Secured Communication over Frequency-Selective Fading Channels: A Practical Vandermonde Precoding

  • Mari Kobayashi
  • Mérouane Debbah
  • Shlomo Shamai (Shitz)
Open Access
Research Article
Part of the following topical collections:
  1. Wireless Physical Layer Security

Abstract

We study the frequency-selective broadcast channel with confidential messages (BCC) where the transmitter sends a confidential message to receiver 1 and a common message to receivers 1 and 2. In the case of a block transmission of Open image in new window symbols followed by a guard interval of Open image in new window symbols, the frequency-selective channel can be modeled as a Open image in new window Toeplitz matrix. For this special type of multiple-input multiple-output channels, we propose a practical Vandermonde precoding that projects the confidential messages in the null space of the channel seen by receiver 2 while superposing the common message. For this scheme, we provide the achievable rate region and characterize the optimal covariance for some special cases of interest. Interestingly, the proposed scheme can be applied to other multiuser scenarios such as the Open image in new window -user frequency-selective BCC with Open image in new window confidential messages and the two-user frequency-selective BCC with two confidential messages. For each scenario, we provide the secrecy degree of freedom (s.d.o.f.) region of the corresponding channel and prove the optimality of the Vandermonde precoding. One of the appealing features of the proposed scheme is that it does not require any specific secrecy encoding technique but can be applied on top of any existing powerful encoding schemes.

Keywords

Toeplitz Matrix Broadcast Channel Secrecy Rate Secrecy Capacity Equal Power Allocation 

1. Introduction

We consider a secured medium such that the transmitter wishes to send a confidential message to its receiver while keeping the eavesdropper, tapping the channel, ignorant of the message. Wyner [1] introduced this model named the wiretap channel to model the degraded broadcast channel where the eavesdropper observes a degraded version of the receiver's signal. In this model, the confidentiality is measured by the equivocation rate, that is, the mutual information between the confidential message and the eavesdropper's observation. For the discrete memoryless degraded wiretap channel, Wyner characterized the capacity-equivocation region and showed that a nonzero secrecy rate can be achieved [1]. The most important operating point on the capacity-equivocation region is the secrecy capacity, that is, the largest reliable communication rate such that the eavesdropper obtains no information about the confidential message (the equivocation rate is as large as the message rate). The secrecy capacity of the Gaussian wiretap channel was given in [2]. Csiszár and Körner considered a more general wiretap channel in which a common message for both receivers is sent in addition to the confidential message [3]. For this model known as the broadcast channel with confidential (BCC) messages, the rate-tuple of the common and confidential messages was characterized.

Recently, a significant effort has been made to opportunistically exploit the space/time/user dimensions for secrecy communications (see, e.g., [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] and references therein). In [4], the secrecy capacity of the ergodic slow fading channels was characterized and the optimal power/rate allocation was derived. The secrecy capacity of the parallel fading channels was given [6, 7] where [7] considered the BCC with a common message. Moreover, the secrecy capacity of the wiretap channel with multiple antennas has been studied in [8, 9, 10, 11, 12, 13, 15] and references therein. In particular, the secrecy capacity of the multiple-input multiple-output (MIMO) wiretap channel has been fully characterized in [5, 11, 12, 14] and more recently its closed-form expressions under a matrix covariance constraint have been derived in [15]. Furthermore, a large number of recent works have considered the secrecy capacity region for more general broadcast channels. In [16], the authors studied the two-user MIMO Gaussian BCC where the capacity region for the case of one common and one confidential message was characterized. The two-user BCC with two confidential messages, each of which must be kept secret to the unintended receiver, has been studied in [17, 18, 19, 20]. In [18], Liu and Poor characterized the secrecy capacity region for the multiple-input single-output (MISO) Gaussian BCC where the optimality of the secret dirty paper coding (S-DPC) scheme was proved. A recent contribution [19] extended the result to the MIMO Gaussian BCC. The multireceiver wiretap channels have been also studied in [21, 22, 23, 24, 25, 26] (and reference therein) where the confidential messages to each receiver must be kept secret to an external eavesdropper. It has been proved that the secrecy capacity region of the MIMO Gaussian multireceiver wiretap channels is achieved by S-DPC [24, 26].

