# Threshold Secret Sharing Transmission against Passive Eavesdropping in MIMO Wireless Networks

Authors: Jungho Myung; Keunyoung Kim; Taehong Kim
Journal: Wireless Communications and Mobile Computing
 (2018)

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## Abstract

We propose a threshold secret sharing scheme for secure communications in multiple input and multiple output wireless networks. In the proposed scheme, the base station divides the secret data intoNmin parts using a polynomial of degree T-1(T≤Nmin) and transmits the divided data to the legitimate user by beamforming with multiple spatial dimensions. Then, at the user, the secret data can be reconstructed with a sufficient number (≥T) of divided parts by using the Lagrange interpolating polynomial. However, it is difficult for the eavesdropper to correctly estimate the T parts due to the difference between the main channel with beamforming and the eavesdropping channel in the physical layer, which results in the failure of secret data reconstruction. The numerical results show that the eavesdropping probability of the proposed scheme is lower than those of conventional schemes. Moreover, we analyze the symbol-error-rate and show that the theoretical result is well aligned with simulation results.


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## Body

## 1. Introduction

Recently, multiple input and multiple output (MIMO) wireless networks have attracted significant attention due to the potential performance improvements; they have been shown to lead to spatial multiplexing or diversity gain [1–5]. The spatial dimensions derived from multiple antennas have initiated many new transmission techniques utilizing space as a new resource other than frequency and time [3–5]. Among the conventional methods, maximum ratio transmission [3] is proposed, in which a single stream is sent for achieving full diversity gain from multiple spatial dimensions. In addition, for throughput enhancement, spatial multiplexing transmission has been proposed to transmit multiple streams simultaneously [4, 5]. However, most of existing works only focus on spatial dimensions by multiple antennas in order to maximize the spatial multiplexing gain or diversity gain without considering security.Wireless transmission is inherently vulnerable to eavesdropping due to the broadcast nature of the wireless medium [6–12]. Although a large number of security measures—from wired equivalent privacy in the wireless link layer to transport layer security in the application layer—have already been developed and widely deployed throughout network layers, the fact remains that it is these very measures that must now confront substantial challenges by attackers or eavesdroppers with immense computing resources acquirable from a cloud or bounded error quantum polynomial time algorithms leveraging quantum computers, to list just a few. As one of new attempts to overcome this problem, physical layer security (PLS) has been introduced to achieve fundamental secrecy in the sense that it does not rely on any intractability assumptions unlike cryptographic algorithms implemented in higher network layers. With a single antenna configuration, Wyner first introduced a wiretap channel and the associated secrecy, the results of which show the feasibility of ultimately secure communication [6]. However, there is a problem in that secrecy cannot be guaranteed if the gain of the eavesdropping channels is higher than the gain of the main channel, that is, the channel of the target user. To overcome this problem, PLS with multiple antennas has been proposed [8–12]. In MIMO wireless networks, by beamforming and jamming techniques, the secrecy can be provided even though the quality of the main channel is worse than the quality of the eavesdropping channel. Most of existing works have only focused on increasing the secrecy rate by beamforming and jamming design, assuming that the transmitter knows the channel state information (CSI) of the eavesdropper. However, it is impossible to obtain the CSI of the eavesdropper due to the passive posture. Therefore, to ensure secrecy against passive eavesdropper, a new transmission technique with multiple spatial dimensions is needed on the condition that the transmitter cannot know the eavesdropping channels.In this paper, we propose a threshold secret sharing transmission for secure communications in the absence of an eavesdropping channel information. In the proposed scheme, the secret data is divided intoNmin parts using a polynomial of degree T-1 and then transmitted to the target user through multiple spatial dimensions by a transmit beamforming. At the user, based on the Lagrange interpolating polynomial, the secret data can be recovered when a sufficient number (≥T) of the divided parts are combined together. We also propose the majority rule for secret reconstruction to overcome fading and noise impairments in wireless channels. At the point of the eavesdropper, it is difficult to correctly estimate the T parts due to the difference between main channel and eavesdropping channel in the physical layer. Therefore, attempt to reconstruct the secret data is unsuccessful. The numerical results show that the eavesdropping probability of the proposed approach is better than those of the conventional approaches. Also, we provide an analysis of SER for the proposed secret sharing scheme and verify that the proposed theoretically derived results well agree with the Monte-Carlo simulation results.Notations. XT, X∗, X†, X, and E[·] denote the transpose, the conjugate transpose, the pseudo inverse, the Euclidean norm of matrix X, and the expectation function, respectively.


