Portal de Periódicos da CAPES

Multi‐antenna joint covert communication with a public communication link over wireless fading channel

2021; Institution of Engineering and Technology; Volume: 15; Issue: 5 Linguagem: Inglês

10.1049/cmu2.12100

1751-8636

Yuda Lin, Liang Jin, Kaizhi Huang, You Zhou,

Cooperative Communication and Network Coding

IET CommunicationsVolume 15, Issue 5 p. 695-707 ORIGINAL RESEARCH PAPEROpen Access Multi-antenna joint covert communication with a public communication link over wireless fading channel Yuda Lin, Corresponding Author Yuda Lin linyuda.ieu@foxmail.com orcid.org/0000-0002-0600-389X PLA Strategic Support Force Information Engineering University, Zhengzhou, China National Digital Switching System Engineering and Technological R&D Center, Zhengzhou, China Correspondence Yuda Lin, PLA Strategic Support Force Information Engineering University, Zhengzhou, China. Email: linyuda.ieu@foxmail.comSearch for more papers by this authorLiang Jin, Liang Jin PLA Strategic Support Force Information Engineering University, Zhengzhou, China National Digital Switching System Engineering and Technological R&D Center, Zhengzhou, ChinaSearch for more papers by this authorKaizhi Huang, Kaizhi Huang PLA Strategic Support Force Information Engineering University, Zhengzhou, China National Digital Switching System Engineering and Technological R&D Center, Zhengzhou, ChinaSearch for more papers by this authorYou Zhou, You Zhou PLA Strategic Support Force Information Engineering University, Zhengzhou, China National Digital Switching System Engineering and Technological R&D Center, Zhengzhou, ChinaSearch for more papers by this author Yuda Lin, Corresponding Author Yuda Lin linyuda.ieu@foxmail.com orcid.org/0000-0002-0600-389X PLA Strategic Support Force Information Engineering University, Zhengzhou, China National Digital Switching System Engineering and Technological R&D Center, Zhengzhou, China Correspondence Yuda Lin, PLA Strategic Support Force Information Engineering University, Zhengzhou, China. Email: linyuda.ieu@foxmail.comSearch for more papers by this authorLiang Jin, Liang Jin PLA Strategic Support Force Information Engineering University, Zhengzhou, China National Digital Switching System Engineering and Technological R&D Center, Zhengzhou, ChinaSearch for more papers by this authorKaizhi Huang, Kaizhi Huang PLA Strategic Support Force Information Engineering University, Zhengzhou, China National Digital Switching System Engineering and Technological R&D Center, Zhengzhou, ChinaSearch for more papers by this authorYou Zhou, You Zhou PLA Strategic Support Force Information Engineering University, Zhengzhou, China National Digital Switching System Engineering and Technological R&D Center, Zhengzhou, ChinaSearch for more papers by this author First published: 04 January 2021 https://doi.org/10.1049/cmu2.12100Citations: 1AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract This paper studies multi-antenna covert communication exploiting a public communication link over the wireless fading channel. In particular, a relatively practical covert communication scenario is considered for the first time, where a warden with radiometer may improve its detection accuracy by jointly detecting uplink covert pilot and downlink covert communication. To this end, a joint covert communication system, jointly considering uplink covert pilot, imperfect channel state information issue and downlink covert communication, is proposed and further studied. Specifically, the detection performance of warden and the covertness performance of system are comprehensively studied, respectively. Afterwards, an optimization problem for covert capacity is formulated to maximize overall performance. Then, with our analysis and simplification, an optimal power allocation algorithm is designed for covert capacity accordingly. Our examination shows a win-win situation has been achieved in the proposed system, that is, covert communication would rarely affect the normal communication of public link and occupy its communication resources. Interestingly enough, we analytically find an invariance property of downlink covert communication that covertness performance is not impacted by noise levels or warden's position. Based on this extraordinary finding, we conclude that the downlink covert communication system can perfectly combat with a powerful warden adopting power trend test method, while the joint covert communication system may resist that through a pilot deception scheme. 