However, very few work have exploited the frequency selectivity nature of the channel for secrecy purposes [27] where the zeros of the channel provide an opportunity to "hide" information. This paper shows the opportunities provided by the broad-band channel and studies the frequency-selective BCC where the transmitter sends one confidential message to receiver 1 and one common message to both receivers 1 and 2. The channel state information (CSI) is assumed to be known to both the transmitter and the receivers. We consider the quasistatic frequency-selective fading channel with Open image in new window paths such that the channel remains fixed during an entire transmission of Open image in new window blocks for an arbitrary large Open image in new window . It should be remarked that in general the secrecy rate cannot scale with signal-to-noise ratio (SNR) over the channel at hand, unless the channel of receiver 2 has a null frequency band of positive Lebesgue measure (on which the transmitter can "hide'' the confidential message). In this contribution, we focus on the realistic case where receiver 2 has a full frequency band (without null subbands) but operates in a reduced dimension due to practical complexity issues. This is typical of current orthogonal frequency division multiplexing (OFDM) standards (such as IEEE802.11a/WiMax or LTE [28, 29, 30]) where a guard interval of Open image in new window symbols is inserted at the beginning of each block to avoid the interblock interference and both receivers discard these Open image in new window symbols. We assume that both users have the same standard receiver, in particular receiver 2 cannot change its hardware structure. Studying secure communications under this assumption is of interest in general and can be justified since receiver 2 is actually a legitimate receiver which can receive a confidential message in other communication periods. Of course, if receiver 2 is able to access the guard interval symbols, it can extract the confidential message and the secrecy rate falls down to zero. Although we restrict ourselves to the reduced dimension constraint in this paper, other constraints on the limited capability at the unintended receiver such as energy consumption or hardware complexity might provide a new paradigm to design physical layer secrecy systems.

In the case of a block transmission of Open image in new window symbols followed by a guard interval of Open image in new window symbols discarded at both receivers, the frequency-selective channel can be modeled as an Open image in new window MIMO Toeplitz matrix. In this contribution, we aim at designing a practical linear precoding scheme that fully exploits the degrees of freedom (d.o.f.) offered by this special type of MIMO channels to transmit both the common message and the confidential message. To this end, let us start with the following remarks. On one hand, the idea of using OFDM modulation to convert the frequency-selective channel represented by the Toeplitz matrix into a set of parallel fading channel turns out to be useless from a secrecy perspective. Indeed, it is known that the secrecy capacity of the parallel wiretap fading channels does not scale with SNR [7]. On the other hand, recent contributions [5, 11, 12, 14, 15] showed that the secrecy capacity of the MIMO wiretap channel grows linearly with SNR, that is, Open image in new window where Open image in new window denotes the secrecy degree of freedom (s.d.o.f.) (to be specified). In the high SNR regime, the secrecy capacity of the MISO/MIMO wiretap channel is achieved by sending the confidential message in the null space of the eavesdropper's channel [10, 11, 14, 15, 18, 19]. Therefore, OFDM modulation is highly suboptimal in terms of the s.d.o.f.

Inspired by these remarks, we propose a linear Vandermonde precoder that projects the confidential message in the null space of the channel seen by receiver 2 while superposing the common message. Thanks to the orthogonality between the precoder of the confidential message and the channel of receiver 2; receiver 2 obtains no information on the confidential message. This precoder is regarded as a single-antenna frequency beamformer that nulls the signal in certain directions seen by receiver 2. The Vandermonde structure comes from the fact that the frequency beamformer is of the type Open image in new window where Open image in new window is one of the roots of the channel seen by receiver 2. Note that Vandermonde matrices [31] have already been considered for cognitive radios [32] and CDMA systems [33] to reduce/null interference but not for secrecy applications. One of the appealing aspects of Vandermonde precoding is that it does not require a specific secrecy encoding technique but can be applied on top of any classical capacity achieving encoding scheme.

For the proposed scheme, we characterize its achievable rate region, the rate-tuple of the common message, the confidential message, respectively. Unfortunately, the optimal input covariances achieving their boundary are generally difficult to compute due to the nonconvexity of the weighted sum rate maximization problem. Nevertheless, we show that there are some special cases of interest such as the secrecy rate and the maximum sum rate point which enable an explicit characterization of the optimal input covariances. In addition, we provide the achievable d.o.f. region of the frequency-selective BCC, reflecting the behavior of the achievable rate region in the high SNR regime, and prove that the Vandermonde precoding achieves this region. More specifically, it enables to simultaneously transmit Open image in new window streams of the confidential message and Open image in new window streams of the common message for Open image in new window simultaneously over a block of Open image in new window dimensions. Interestingly, the proposed Vandermonde precoding can be applied to multiuser secure communication scenarios: (a) a Open image in new window -user frequency-selective BCC with Open image in new window confidential messages and one common message, (b) a two-user frequency-selective BCC with two confidential messages and one common message. For each scenario, we characterize the achievable s.d.o.f. region of the corresponding frequency-selective BCC and show the optimality of the Vandermonde precoding.