## 2. System Model

As shown in Figure1, we consider MIMO wireless networks with a base station (BS) with Nt transmit antennas, a target user (TU) with Nr receiving antennas, and an eavesdropper (EA) with Ne receive antennas. When BS transmits a secret data over the channel matrix H to the TU, the radio signal is exposed to the EA over the cross channel G. The channel coefficients are assumed to be mutually independent Rayleigh flat fading with additive white Gaussian noise (AWGN) having zero mean and unit variance.Figure 1
MIMO wireless networks with an eavesdropper.The received signal at the target user can be written as(1)y=Hx+n,where y,x, and n denote the received signal vector, the transmitted signal vector, and the AWGN vector, respectively. Also, the eavesdropping signal can be written as(2)z=Gx+w,where z and w denote the received signal vector and the AWGN vector at the EA, respectively.


## 3. Threshold Secret Sharing Transmission in MIMO Wireless Networks

### 3.1. Threshold Secret Sharing Generator

Threshold secret sharing is a well-known scheme in cryptography introduced by Adi Sharmir [13]. In the scheme, a secret data is divided into N parts by a polynomial of degree T-1. To reconstruct the secret data, a minimum number of parts (T) is required for solving the polynomial problem correctly. In the threshold scheme, this number is less than the total number of parts (T≤N). Therefore, this is called the (T,N)-threshold secret sharing scheme [14–16].For spectral efficiency and throughput enhancement in MIMO networks, a BS transmits independent and separately encoded signals, so-called streams, from each of the multiple transmit antennas. Considering our system model, the number of streams (that is, the spatial multiplexing order) can be obtained as(3)Nmin=min⁡Nt,Nr.It means that there are Nmin independent wireless paths between the BS and the TU by beamforming.With multiple paths and the threshold scheme, the BS divides the secret data intoNmin parts using a polynomial of degree T-1 (T≤Nmin). Then, the ith part with a polynomial can be calculated as(4)Si=fi=a0+a1i+⋯+aT-1iT-1mod⁡p,where coefficient a0 is the original secret data while the other coefficients a1,a2,…,aT-1 are all randomly chosen at the BS. In addition, p, T, and mod denote a large prime number greater than any of the coefficients, a parameter controlling the balance between the symbol-error-rate (SER) and eavesdropping probability (EP), and the p-modulo operation, respectively. We assume that the value p and T are preshared with BS and TU.For example, let us examine how the proposed scheme works in MIMO wireless networks withNt=Nr=5. In the given antenna configuration, the value of Nmin is determined to be 5 by (3). Also, with preshared parameter (ex. p=17 and T=3), BS randomly chooses the remaining coefficients as a number smaller than p (ex. a1=4, a2=2). Then, Si for secret data can be obtained as(5)Si=fi=4︸secretdata+4i+2i2mod⁡17,and S1=10, S2=3, S3=0, S4=1, and S5=6.The divided parts in (4) are modulated (ex. QPSK, QAM) and then the modulated signals are transmitted to the target user through spatial dimensions with the transmit beamforming matrix V as(6)y=HVx+n,where x=[S1~,S2~,…,S~Nmin]T and Si~ denotes the i-th modulated signal. Also, the eavesdropping signal with transmit beamforming is rewritten as(7)z=GVx+w.