1 INTRODUCTION As a tremendous amount of confidential and sensitive data is transferred over the open wireless networks, the guarantee of security and privacy has become an ongoing challenge in wireless communications [1, 2]. Traditional cryptography as well as the physical layer security technologies [3, 4] has been developed to ensure the safety of transmission, while they cannot prevent the transmission from being detected by a warden owing to the broadcast nature of wireless communications. Sometimes it is often more significant to prevent the wireless signal from being intercepted to a certain extent than against the information eavesdropping by an unknown third party. Especially in the scenarios with high security level, such as overseas bases, embassies, and modern battlefields implementing the strategy named 'discover and destroy' etc. only the exposure of communication itself may bring enormous and unpredictable risks or even fire strike. Against this background, covert communication, also referred to as low probability of detection (LPD) communication, is emerging as a cutting-edge wireless communication security technique for safeguarding the privacy of communication behavior. 1.1 Related works and motivations In fact, spread spectrum techniques, as a typical example of covert communication, have been widely used since the early 20th century [5]. Nevertheless, the fundamental limits of covert communication have never been thoroughly examined until the square root law derived in [6] recently, which concludes that no more than O ( n ) bits of information can be reliably and covertly transmitted in n channel uses over additive white Gaussian noise (AWGN) channels. Further similar work in this regard has been studied and expanded to different wireless channels, such as binary symmetric channels (BSCs) [7], discrete memoryless channels (DMCs) [8], multiple access channels (MACs) [9], Classical-Quantum Channels (CQCs) [10] etc. With the square root law indicating that covert capacity is asymptotically zero, a positive covert rate has been proved to be achievable through creating the uncertainties of warden's received signals. The common methods for providing uncertainties are to exploit the randomness of transmitter power [11-13] and the cooperative jamming signal [14-18], the sample randomness caused by a finite channel use [19-22], and the uncertainty of the noise power [23-26], which are collectively called covertness sources here. However, the first two covertness sources are considered to be under potential security threats made by deep learning due to their artificial randomness of fixed distributions. And the last two covertness sources are usually poor in randomness, since the receiving uncertainties of warden will be greatly reduced once he collects sufficient samples or the noise uncertainty becomes sufficiently small. In particular, considering the wireless fading channel as the only covertness source, this paper takes advantage of its inherent characteristics, including uniqueness, time-varying nature and randomness, to ensure the uncertainty and security of covertness source, so as to make up for the deficiencies above. To date, the vast majority of existing works have exploited cooperative jamming as their covert communication covers e.g. added cooperative jamming node [14-16], cooperative jamming relay [11, 17], cooperative jamming by a full-duplex receiver [12, 21] etc. Obviously, cooperative jamming scheme should be a priority for covert communication due to its simple design and flexible control [2]. However, due to the actual problems such as the security of jamming nodes themselves and the cooperation between nodes, the application of cooperative jamming in practical scenarios may be relatively limited. Besides, the behavior of transmitting jamming itself may also expose the possibility of undergoing covert communication to some extent. Hence, in view of the increasingly complicated and heterogeneous wireless communication environments nowadays, this paper explores how to make use of existing public communication link to achieve covert communication. It is worth mentioning the existing works [18, 27–29] related to this paper. Specifically, public communication user was first used to provide cover for a secret action in [27], while the relatively difficult multi-antenna system has not been considered. Further, the recent work [18] has studied multi-antenna-aided covert communications with Poisson field of interferers and