Front. Comms. Net Frontiers in Communications and Networks Front. Comms. Net 2673-530X Frontiers Media S.A. 1370496 10.3389/frcmn.2024.1370496 Communications and Networks Original Research Secure authentication in MIMO systems: exploring physical limits Abdrabou and Gulliver 10.3389/frcmn.2024.1370496 Abdrabou Mohammed * Gulliver T. Aaron Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, Canada

Edited by: Pantea Nadimi Goki, Sant’Anna School of Advanced Studies, Italy

Reviewed by: Lei Guan, Xidian University, China

Chanapha Bast, Udon Thani Rajabhat University, Thailand

 *Correspondence: Mohammed Abdrabou, abdrabou@outlook.com 03 05 2024 2024 5 1370496 14 01 2024 05 03 2024 Copyright © 2024 Abdrabou and Gulliver. 2024 Abdrabou and Gulliver

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Multiple-input multiple-output (MIMO) technology is employed to improve the reliability and capacity of wireless communication systems. However, the wireless communication environment creates vulnerabilities to spoofing attacks. Furthermore, the authentication challenges posed by the heterogeneous characteristics of wireless applications increase as diverse technologies facilitate the growing number of Internet of Things (IoT) devices. To address these challenges, adaptive physical-layer authentication (PLA) leveraging the inherent antenna diversity in MIMO systems is examined, and an information-theoretic perspective on PLA in MIMO systems is given. The real and imaginary components of the received reference signals are used as attributes with a single-class classification support vector machine (SCC-SVM). It is shown that the authentication performance improves with the number of antennas, and the proposed scheme provides robust authentication.

 physical-layer authentication single-class classification information theory machine learning MIMO systems section-at-acceptance Security, Privacy and Authentication 1 Introduction

The number of wireless devices has increased significantly as a result of Internet of Things (IoT) development. However, the limited bandwidth for applications and limited device capabilities make it challenging to deliver reliable services in wireless networks. Multiple-input multiple-output (MIMO) technology can improve the effectiveness and capacity of wireless communication networks without increasing bandwidth resources (Zhang et al., 2020).

Information-theoretic security metrics for secrecy, privacy, authentication, stealth, and covertness have been developed and fundamental limits derived (Liang et al., 2008; Bloch et al., 2021). Shannon (1949) presented the first information-theoretic study of a cryptography model. Cryptography is the science of securing communications. It involves the transformation of plaintext into ciphertext, which can be intercepted by an attacker. A key K is a secret parameter used in conjunction with an algorithm to encrypt or decrypt data. The message M represents the data that need to be protected. For encryption, K is used to transform M into a ciphertext E. The goal is to maximize the secret information rate between legitimate users. Wyner (1975) presented information-theoretic security limits for a discrete memoryless wiretap channel. It was shown that security is possible if the capacity C m between legitimate users is higher than the wiretap channel capacity C w between the transmitter and attacker. Abdi et al. (2001) extended the model proposed by Wyner (1975) to a Gaussian wiretap channel. It was shown that secure communication can be achieved if C s = C m − C w > 0. However, C s is degraded by wireless channels due to factors such as fading (Zou et al., 2016).

The model proposed by Wyner (1975) was extended to MIMO systems (Khisti and Wornell, 2010a), and this was improved by Khisti and Wornell (2010b). It was shown that increasing the number of antennas can improve C s . Bloch et al. (2008) considered quasi-static fading. The outage probability P out was shown to depend on the user location and path loss, where P out = P(C s < R s ) and R s is the secrecy rate. Information-theoretic security for MIMO systems was considered by Oggier and Hassibi (2011), but the channel state information (CSI) was assumed to be known. Security without knowledge about the attacker CSI was considered by He et al. (2011).

A MIMO scheme for practical physical-layer security was presented by Ishikawa et al. (2021). It employs chaos-based unitary matrices to eliminate the need for channel estimation. Unlike conventional approaches, an imperfect key agreement in high-mobility scenarios was considered, and an algorithm for reconciling chaotic sequences between legitimate parties was given. Simulation results were presented which show that the performance is superior to that with conventional chaos-based MIMO schemes assuming perfect CSI.

