not been systematically mentioned in this scheme. Therefore, the next research will start from the physical layer and consider the confidentiality of information transmitted in the physical layer.



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*Source: 9951742-2021-05-04.xml*