maximal ratio transmitting (MRT) precoding. However, the covertness of pilot signals has not been examined in its system. Then, in view of pilot covertness, a covert pilot spoofing attack scheme was proposed in proactive eavesdropping paradigm [28], while it focused on the legitimate eavesdropping performance rather than covert communication performance. In addition, to the best of our knowledge, almost all the current results in the context of covert communications have not jointly considered the covertness of both pilot signals and downlink signals. Most importantly, all of them cannot effectively defend against the power trend test (PTT) method [29], which can be adopted by an active warden relying on the large scale fading of received signals. In general, the research in this paper is motivated together by the lack of cross research and the weaknesses mentioned above, especially the urgent need for an extraordinary covert communication system tackling the challenge of warden's PTT. 1.2 Contributions In this work, we utilize a public communication link combined with multi-antenna technique in wireless fading channel to achieve joint covert communication. First of all, according to the inherent characteristics of wireless fading channel, the uncertainty of warden's received signals is provided and almost impossible to neutralize. Moreover, multi-antenna beamforming technology is adopted to reduce the leakage of covert signal to unintended users, further confusing warden's detection. Last but not least, the large scale fading of public signal is used to provide a cover for that of covert signal, attempting to confound warden's powerful PTT. In general, our main contributions of this work can be summarized as below. We focus on a relatively practical covert communication scenario for the first time, where a powerful warden may jointly detect uplink covert pilot (ULCP) and downlink covert communication (DLCC) to improve its detection accuracy. Furthermore, we consider such issues as limited pilot transmission, channel estimation error and the accompanying imperfect channel state information (CSI). To this end, a joint covert communication (JTCC) system is proposed and further studied We systematically evaluate the detection performance of warden, the covertness and overall performance of the proposed system. Specifically, we derive the optimal detection threshold and the minimal average covert probability of DLCC, ULCP and JTCC, respectively. Then, an optimization framework for covert capacity is provided subject to the covertness constraint, the data transmission rate constraint of public user and the transmission power constraint of source transmitter. After analyzing and simplifying the optimization problem, an optimal power allocation algorithm is designed to achieve covert capacity Finally, various valuable insights are provided by our theoretical analyses and simulation results. We first find that the covert communication in the proposed system has the significant advantages of rarely affecting the public communication or occupying its communication resources. Besides, it is particularly found that the covertness performance of DLCC is not influenced by noise levels or warden's position, which can perfectly resist warden's PTT method. Furthermore, the design guideline on optimal nodes deployment is explored, and the overall performance of JTCC system and DLCC system is also briefly compared 1.3 Organization and notations The rest of this paper is organized as follows: Section 2 first details the system model. Section 3 examines the performance of detection and covertness. Section 4 studies optimal design for overall performance of the JTCC system. Section 5 presents the simulation results, and Section 6 concludes the paper. Notation: C N ( μ , σ 2 ) denotes the complex Gaussian distribution with mean μ and variance σ2, Γ ( α , β ) denotes the normalized gamma random function with shape parameter α and scale parameter β, | · | denotes absolute value, ∥ · ∥ denotes Euclidean norm, P [ · ] denotes probability measure and E { · } denotes expectation measure. 