Downlink security in cell-free massive MIMO systems with imperfect channel estimation was investigated by Tubail et al. (2023). Two power allocation algorithms using artificial noise were introduced to combat passive eavesdropping. Imperfect channel estimation was considered as it leads to artificial noise leakage which impacts performance. The results show that the proposed algorithm improves the security through robust authentication.

In this paper, physical-layer authentication (PLA) for MIMO systems is examined from an information-theoretic perspective. The authentication boundary is determined using machine learning (ML). For training, a single-class classification support vector machine (SCC-SVM) is used with only legitimate user data. The real and imaginary components of the received reference signals are employed as attributes. These attributes vary slowly over time as a result of factors such as mobility (Hou et al., 2014; Ferrante et al., 2015; Wang et al., 2015; Wang et al., 2016; Fang et al., 2020). As a result, the SCC-SVM authentication boundary is updated to ensure reliable authentication. The authentication rate (AR) is shown to improve with antenna diversity. The contributions of this paper are as follows.

• PLA for MIMO systems is examined from an information-theoretic perspective.

• The real and imaginary components of the received signals are used as attributes to improve PLA performance.

• A practical approach is presented to extract attributes from cellular network reference signals.

• The effectiveness of the proposed PLA scheme is evaluated and validated in urban environments.

The remainder of this paper is organized as follows: the system model is presented in Section 2; Section 3 introduces SCC-SVM and the proposed PLA scheme; the performance evaluation metrics are given in Section 4; Section 5 provides the simulation results; and Section 6 provides some concluding remarks.

 2 System model

The system model for the proposed authentication scheme is shown in Figure 1. In this scheme, Alice (A) represents a user requiring authentication from Bob (B), while the spoofer (S) is an attacker attempting to impersonate A. B must ascertain the legitimacy of A and also reject S. Consequently, B must decide between the two hypotheses H 0 : A  is transmitting H 1 : S  is transmitting . Thus, H 0 indicates that the signal originates from A, while H 1 indicates that it is from S. Both A and S are assumed to be mobile, while B is stationary. A and S have either one or M transmit antennas, and B has N receive antennas.

System model.

Reference signals are transmitted in cellular networks to facilitate CSI estimation. In particular, sounding reference signals are repeated every 28 symbols or 56 symbols (GPP, version 15.2.0, 2018). The received signal at B can be expressed as Y = H x + W , where x is the transmitted reference signals, H is the channel matrix, and W is independent additive white Gaussian noise (AWGN). The channel matrix for the M × N MIMO system can be partitioned according to the transmit and receive antennas as follows: H = h 11 h 12 … h 1 M h 21 h 22 … h 2 M ⋮ h N 1 h N 2 … h N M , where h nm represents the channel from the mth transmit antenna to the nth receive antenna. H can be estimated from Y corresponding to the known reference signals (Schindler and Mellein, 2011; Germain et al., 2020). Y is used to obtain the data for SCC-SVM training and testing.

The proposed scheme is implemented in the uplink and has an initial phase (T 1), followed by subsequent phases (T 2, T 3, … , T n ). During the initial phase, upper-layer authentication (ULA) is conducted, and SCC-SVM training is performed at B using the received reference signals from A. The real and imaginary components of these signals for each transmit–receive antenna pair are obtained to provide 2 × M attributes per antenna for a total of 2 × M × N attributes. In subsequent phases, SCC-SVM testing is conducted at B to validate the legitimacy of the received signals, which may originate from either A or a spoofer S. Hence, the received signals during these phases are treated as being from an unknown user U. If successful, the attributes are updated, and SCC-SVM training is repeated. Conversely, if testing fails, the connection is terminated.

SCC-SVM training is used in the initial phase to establish an authentication boundary. In subsequent phases, SCC-SVM testing is used to determine whether the received signal attributes fall within this boundary. In this case, the test is successful, and authentication continues. The received signal is rejected if the attributes fall outside the boundary. The proposed scheme leverages the spatial independence of A and S as the physical-layer attributes for authentication are independent. Furthermore, transmit and receive antenna diversity is exploited to generate a large number of attributes. These attributes vary slowly over time due to factors such as mobility (Hou et al., 2014; Ferrante et al., 2015; Wang et al., 2015; Wang et al., 2016; Fang et al., 2020). Consequently, the SCC-SVM authentication boundary is dynamically updated in each subsequent phase to ensure reliable authentication.