### 3.2. Spatial Dimension with Beamforming

For the design of an efficient transmit beamforming and receive combining efficiently, we assume that the perfect CSI ofH is available at the BS by channel reciprocity or feedback. If the BS also has the perfect CSI of the eavesdropping channel G, a secure signal transmission can be possible using transmit beamforming (e.g., Zero-Forcing Beamforming) to nullify the eavesdropping channel. However, due to the passive characteristic of the EA, it is hard to obtain the eavesdropping CSI at the BS. Therefore, without any information about the eavesdropping channel, transmit beamforming is generally designed for maximizing the spectral efficiency or improving the reliability of the target user.With the perfect CSI ofH, the spatial dimensions in the MIMO networks are obtained by singular value decomposition (SVD). The channel H can be decomposed to UDV∗ by using SVD, where U∈CNr×Nr, V∈CNt×Nt are unitary matrices, and D∈CNr×Nt is a diagonal matrix whose nonzero entries λi are the square roots of the eigenvalues of H∗H. With the transmit beamforming matrix V and the receive combining matrix U∗, the combined signal at the target user can be rewritten as(8)U∗·y=U∗·U︸IDV∗·V︸Ix+U∗·n,y~=Dx+n~.SinceU is a unitary matrix, the noise vectors n~ and n have the same distribution. Then, the i-th signal of y~ can be obtained as(9)yi~=λixi+n~i,i=1,…,Nmin.


### 3.3. Reconstruction of the Secret

With the receive combining matrixU∗ and the channel compensation of the received signals, the demodulated data [S1^,S2^,…,S^Nmin] can be obtained at the target user. Then, the target user with Nmin demodulated data randomly chooses a subset composed of T data (ex. [S1~,S2~,…,S~T]) and estimates the Lagrange interpolating polynomial as(10)f^i=∑j=1TSj^·∏k=1,k≠jTi-ikij-ikmod⁡p,where ij,ik means the index parameter of a subset. Then, the secret data can be obtained by f^(0)=K^. For example, with Nmin=5, T=3, p=17, and the demodulated data S^1=10, S^2=3, S^4=1, and the secret data and a polynomial of degree T-1 can be reconstructed by (10) as(11)f^i=∑j=13Sj^·∏k=1,k≠j3i-ikij-ikmod⁡17,=10·i-21-2·i-41-4+3·i-12-1·i-42-4+1·i-14-1·i-24-2mod⁡17,=21-13i+2i2mod⁡17=4︸secretdata+4i+2i2.However, since modulated signals are transmitted with fading channel and noise in MIMO wireless networks, the reconstruction of the secret must consider the demodulation error. In other words, the derived result in (10) can change depending on which subset a user chooses. Considering the proposed secret sharing scheme, the number of the subset (Ns) is(12)Ns=CNminT,where Cab denotes the number of possible combinations of b objects from a set of a objects. Then, a set (℧) of the estimated secret data K^ from (10) and (12) is obtained as(13)℧=K^1,K^2,…,K^Ns,and the secret data is finally determined by majority rule of ℧ to reduce the effect of the demodulation error. Figure 2 shows a simplified block diagram of the proposed threshold secret sharing system in MIMO wireless networks.Figure 2
A simplified block diagram of the proposed threshold secret sharing system in MIMO wireless networks.