2 SYSTEM MODEL The system model for covert communications is illustrated in Figure 1, where a transmitter (Alice) attempts to deliver sensitive information to a destination (Bob) with a low probability of being detected by a warden (Willie). In particular, an existing public communication link from Alice to the public user (Carol) is exploited as the cover of covert communication. Assume that the public communication behavior is mastered by all nodes, thus the covert transmission would possibly occur only when Carol is communicating. Consider a time-slotted system where the locations of all the nodes remain static in a time slot and each slot contains N symbol periods. We further assume that Alice is equipped with M antennas while all the other nodes have a single antenna. In a word, covert communication is going to be achieved specifically by signals design, multi-user beamforming and transmission power allocation. FIGURE 1Open in figure viewerPowerPoint Multi-antenna covert communication system with a public communication link 2.1 Channel model All the wireless channels are subject to independent quasi-static Rayleigh fading, which indicates that the channel coefficients are invariant within a time slot, but independently change from one slot to another. Denote the channel coefficients by h a z , where h a z = { h a z m } , m = 1 , 2 , … , M , z ∈ { b , c } , and the subscripts a, b, c and w denote Alice, Bob, Carol and Willie, respectively. When DLCC is on, the received signal at Bob or Carol for each symbol period is uniformly given by y z [ n ] = P a b d a z − α h a z w b x b [ n ] + P a c d a z − α h a z w c x c [ n ] + r z [ n ] , (1)where x b [ n ] and x c [ n ] denote the corresponding signals, n = 1 , 2 , … , N denotes symbol index, P a b and P a c , respectively, denote the transmission power of Alice-Bob and Alice-Carol, d a z denotes distance between the corresponding nodes, α denotes the path loss exponent, w b and w c denote the respective precoding vectors, r z [ n ] denotes the thermal noise with variance σ 0 2 . For simplicity, it is assumed that x b [ n ] ∼ C N ( 0 , 1 ) , x c [ n ] ∼ C N ( 0 , 1 ) , h a z m ∼ C N ( 0 , 1 ) and r z [ n ] ∼ C N ( 0 , σ 0 2 ) . Intuitively, due to the pilot covertness and the accompanying limitation of pilot transmission considered in this paper, the channel estimation error cannot be ignored, which is given by [30] ϕ z = 1 + L γ b M − 1 , (2)where ϕ z ∈ ( 0 , 1 ) , L denotes pilot sequence length, P z a denotes pilot transmission power and γ z = P z a d a z − α σ 0 2 denotes signal to noise ratio (SNR) of received pilot. Then, the channel coefficient is further given by [31] h a z = h ̂ a z + h ∼ a z , (3)where h ̂ a z ∼ C N ( 0 , ( 1 − ϕ z ) I M ) and h ∼ a z ∼ C N ( 0 , ϕ z I M ) represent the estimation part and the error part of the real channel h a z , respectively. 2.2 Beamforming and detection schemes As beamforming technology could not only enhance the received signal at intended users but also suppress the transmitted signal to unintended users, first we should attach great importance to the beamforming design. Besides, since pilot signal is necessary for channel estimation and subsequent beamforming, a possible utilization of pilot transmission behavior at Willie is considered for the first time. Beamforming design: The essential part of beamforming design is the selection or exploration of precoding methods in order to meet the different system requirements. Specifically, on the one hand, the performance of covert communication needs to be considered, utilizing the multi-antenna technologies and the public communication cover. On the other hand, the performance of public communication is also required so as not to arouse suspicion. In practice, it would be best to realize the effect that the covert communication is completely insensitive to the public communication. Based on the above considerations, zero force (ZF) precoding is employed by Alice, which is a simple and practical multi-user linear precoding that could eliminate the interference between different users [32]. Then, the overall precoding vector W with imperfect CSI is given by W = w b , w c = H ̂ H H ̂ H ̂ H − 1 H ̂ H H ̂ H ̂ H − 1 , (4)where H ̂ = [ h ̂ a b T , h ̂ a c T ] T denotes the overall channel matrix. Detection schemes: It is clear that Willie will make every effort to detect the DLCC from Alice-Bob so as to further obtain sensitive information. However as described earlier, pilot transmission process may expose the possibility of covert communication to a certain extent, thus Willie may improve his detection accuracy by detecting the presence of Bob's uplink pilot. To this end, the JTCC system is proposed for taking into account the covertness of ULCP. In addition, it is not feasible for Willie to determine the success of covert communication detection simply by detecting ULCP from Bob-Alice. This is because Bob can easily