 3 The proposed scheme 3.1 Single-class classification support vector machine

SCC is a machine learning technique that can be employed to distinguish between A and S using training data from A. The goal is to find the optimal authentication boundary that surrounds most of the training data from A (Tax and Duin, 2004). In this paper, the SCC-SVM algorithm proposed by Schölkopf et al. (2001) is employed. SCC-SVM computes a decision function f which encompasses the majority of the training data [Senigagliesi et al. (2020); Abdrabou and Gulliver (2022c); Abdrabou and Gulliver, 2022a; Abdrabou and Gulliver, 2023]. First, the following optimization problem is solved (Hoang et al., 2021); (Schölkopf et al., 2001): min w , s , ρ 1 2 ‖ w ‖ 2 + 1 η ℓ ∑ i = 1 ℓ s i − ρ , subject to w ⋅ Φ g i ≥ ρ − s i , s i ≥ 0 , where w is the weight vector, ρ is the distance from the origin to the boundary, Φ is a feature mapping determined based on the kernel employed, g i is the ith feature vector used for SCC-SVM training, s i is the corresponding slack variable, ℓ is the number of training samples, and η is the percentage of data considered as outliers (Senigagliesi et al., 2020). SCC-SVM maps data to a feature space using kernels and then separates the features using a boundary. The optimization problem in Eq. 1 provides w and ρ which determine the boundary used in the decision function Eq. 3. Using Lagrange multipliers p i , q i ≥ 0 yields (Schölkopf et al., 2001) L w , s , p , q , ρ = 1 2 ‖ w ‖ 2 + 1 η ℓ ∑ i = 1 ℓ s i − ρ − ∑ i = 1 ℓ p i w ⋅ Φ g i − ρ + s i − ∑ i = 1 ℓ q i s i . Setting the derivatives with respect to w, s, and ρ as zero yields (Schölkopf et al. 2001) w = ∑ i = 1 ℓ p i Φ g i , p i = 1 η ℓ − q i ≤ 1 η ℓ , ∑ i = 1 ℓ p i = 1 .

The decision function used to test a new sample t is expressed as follows (Senigagliesi et al. 2020); (Hoang et al. 2021): f t = sgn w ⋅ Φ t − ρ , and substituting w from Eq. 2 yields f t = sgn ∑ i p i Φ g i ⋅ Φ t − ρ . The kernel expansion is defined as (Schölkopf et al. 2001) k g i , t = Φ g i ⋅ Φ t , so the decision function is f t = sgn ∑ i p i k g i , t − ρ . A test sample t is accepted if f(t) > 0, which indicates that it is within the authentication boundary (Senigagliesi et al., 2020).

 3.2 Proposed scheme

The proposed scheme employs SCC-SVM with antenna diversity. The received signal at the nth receive antenna from the mth transmit antenna is y n m = x m h n m + n n m , where x m is the transmitted reference symbol and h nm and n nm are the corresponding channel coefficients and AWGN, respectively. The WINNER II channel model for non-line-of-sight (NLOS) urban environments (Kyösti et al., 2007) is considered here. The received MIMO signals can be expressed as Y = y 11 y 12 … y 1 M y 21 y 22 … y 2 M ⋮ y N 1 y N 2 … y N M . This matrix is employed to obtain features for SCC-SVM training and testing. Binary phase-shift keying (BPSK) modulation is assumed, so the transmitted signal is (Chatzidiamantis et al., 2011) x t = ∑ k g t − k T s cos 2 π f s t + φ k , where g(t) is the pulse shaping function, 0 ≤ t ≤ T s, T s is the symbol time, f s is the carrier frequency, and φ k ∈ [0, π] is the phase of the kth symbol. The real (R) and imaginary (I) components are used as attributes so that Y = R I 11 R I 12 … R I 1 M R I 21 R I 22 … R I 2 M ⋮ R I N 1 R I N 2 … R I N M .

Figure 2 illustrates the proposed scheme. In the initial phase T 1 following ULA authentication, data are obtained from the legitimate user (A) for SCC-SVM training. In subsequent phases, reference signal data from U are utilized by B for testing and training. If testing is successful in a given phase, the corresponding data are employed to update the attributes for training. A sliding window is used so the oldest data are discarded. Conversely, if testing fails, the connection is terminated.