### 3.4. Performance Analysis

In this subsection, we provide an analysis of SER to show how SER is affected byT, Nmin, and signal-to-noise ratio (SNR). In our MIMO networks, the i-th modulated signal Si~ is transmitted to the user through the i-th spatial dimension in (9) by beamforming technique in Section 3.2. To find the distribution of the i-th eigenvalue, the joint probability density function (EJPDF) of the Nmin-eigenvalues of Wishart matrices, λ1≥λ2≥⋯≥λNmin≥0, is defined [17] as(14)EJPDF=1Km,ne-1/2∑i=1Nminλi∏i=1Nminλi1/2Nmax-Nmin-1∏i<jλi-λjdλ1⋯dλNmin,where(15)Km,n=2NmaxπNmin/2∏i=1NminΓNmax-i+12ΓNmin-i+12,Nmax=max⁡Nt,Nr.Then, the density of thei-th eigenvalue can be obtained by integrating (14) as(16)EPDFi=∫R+Nmin-1EJPDFdΩ,where dΩ=dλ1⋯dλi+1dλi-1⋯dλNmin and the integration takes place over the positive orthant, R+Nmin-1={(λ1,…,λi+1,λi-1,…,λNmin):λ≥0}.Also, iflog2M is even integer, SER of M-QAM in AWGN channel can be defined [18] as PSER=1-(1-p)2 with(17)p=1-1Merfc32M-1EsNo,where(18)erfcx=2π∫x∞e-t2dt,and Es/No means the signal-to-noise ratio. Then, SER of M-QAM in fading channel can be obtained as(19)Ps=2Ep-Ep2.Also, withNmin spatial dimensions in our MIMO network, SER of the i-th dimension is rewritten as(20)Psi=2Eip-Eip2,where(21)Eip=∫0∞p¯·EPDFidλi,Eip2=∫0∞p¯2·EPDFidλi,p¯=1-1Merfc3λi2M-1EsNo.Since we cannot find any closed-form expression in our best knowledge,Psi needs to be calculated by numerically integrating (21). Then, based on the majority rule in Section 3.3, SER of the proposed threshold sharing scheme can be obtained by (10) and (20) as(22)Ps=1-∑sumk≤Nmin-T-1∏i=1NminPsiki·1-Psi1-ki,where(23)k=k1,…,ki,…,kNmin,sumk=∑i=1Nminki,ki∈0,1.


## 3.1. Threshold Secret Sharing Generator

Threshold secret sharing is a well-known scheme in cryptography introduced by Adi Sharmir [13]. In the scheme, a secret data is divided into N parts by a polynomial of degree T-1. To reconstruct the secret data, a minimum number of parts (T) is required for solving the polynomial problem correctly. In the threshold scheme, this number is less than the total number of parts (T≤N). Therefore, this is called the (T,N)-threshold secret sharing scheme [14–16].For spectral efficiency and throughput enhancement in MIMO networks, a BS transmits independent and separately encoded signals, so-called streams, from each of the multiple transmit antennas. Considering our system model, the number of streams (that is, the spatial multiplexing order) can be obtained as(3)Nmin=min⁡Nt,Nr.It means that there are Nmin independent wireless paths between the BS and the TU by beamforming.With multiple paths and the threshold scheme, the BS divides the secret data intoNmin parts using a polynomial of degree T-1 (T≤Nmin). Then, the ith part with a polynomial can be calculated as(4)Si=fi=a0+a1i+⋯+aT-1iT-1mod⁡p,where coefficient a0 is the original secret data while the other coefficients a1,a2,…,aT-1 are all randomly chosen at the BS. In addition, p, T, and mod denote a large prime number greater than any of the coefficients, a parameter controlling the balance between the symbol-error-rate (SER) and eavesdropping probability (EP), and the p-modulo operation, respectively. We assume that the value p and T are preshared with BS and TU.For example, let us examine how the proposed scheme works in MIMO wireless networks withNt=Nr=5. In the given antenna configuration, the value of Nmin is determined to be 5 by (3). Also, with preshared parameter (ex. p=17 and T=3), BS randomly chooses the remaining coefficients as a number smaller than p (ex. a1=4, a2=2). Then, Si for secret data can be obtained as(5)Si=fi=4︸secretdata+4i+2i2mod⁡17,and S1=10, S2=3, S3=0, S4=1, and S5=6.The divided parts in (4) are modulated (ex. QPSK, QAM) and then the modulated signals are transmitted to the target user through spatial dimensions with the transmit beamforming matrix V as(6)y=HVx+n,where x=[S1~,S2~,…,S~Nmin]T and Si~ denotes the i-th modulated signal. Also, the eavesdropping signal with transmit beamforming is rewritten as(7)z=GVx+w.