deceive Willie through a pilot deception scheme, stating in detail, Bob may continuously transmit pilot sequence but does not communicate in downlink. Besides, only the detection of ULCP cannot facilitate Willie any further decryption of confidential information. Therefore, Willie would generally adopt the schemes for detecting DLCC only or jointly detecting ULCP as well. 2.3 Hypothesis testing and performance metrics In order to detect the presence of covert communication, Willie faces a binary hypothesis testing problem on whether Alice is transmitting to Bob or not. Take DLCC detection as an example, the two hypotheses could be denoted as below H 0 : y w [ n ] = P a c d a w − α h a w w c x c [ n ] + r w [ n ] , (5) H 1 : y w [ n ] = P a b d a w − α h a w w b x b [ n ] + P a c d a w − α h a w w c x c [ n ] + r w [ n ] , (6)where the null hypothesis H 0 represents that Alice did not transmit to Bob while the alternative hypothesis H 1 represents that Alice did transmit. We assume that Willie could synchronize each communication slot between Alice and Bob. Then, with a radiometer adopted as detector [14], Willie makes the decision as below T ( y w ) = 1 N ∑ n = 1 N y w [ n ] 2 ≷ D 0 D 1 υ , (7)where the test statistic T ( y w ) denotes the average received power at Willie in a slot, υ > 0 denotes the predefined threshold of Willie's detector. D0 and D1 denote the binary decisions in favor of H 0 and H 1 , respectively. As per the mechanism of the radiometer, Willie will make the decision of D0 if T ( y w ) ≤ υ , and the decision of D1 if T ( y w ) > υ . Then, the false alarm probability and the misdetection probability are, respectively, defined as α = P [ D 1 | H 0 ] and β = P [ D 0 | H 1 ] . Considering that Willie would not have any prior knowledge about the exact transmission time of Alice, without loss of generality, we suppose the a priori probability of either hypothesis is 0.5 [33]. Then, the detection error probability at Willie is defined as ξ = α + β , (8)where 0 ≤ ξ ≤ 1 . Specifically, ξ = 0 indicates that Willie can perfectly detect the transmission without error. In contrast, ξ = 1 indicates that the detection result of Willie is equivalent to random guess and judgment, which is not convincing at all. Hence, ξ also represents the covert probability of the communication system actually. In this work, we focus on the case that Willie is allowed to observe an infinite number of samples in a slot [34] i.e. N → ∞ . Then, the covert probability ξ is rewritten as ξ ( υ , P w , P I ) = 0 , P I + σ 0 2 ≤ υ ≤ P w + P I + σ 0 2 , 1 , otherwise , (9)where P w = | h a w w b | 2 P a b d a w − α and P I = | h a w w c | 2 P a c d a w − α , respectively, denote the received signal power from Alice-Bob and the received interference power from Alice-Carol. Now, we introduce minimum average covert probability and covert capacity to measure system performance. Minimal average covert probability: Note that both P w and P I are random variables owing to the randomness of fading channels; thus, ξ is a Bernoulli distributed random variable for any given detection threshold υ. Similar to the existing works [16, 24], we adopt the minimal average covert probability ξ ¯ to measure the average covertness performance of the system i.e. ξ ¯ ∗ = min υ > 0 ∫ 0 ∞ ∫ 0 ∞ ξ υ , P w , P I f P I P I d P I f P w P w d P w , (10)where f P w ( P w ) and f P I ( P I ) denote the distribution of P w and P I , respectively. min υ indicates that Willie would adjust υ to minimize ξ ¯ . Covert capacity: Instead of employing covert throughput to capture rate efficiency as in [16, 18], we adopt covert capacity to quantize the overall system performance. As such, reliability constraint is considered here due to the adoption of average communication rate. Under covertness constraint, the data transmission rate constraint of Carol and the transmission power constraint of Alice, covert capacity is given by η = max R b , s . t . ξ ¯ ∗ ≥ 1 − ε , R c ≥ R c l and P a b + P a c ≤ P max , (11)where R b denotes the average data transmission rate of Bob, 0 ≤ ε ≤ 1 denotes covertness requirement, R c l denotes the data transmission rate requirement of Carol and P max denotes the maximum transmission power requirement of Alice. 