Authentication scheme flowchart.

The initial authentication is performed in phase T 1 with A through ULA. Then, the reference signal data from A are used to construct the training sample vector d i . The ith data sample is then d i = R I 11 , … , R I 1 M , R I 21 , … , R I N 1 , … , R I N M . SCC-SVM training is performed using the vectors d i , where i = 1, 2, … , ℓ, to obtain the authentication boundary. In subsequent phases, SCC-SVM is utilized to test new samples t, as shown in Eq. 4, originating from U, where U could be A or S. If testing is successful, U is accepted and the attributes are updated, followed by SCC-SVM training. Otherwise, the connection is terminated.

The ℓ training samples from A in the initial phase are d i = R i I i 11 , … , R i I i 1 M , R i I i 21 , … , R i I i N 1 , … , R i I i N M , i = 1,2 , … , ℓ , and the testing sample from U is t = R I U 11 , … , R I U 1 M , R I U 21 , … , R I U N 1 , … , R I U N M ,

Figure 3 illustrates the sliding window update process for the attributes. In phase T 1, the training data from A consist of ℓ matrices Y, where each matrix is represented by a vector as shown in Eq. 5. Subsequently, in phase T 2 a new data vector t is tested (following scaling), and if accepted, the data matrix is updated by discarding the first row d 1 and adding the new data vector as row ℓ + 1. Consequently, if the first e new data vectors are accepted, the training data matrix is M e = d 1 + e d 2 + e ⋮ d ℓ + e , so ℓ vectors are used for training.

Sliding window for attribute updates.

Each attribute undergoes separate minimum–maximum scaling m r = m 1 , r m 2 , r … m ℓ , r T , r = 1,2 , … , 2 M N , to obtain g r = g 1 , r g 2 , r … g ℓ , r T , r = 1,2 , … , 2 M N , where g i , r = m i , r − m min,r m max,r − m min,r , and m min,r is the minimum value in m r and m max,r is the maximum value in m r . The feature matrix for training is then G = g 1 g 2 … g 3 M N , and the elements of t are scaled using m min,r and m max,r from the training data as follows: t r = b r − m min,r m max,r − m min,r to form the test vector t. The proposed scheme is summarized in Algorithm 1.

Proposed scheme.

 ULA for A.

 Obtain the received reference signals from each antenna.

 Extract the real (R) and imaginary (I) components from the received reference signals.

 Construct the training samples d i using Eq. 5.

 Train SCC-SVM to establish the authentication boundary.

 Test SCC-SVM with the sample t as in Eq. 6.

 if (f(t) > 0) then

  Accept the user.

  Update the attributes.

  Retrain SCC-SVM.

 else if (f(t) ≤ 0) then

  Terminate the connection.

 end if

3.3 Information-theoretic PLA

The authentication performance can be improved with antenna diversity (Abdrabou and Gulliver, 2022b; 2024). This is indicated by the mutual information between the training and testing attributes from A I(Tr A ; Te A ), which is greater than the mutual information between the training attributes from A and testing attributes from S I(Tr A ; Te S ). The mutual information authentication rate I AR can be defined as I A R = I T r A ; T e A − I T r A ; T e S . First, a single-input multiple-output system (SIMO) and independent channels based on user location and mobility are considered. The mutual information that characterizes the legitimacy of A at the receiver is a function of the wireless link at two consecutive time instances (training and testing). The mutual information between the transmit antenna and receive antennas is I i SIMO = I T x ; R x 1 I T x ; R x 2 ⋮ I T x ; R x N , i ∈ A , S . Each element of this matrix characterizes the received reference signal at times t − 1 (training) and t (testing), so it can be reformulated as I i SIMO = I 1 y 1 t ; y 1 t − 1 I 2 y 2 t ; y 2 t − 1 ⋮ I N y N t ; y N t − 1 , where y i (t − 1) is the training data from A and y i (t) is the testing data from U, so then, I A R SIMO = I A SIMO − I S SIMO .