## 3.2. Spatial Dimension with Beamforming

For the design of an efficient transmit beamforming and receive combining efficiently, we assume that the perfect CSI ofH is available at the BS by channel reciprocity or feedback. If the BS also has the perfect CSI of the eavesdropping channel G, a secure signal transmission can be possible using transmit beamforming (e.g., Zero-Forcing Beamforming) to nullify the eavesdropping channel. However, due to the passive characteristic of the EA, it is hard to obtain the eavesdropping CSI at the BS. Therefore, without any information about the eavesdropping channel, transmit beamforming is generally designed for maximizing the spectral efficiency or improving the reliability of the target user.With the perfect CSI ofH, the spatial dimensions in the MIMO networks are obtained by singular value decomposition (SVD). The channel H can be decomposed to UDV∗ by using SVD, where U∈CNr×Nr, V∈CNt×Nt are unitary matrices, and D∈CNr×Nt is a diagonal matrix whose nonzero entries λi are the square roots of the eigenvalues of H∗H. With the transmit beamforming matrix V and the receive combining matrix U∗, the combined signal at the target user can be rewritten as(8)U∗·y=U∗·U︸IDV∗·V︸Ix+U∗·n,y~=Dx+n~.SinceU is a unitary matrix, the noise vectors n~ and n have the same distribution. Then, the i-th signal of y~ can be obtained as(9)yi~=λixi+n~i,i=1,…,Nmin.


## 3.3. Reconstruction of the Secret

With the receive combining matrixU∗ and the channel compensation of the received signals, the demodulated data [S1^,S2^,…,S^Nmin] can be obtained at the target user. Then, the target user with Nmin demodulated data randomly chooses a subset composed of T data (ex. [S1~,S2~,…,S~T]) and estimates the Lagrange interpolating polynomial as(10)f^i=∑j=1TSj^·∏k=1,k≠jTi-ikij-ikmod⁡p,where ij,ik means the index parameter of a subset. Then, the secret data can be obtained by f^(0)=K^. For example, with Nmin=5, T=3, p=17, and the demodulated data S^1=10, S^2=3, S^4=1, and the secret data and a polynomial of degree T-1 can be reconstructed by (10) as(11)f^i=∑j=13Sj^·∏k=1,k≠j3i-ikij-ikmod⁡17,=10·i-21-2·i-41-4+3·i-12-1·i-42-4+1·i-14-1·i-24-2mod⁡17,=21-13i+2i2mod⁡17=4︸secretdata+4i+2i2.However, since modulated signals are transmitted with fading channel and noise in MIMO wireless networks, the reconstruction of the secret must consider the demodulation error. In other words, the derived result in (10) can change depending on which subset a user chooses. Considering the proposed secret sharing scheme, the number of the subset (Ns) is(12)Ns=CNminT,where Cab denotes the number of possible combinations of b objects from a set of a objects. Then, a set (℧) of the estimated secret data K^ from (10) and (12) is obtained as(13)℧=K^1,K^2,…,K^Ns,and the secret data is finally determined by majority rule of ℧ to reduce the effect of the demodulation error. Figure 2 shows a simplified block diagram of the proposed threshold secret sharing system in MIMO wireless networks.Figure 2
A simplified block diagram of the proposed threshold secret sharing system in MIMO wireless networks.


## 3.4. Performance Analysis

In this subsection, we provide an analysis of SER to show how SER is affected byT, Nmin, and signal-to-noise ratio (SNR). In our MIMO networks, the i-th modulated signal Si~ is transmitted to the user through the i-th spatial dimension in (9) by beamforming technique in Section 3.2. To find the distribution of the i-th eigenvalue, the joint probability density function (EJPDF) of the Nmin-eigenvalues of Wishart matrices, λ1≥λ2≥⋯≥λNmin≥0, is defined [17] as(14)EJPDF=1Km,ne-1/2∑i=1Nminλi∏i=1Nminλi1/2Nmax-Nmin-1∏i<jλi-λjdλ1⋯dλNmin,where(15)Km,n=2NmaxπNmin/2∏i=1NminΓNmax-i+12ΓNmin-i+12,Nmax=max⁡Nt,Nr.Then, the density of thei-th eigenvalue can be obtained by integrating (14) as(16)EPDFi=∫R+Nmin-1EJPDFdΩ,where dΩ=dλ1⋯dλi+1dλi-1⋯dλNmin and the integration takes place over the positive orthant, R+Nmin-1={(λ1,…,λi+1,λi-1,…,λNmin):λ≥0}.Also, iflog2M is even integer, SER of M-QAM in AWGN channel can be defined [18] as PSER=1-(1-p)2 with(17)p=1-1Merfc32M-1EsNo,where(18)erfcx=2π∫x∞e-t2dt,and Es/No means the signal-to-noise ratio. Then, SER of M-QAM in fading channel can be obtained as(19)Ps=2Ep-Ep2.Also, withNmin spatial dimensions in our MIMO network, SER of the i-th dimension is rewritten as(20)Psi=2Eip-Eip2,where(21)Eip=∫0∞p¯·EPDFidλi,Eip2=∫0∞p¯2·EPDFidλi,p¯=1-1Merfc3λi2M-1EsNo.Since we cannot find any closed-form expression in our best knowledge,Psi needs to be calculated by numerically integrating (21). Then, based on the majority rule in Section 3.3, SER of the proposed threshold sharing scheme can be obtained by (10) and (20) as(22)Ps=1-∑sumk≤Nmin-T-1∏i=1NminPsiki·1-Psi1-ki,where(23)k=k1,…,ki,…,kNmin,sumk=∑i=1Nminki,ki∈0,1.