3 PERFORMANCE EVALUATION OF DETECTION AND COVERTNESS In this section, we comprehensively study the optimal detection at Willie and the covertness performance of the JTCC system. Specifically, the optimal detection thresholds at Willie are analyzed and proved first. Then, we derive the minimal average covert probabilities of DLCC, ULCP and JTCC, which are used to measure the respective covertness performance. Furthermore, we analytically find some interesting invariance properties of covertness performance in respect of noise levels and the distances from Alice-Willie. 3.1 Detection of DLCC at Willie We first study the detection of DLCC at Willie. Before deriving the optimal detection threshold and the minimal average covert probability of DLCC, we need to compute the expression of average covert probability. According to the leakage beam gain | h a w w z | 2 ∼ exp ( 1 ) with imperfect CSI [35], the distributions of P w and P I are, respectively, given by f P w ( x ) = 1 P a b d a w − α exp − x P a b d a w − α , x ≥ 0 , (12) f P I ( x ) = 1 P a c d a w − α exp − x P a c d a w − α , x ≥ 0 . (13) Then, the average covert probability in Equation (9) is calculated as ξ ¯ D L = ∫ 0 ∞ ∫ 0 ∞ ξ υ D L , P w , P I P a b P a c d a w − 2 α exp − P I P a c d a w − α − P w P a b d a w − α × d P I d P w . (14) From the perspective of robust design, optimal detection performance at Willie is considered, or more precisely, the optimal detection threshold υ D L ∗ is set for the minimal average covert probability ξ ¯ D L ∗ . Theorem 1.Using a radiometer for detecting DLCC, the optimal detection threshold for Willie's detector is υ D L ∗ = P a b d a w − α + σ 0 2 , P a b = P a c , ( ln P a b − ln P a c ) P a b P a c d a w − α P a b − P a c + σ 0 2 , P a b ≠ P a c . (15) Proof: Following the covert probability ξ ( υ , P w , P I ) in Equation (9), the average covert probability of DLCC ξ ¯ D L in Equation (14) can be calculated as below ξ ¯ D L υ D L = ( ∫ 0 ∞ ∫ υ D L − σ 0 2 ∞ + ∫ 0 υ D L − σ 0 2 ∫ 0 υ D L − P w − σ 0 2 ) × 1 P a b P a c d a w − 2 α exp − P I P a c d a w − α − P w P a b d a w − α d P I d P w = 1 P a b d a w − α ∫ 0 ∞ exp − P w P a b d a w − α exp − υ D L − σ 0 2 P a c d a w − α d P w + ∫ 0 υ D L exp − P w P a b d a w − α ( 1 − exp − υ D L − P w − σ 0 2 P a c d a w − α d P w . (16)Due to Willie's detection principle of minimizing its average detection error probability, the optimal detection threshold is denoted by υ D L ∗ = arg min υ D L ξ ¯ D L υ D L . (17)We next analyze the two possible cases in Equation (16) separately and find the optimal detection threshold υ D L ∗ .Case I: P a b ≠ P a c . Then, the integral in Equation (16) is calculated as below ξ ¯ D L υ D L = 1 − P a b P a b − P a c ( exp ( − υ D L − σ 0 2 P a b d a w − α ) − exp ( − υ D L − σ 0 2 P a c d a w − α ) ) . (18)In order to determine υ D L ∗ , we set the first derivative of ξ ¯ D L ( υ D L ) w.r.t. υ D L equal to zero i.e. ∂ ξ ¯ D L υ D L ∂ υ D L = P a b P a b − P a c d a w − α ( 1 P a b exp ( − υ D L − σ 0 2 P a b d a w − α ) − 1 P a c exp ( − υ D L − σ 0 2 P a c d a w − α ) ) = 0 . (19)After a simple calculation, we can derive υ D L × = ( ln P a b − ln P a c ) P a b P a c d a w − α P a b − P a c + σ 0 2 . (20)Further substituting υ D L × into the second derivative of ξ ¯ D L ( υ D L ) , we have ∂ 2 ξ ¯ D L υ D L ∂ υ D L 2 υ D L = υ D L × = 1 P a b P a c d a w − 2 α exp − υ D L × P a b d a w − α > 0 . (21)Hence, in the case P a b ≠ P a c , the optimal detection threshold is obtained as below υ D L ∗ = υ D L × = ( ln P a b − ln P a c ) P a b P a c d a w − α P a b − P a c + σ 0 2 . (22)Case II: P a b = P a c . Now Equation (16) is simplified as ξ ¯ D L υ D L = 1 − υ D L − σ 0 2 P a b d a w − α exp ( − υ D L − σ 0 2 P a b d a w − α ) . (23)Similarly to Case I, after the calculation about the first and the second derivative of ξ ¯ D L ( υ D L ) coupled with a few simple manipulations, we could derive the optimal detection threshold as below υ D L ∗ = P a b d a w − α + σ 0 2 . (24)Combining these two cases completes the proof. □ Theorem 2.The minimal average covert probability of DLCC is given by ξ ¯ D L ∗ ρ = 1 − ρ 1 1 − ρ , ρ > 0 and ρ ≠ 1 , 1 − exp ( − 1 ) , ρ = 1 , (25)and ξ ¯ D L ∗ ( ρ ) decreases with ρ ∈ ( 0 , ∞ ) , where ρ = P a b P a c .Proof: Please refer to Appendix 1. □ Figure 2 depicts the average covert probability ξ ¯ D L versus the detection threshold υ D L for varying d a w , P a b and σ 0 2 in DLCC. It can be seen that the simulation results are well fit with our theoretical results above. Specifically, ξ ¯ D L decreases first and then increases with υ D L , thus an optimal value of υ D L always exists to minimize ξ ¯ D L , which is corresponding to the optimal detection threshold υ D L ∗ in Theorem 1. Besides, unless varying the transmission power rat