Now, a MIMO system is considered. The mutual information between the transmit antennas and receive antennas can be expressed as I i MIMO = I T x 1 ; R x 1 , … , I T x M ; R x 1 I T x 1 ; R x 2 , … , I T x M ; R x 2 I T x 1 ; R x 3 , … , I T x M ; R x 3 ⋮ I T x 1 ; R x N , … , I T x M ; R x N , i ∈ A , S . Each element of this matrix characterizes the received reference signal at times t − 1 and t, so I i MIMO = I 11 y 11 t ; y 11 t − 1 , … , I 1 M y 1 M t ; y 1 M t − 1 I 21 y 21 t ; y 21 t − 1 , … , I 2 M y 2 M t ; y 2 M t − 1 I 31 y 31 t ; y 31 t − 1 , … , I 3 M y 3 M t ; y 3 M t − 1 ⋮ I N 1 y N 1 t ; y N 1 t − 1 , … , I N M y N M t ; y N M t − 1 , where y*(t − 1) is the training data from A and y*(t) is the testing data from U, and therefore, I A R MIMO = I A MIMO − I S MIMO .

Since mutual information is greater than or equal to zero [Cover and Thomas, 2006, (2.90)], I i j ≥ 0 , i ∈ A , S , j ∈ S I M O , M I M O . The joint mutual information for independent channels in a SIMO system can be formulated as I T x 1 ; R x 1 R x 2 = I T x 1 ; R x 1 + I T x 1 ; R x 2 , ⋮ I T x 1 ; R x 1 … R x N = I T x 1 ; R x 1 + … + I T x 1 ; R x N , so I T x 1 ; R x 1 ≤ I T x 1 ; R x 1 R x 2 ≤ I T x 1 ; R x 1 R x 2 R x 3 ⋮ ≤ I T x 1 ; R x 1 … R x N − 1 ≤ I T x 1 ; R x 1 … R x N . For a MIMO system, this is I T x 1 T x 2 ; R x 1 R x 1 = I T x 1 ; R x 1 + I T x 2 ; R x 1 + I T x 1 ; R x 2 + I T x 2 ; R x 2 , ⋮ I T x 1 … T x M ; R x j … R x N = I T x 1 ; R x 1 + … + I T x M ; R x 1 + I T x 1 ; R x 2 + … + I T x M ; R x 2 ⋮ I T x 1 ; R x N + … + I T x M ; R x N , so I T x 1 ; R x 1 … R x N ≤ I T x 1 T x 2 ; R x 1 … R x N ≤ I T x 1 T x 2 T x 3 ; R x 1 … R x N ⋮ ≤ I T x 1 T x 2 T x 3 … T x M − 1 ; R x 1 … R x N ≤ I T x 1 T x 2 T x 3 … T x M ; R x 1 … R x N , . These results show that the authentication performance improves with the number of receive and transmit antennas so that I A R u v ≥ I A R p q , u ≥ p , v ≥ q , where u and p are the number of receive antennas and v and q are the number of transmit antennas.

 3.4 Channel capacity for PLA

The channel capacity C is defined as the maximum mutual information (Cover and Thomas, 2006), C = max P x i I X ; Y . The capacity of a MIMO system depends on several factors such as the signal-to-noise ratio (SNR). It can be obtained using singular value decomposition (SVD) to decompose the channel matrix H and obtain the singular values and unitary matrices for the transmitter and receiver as (Golub and Van Loan, 2013) H = U Σ V H , where U is the unitary matrix corresponding to the transmitter, Σ is a diagonal matrix containing the singular values, and V H is the conjugate transpose of the unitary matrix corresponding to the receiver. The singular values are Λ = λ 1 , λ 2 , … , λ n , and the corresponding channel capacity is (Tse and Viswanath, 2005) C = ∑ i = 1 n log 2 1 + S N R ⋅ λ i 2 , where n is the number of singular values and λ i is the ith singular value of the channel matrix.

 4 Performance evaluation

The performance is evaluated using the confusion matrix which includes true positive (TP), the acceptance of A, true negative (TN), the rejection of S, false negative (FN), the erroneous rejection of A, and false positive (FP), the erroneous acceptance of S. The evaluation metrics are the missed detection rate (MDR), false alarm rate (FAR), precision or positive predictive value (PPV), false discovery rate (FDR), and authentication rate (AR) (Senigagliesi et al., 2020; Abdrabou and Gulliver, 2022b): M D R = F P F P + T N , F A R = F N F N + T P , P P V = T P T P + F P , F D R = F P T P + F P , A R = T P + γ × T N T P + F N + γ × T N + F P , where γ is used to balance between A and S and is given by γ = T P + F N T N + F P . The AR improves with the number of antenna (Abdrabou and Gulliver, 2022b), and the corresponding PPV increases, so I A ∝ P P V , and the FDR decreases, so I S ∝ F D R , and therefore, I A R ∝ P P V − F D R .