## 4. Eavesdropper Behavior

In this section, we explain the eavesdropper’s behavior against the threshold secret sharing system.


### 4.1. With onlyG

If EA is a legitimate user and an internal eavesdropper in wireless network, EA can easily obtain the channel state information ofG by channel estimation using pilot and preamble signals of BS [19]. With only information G, the eavesdropping signal in (7) is combined by the receive combining matrix UeG to eliminate interference caused by channel G as(24)UeG·z=UeG·GVx+UeG·w=V~x+w~G,where(25)UeG=G†G.In this case, since there is no information of the transmit beamforming matrixV and the eavesdropper fails to decode the received signal correctly, except for the transmit beamforming matrix V=I.


### 4.2. WithG and H

For efficient eavesdropping, channel information aboutH and G is needed at the EA. First, based on the channel estimation technique, EA knows G. We also assumed that smart EA can be obtained H and p by eavesdropping the TU’s channel feedback and the exchanged information for secure data transmission.Then, with the channel information ofH, EA predicts V for the target user by SVD of H. Therefore, for efficient eavesdropping, the eavesdropping signal can be combined by the receive matrix UeGH considering channel compensation as(26)UeGH·z=UeGH·GVx+UeGH·w=D~ex+w~,where(27)UeGH=G·V†G·V),and D~e denotes the diagonal matrix through the inverse matrix operation. Then, the transmitted data are estimated by the compensation of D~e. Finally, the secret data can be obtained through the same procedure as that of the secret reconstruction described in Section 3.3.


## 4.1. With onlyG

If EA is a legitimate user and an internal eavesdropper in wireless network, EA can easily obtain the channel state information ofG by channel estimation using pilot and preamble signals of BS [19]. With only information G, the eavesdropping signal in (7) is combined by the receive combining matrix UeG to eliminate interference caused by channel G as(24)UeG·z=UeG·GVx+UeG·w=V~x+w~G,where(25)UeG=G†G.In this case, since there is no information of the transmit beamforming matrixV and the eavesdropper fails to decode the received signal correctly, except for the transmit beamforming matrix V=I.


## 4.2. WithG and H

For efficient eavesdropping, channel information aboutH and G is needed at the EA. First, based on the channel estimation technique, EA knows G. We also assumed that smart EA can be obtained H and p by eavesdropping the TU’s channel feedback and the exchanged information for secure data transmission.Then, with the channel information ofH, EA predicts V for the target user by SVD of H. Therefore, for efficient eavesdropping, the eavesdropping signal can be combined by the receive matrix UeGH considering channel compensation as(26)UeGH·z=UeGH·GVx+UeGH·w=D~ex+w~,where(27)UeGH=G·V†G·V),and D~e denotes the diagonal matrix through the inverse matrix operation. Then, the transmitted data are estimated by the compensation of D~e. Finally, the secret data can be obtained through the same procedure as that of the secret reconstruction described in Section 3.3.


## 5. Numerical Results

In this section, we provide the simulation results of the symbol-error-rate (SER) and the eavesdropping probability of the proposed scheme. For simulation, MIMO wireless channels are considered as shown in Figure1, where the channel coefficients are assumed to be flat Rayleigh fading with mutually independent and additive white Gaussian noise terms having zero mean and equivalent variance CN(0,σ2). We compare the (T,Nmin)-threshold secret sharing scheme (TS-(T,Nmin)) with spatial multiplexing transmission (SM) and diversity transmission (Div). In SM, for full multiplexing gain, we assumed that the BS transmits Nmin independent data to the TU by beamforming. Therefore, an instantaneous EP at the EA is defined by Nsm over Nmin, where Nsm is the number of successfully decoded data. Also, since a single data transmission is assumed for full diversity gain in Div, an instantaneous EP is defined by eavesdropping success (1) or failure (0). The specific parameters are indicated at each figure.In Figure3, SER versus SNR at the TU is evaluated for different approaches. With full diversity gain, Div shows the best SER performance. On the other hand, SM shows the worst SER performance, because Nmin stream is simultaneously transmitted through spatial dimensions with an equal transmit power constraint. In general, we can see that TS achieves better performance than SM. In the high SNR region in particular, it achieves the diversity gain as Div because the TS can recover the secret data by majority rule, even though there are some miss-decoded parts due to the low eigenvalues in (9). On the contrary, in a low SNR region, due to the error propagation of majority rule, the TS shows the worst SER performance. In addition, through the slope of the graph, we can see that SER performance is determined by a gap between T and Nmin. With the fixed Nmin, the performance of the TS approaches that of the SM as T approaches to Nmin, because the secret data can be reconstructed only when all parts are successfully decoded to solve the polynomial. In addition, if T approaches 1, the performance of the TS approaches that of Div, because the secret data can be easily obtained by solving the polynomial with a small number of parts. It is noticeable that the theoretically derived result in Section 3.4 is well matched with simulation results.Figure 3
Symbol error rate vs. SNR (dB) at the target user, whereNt=Nr={8,5} and T={4,3,2}.In Figure4, we evaluate the performance of EP at the eavesdropper under different SNR. With full diversity gain, Div shows the most vulnerable performance to eavesdropping. On the other hand, the proposed TS shows good performance against eavesdropping compared to SM and Div, especially in the low SNR region. If only G is available, the eavesdropping fails due to the intersymbol interference caused by beamforming matrix V. Even though the smart EA has the perfect CSIs of both G and H, it is hard to estimate the sufficient number (≥T) of parts correctly, and it fails to reconstruct the secret data because the gain of the effective eavesdropping channel GV is degraded compared to the gain of the effective main channel HV in physical layer. We can also see that EP is determined by the threshold value. When the threshold value is set to be Nmin, the most secure communication against eavesdropping is possible. Therefore, an appropriate threshold value should be set according to the user’s purpose because there is a performance trade-off between SER and DP depending on the threshold value.Figure 4
Eavesdropping probability vs. SNR(dB) at the eavesdropper, whereNt=Nr=Ne={8,5} and T={4,3}.


## 6. Conclusion

In this paper, we considered a threshold secret sharing to enhance physical layer security against an eavesdropper. In the proposed scheme, withNmin spatial dimensions by beamforming, the secret data is divided into Nmin parts using a unique polynomial of degree T-1 and then transmitted to the user. Then, the user can reconstruct the secret data with a sufficient number (≥T) of parts by using the Lagrange interpolating polynomial. However, at the eavesdropper, the reconstruction of secret data fails due to the difference between the main channel and the eavesdropping channel in the physical layer. The simulation shows that the eavesdropping probability of the proposed scheme is better than those of the conventional approaches. Moreover, we found that the threshold value plays an important role in our scheme. Therefore, in the future, we would like to extend our current scheme with a fixed threshold value to a dynamic scheme where the base station can determine their own threshold value to simultaneously maximize the user’s symbol error rate and to minimize the eavesdropping probability with imperfect CSI.



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*Source: 4143061-2018-10-22.xml*