5 Simulation results

In this section, the performance of the proposed scheme is evaluated in multipath fading channels via Monte Carlo simulation. BPSK modulation is employed with the WINNER II channel model for NLOS urban environments (Kyösti et al., 2007). The single-user scenario has one Alice and three spoofers, while the multiple-user scenario has three Alice and three spoofers, as shown in Figures 4, 5, respectively. In these scenarios, Bob is stationary, while Alice and spoofers move arbitrarily and independently. SCC-SVM is implemented using the scikit-learn library in Python. For the single-user scenario, γ = 1 3 , and for the multiple-user scenario, γ = 1. The simulation parameters are given in Table 1.

Single-user scenario.

Multiple-user scenario.

Simulation parameters.

Parameter Value
Sampling rate 20 MHz
Urban model frequency band 2–6 GHz
Carrier frequency 5 GHz
Shadow fading standard deviation 4 dB
Number of transmit antennas for Alice/spoofer 1, 2, and 4
Number of receive antennas for Bob 1, 4, and 8
Antenna height for Alice/spoofer 1 m
Velocity 0.4 km/h
Signal-to-noise ratio (SNR) 8 dB
Position of Bob 1,000 m and 1,000 m
Reference signal interval 28 symbols
Alice 1 initial position 1,500 m and 1,200 m
Alice 2 initial position 500 m and 900 m
Alice 3 initial position 1,000 m and 600 m
Spoofer 1 initial position 1,400 m and 700 m
Spoofer 2 initial position 600 m and 1,250 m
Spoofer 3 initial position 500 m and 500 m

Figure 6 presents the MDR, FAR, and AR for the single-user scenario with an SNR of 8 dB, velocity of 0.4 km/h, and 1, 4, and 8 receive antennas. This shows that MDR, FAR, and AR improve with an increase in the number of antennas. For instance, with a single transmit antenna, the MDR decreases from 26.0% with 1 receive antenna to 17.8% with four receive antennas and 2.9% with eight receive antennas. Similarly, with four receive antennas, the MDR decreases from 17.8% with one transmit antenna to 4.0% with two transmit antennas and 1.7% with four transmit antennas. The FAR with a single transmit antenna decreases from 11.9% with one receive antenna to 1.9% with four receive antennas and 0.3% with eight receive antennas. In the case of four receive antennas, the FAR decreases from 1.9% with one transmit antenna to 0.2% with two transmit antennas and only 0.1% with four transmit antennas. The AR increases with an increase in the number of antennas. For example, with four receive antennas, the AR increases from 90.2% with one transmit antenna to 97.9% with two transmit antennas and 99.1% with four transmit antennas. These results illustrate the impact of antenna diversity on performance.

MDR, FAR, and AR versus the number of receive antennas for the single-user scenario with a signal-to-noise ratio (SNR) of 8 dB, velocity of 0.4 km/h, and 1, 4, and 8 receive antennas.

Figure 7 presents C, PPV, FDR, and PPV − FDR versus the number of receive antennas in the single-user scenario with an SNR of 8 dB, velocity of 0.4 km/h, and 1, 4, and 8 receive antennas. This shows that increasing the number of antennas improves C and thus, the information authentication rate I AR . With a single transmit antenna, C increases from 1.15 bps with one receive antenna to 4.75 bps with four receive antennas and 5.95 bps with eight receive antennas. Furthermore, the PPV increases, FDR decreases, and PPV − FDR increases with an increase in the number of antennas. For instance, with one transmit antenna, the PPV increases from 77.2% with one receive antenna to 84.6% with four receive antennas and 97.2% with eight receive antennas, while with four receive antennas, the PPV increases from 84.6% with one transmit antenna to 96.2% with two transmit antennas and 98.3% with four transmit antennas. In addition, with four receive antennas, the FDR decreases from 15.4% with one transmit antenna to 3.8% with two transmit antennas and 1.7% with four transmit antennas. The PPV − FDR also improves with an increase in the number of antennas as with four receive antennas, it increases from 69.2% with one transmit antenna to 92.3% with two transmit antennas and 96.6% with four transmit antennas. These results further illustrate the impact of antenna diversity on performance.

C, PPV, FDR, and PPV − FDR versus the number of receive antennas for the single-user scenario with an SNR of 8 dB, velocity of 0.4 km/h, and 1, 4, and 8 receive antennas.

Figure 8 presents the MDR, FAR, and AR versus the number of receive antennas with an SNR of 8 dB, velocity of 0.4 km/h, and one transmit antenna for the single- and multiple-user scenarios. This shows that the MDR, FAR, and AR increase with the number of receive antennas in both scenarios. For instance, in the multiple-user scenario, the MDR decreases from 44.9% with one receive antenna to 40.1% with four receive antennas and 31.9% with eight receive antennas. The corresponding FAR decreases from 17.7% with 1 receive antenna to 3.7% with four receive antennas and 2.0% with eight receive antennas, while the AR increases from 68.7% with one receive antenna to 78.1% with four receive antennas and 83.1% with eight receive antennas. These results also indicate that the MDR, FAR, and AR are better in the single-user scenario than in the multiple-user scenario. For instance, with four receive antennas, the MDR for the single- and multiple-user scenarios is 17.8% and 40.1%, respectively, while the corresponding FAR and AR are 1.9% and 3.7%, and 90.2% and 78.1%, respectively.

MDR, FAR, and AR versus the number of receive antennas with an SNR of 8 dB, velocity of 0.4 km/h, and 1 transmit antenna for the single- and multiple-user scenarios.

Figure 9 presents the MDR, FAR, and AR versus the number of receive antennas with an SNR of 8 dB, velocity of 0.4 km/h, and one transmit antenna for the single-user scenario, and η = 0.2 and 0.8. An increase in η means that a higher percentage of the data is treated as outliers, which decreases the MDR, so the probability of accepting spoofers is reduced. For example, with two receive antennas, the MDR is 26.6% for η = 0.2 and decreases to 10.9% for η = 0.8. However, an increase in η also results in a higher FAR, which increases the likelihood of rejecting Alice. For instance, with one receive antenna, the FAR is 8.7% for η = 0.2 and increases to 22.5% for η = 0.8. These results indicate that an increase in η will increase the AR. For example, with four receive antennas, the AR is 87.3% for η = 0.2 and increases to 92.0% for η = 0.8. Thus, η has a significant impact on the authentication performance and, therefore, should be chosen appropriately.

MDR, FAR, and AR versus the number of receive antenna with an SNR of 8 dB, velocity of 0.4 km/h, and 1 transmit antenna for the single-user scenario and η = 0.2 and 0.8.

Figure 10 presents the AR versus the velocity with an SNR of 8 dB, 1 transmit antenna, and 4 receive antennas using the real and imaginary components of the received signals as features (proposed scheme) and using only the magnitude, as in Pei et al. (2014). This shows that the AR with the proposed scheme is higher for all velocities. For example, the AR at a velocity of 0.4 km/h using the real and imaginary components of the signals as features is 89.2%, but it is only 56.7% using only the magnitude.

AR versus velocity with an SNR of 8 dB, 1 transmit antenna, and 4 receive antennas using the real and imaginary components of the received signals as features (proposed scheme) and using only the magnitude, as in Pei et al. (2014).

 6 Conclusion

An adaptive PLA scheme was proposed that leverages ML and the antenna diversity in MIMO communication systems. Authentication robustness is achieved via a sliding window to continually update the attributes. Furthermore, an information-theoretic perspective was given for PLA in MIMO systems. The results presented illustrate the relationship between authentication performance and the number of antennas; in particular, the mutual information authentication rate I AR improves with the number of antennas.

 Data availability statement

The original contributions presented in the study are included in the article; further inquiries can be directed to the corresponding author.

 Author contributions

MA: writing–original draft and writing–review and editing. TAG: writing–original draft and writing–review and editing.

 Funding

The author(s) declare that no financial support was received for the research, authorship, and/or publication of this article.

 Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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