The paper is organized as follows. Section 2 presents the frequency-selective fading BCC. Section 3 introduces the Vandermonde precoding and characterizes its achievable rate region as well as the optimal input covariances for some special cases. Section 4 provides the application of the Vandermonde precoding to the multiuser secure communications scenarios. Section 5 shows some numerical examples of the proposed scheme in the various settings, and finally Section 6 concludes the paper.

Notation. In the following, upper (lower boldface) symbols will be used for matrices (column vectors) whereas lower symbols will represent scalar values, Open image in new window will denote transpose operator, Open image in new window conjugation, and Open image in new window hermitian transpose. Open image in new window , Open image in new window represent the Open image in new window identity matrix, Open image in new window zero matrix. Open image in new window denote a determinant, rank, trace of a matrix Open image in new window , respectively. Open image in new window denotes the sequence Open image in new window . Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window denote the realization of the random variables Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window . Finally, " Open image in new window '' denotes less or equal to in the positive semidefinite ordering between positive semidefinite matrices, that is, we have Open image in new window if Open image in new window is positive semidefinite.

2. System Model

We consider the quasistatic frequency-selective fading BCC illustrated in Figure 1. The received signal Open image in new window of receivers 1, 2 at block Open image in new window is given by
where Open image in new window denote an Open image in new window Toeplitz matrix with the Open image in new window -path channel vector Open image in new window of user 1, Open image in new window of user 2, respectively, Open image in new window denotes the transmit vector, and finally Open image in new window are mutually independent additive white Gaussian noise (AWGN). The input vector is subject to the power constraint given by
where we let Open image in new window . The structure of Open image in new window is given by
Figure 1

Frequency-selective broadcast channels with confidential messages.

We assume that the channel matrices Open image in new window , Open image in new window remain constant for the whole duration of the transmission of Open image in new window blocks and are known to all terminals. At each block Open image in new window , we transmit Open image in new window symbols by appending a guard interval of size Open image in new window larger than the delay spread, which enables to avoid the interference between neighbor blocks.

The transmitter wishes to send a common message Open image in new window to two receivers and a confidential message Open image in new window to receiver 1. A Open image in new window code consists of the following: (1) two message sets Open image in new window and Open image in new window with the messages Open image in new window uniformly distributed over the sets Open image in new window , Open image in new window , respectively; (2) a stochastic encoder that maps each message pair Open image in new window to a codeword Open image in new window ; (3) one decoder at receiver 1 that maps a received sequence Open image in new window to a message pair Open image in new window and another at receiver 2 that maps a received sequence Open image in new window to a message Open image in new window . The average error probability of a Open image in new window code is defined as
where Open image in new window denotes the error probability when the message pair Open image in new window is sent defined by
The secrecy level of the confidential message Open image in new window at receiver 2 is measured by the equivocation rate Open image in new window defined as

which is the normalized entropy of the confidential message conditioned on the received signal at receiver 2 and available CSI.

A rate-equivocation tuple Open image in new window is said to be achievable if for any Open image in new window there exists a sequence of codes Open image in new window such that we have

In this paper, we focus on the perfect secrecy case where receiver 2 obtains no information about the confidential message Open image in new window , which is equivalent to Open image in new window . In this setting, an achievable rate region Open image in new window of the general BCC (expressed in bit per channel use per dimension) is given by [3]

where the union is over all possible distribution Open image in new window , Open image in new window , Open image in new window satisfying [20, Lemma  1]
where Open image in new window might be a deterministic function of Open image in new window . Recently, the secrecy capacity region Open image in new window of the two-user MIMO-BCC (1) was characterized in [16] and is given by all possible rate tuples Open image in new window satisfying
for some Open image in new window with Open image in new window denotes the input covariance satisfying Open image in new window and Open image in new window , Open image in new window denotes the channel matrix of receiver 1, 2, respectively. Obviously, when only the confidential message is transmitted to receiver 1, the frequency-selective BCC (1) reduces to the MIMO flat-fading wiretap channel whose secrecy capacity has been characterized in [10, 11, 12, 14, 15]. In particular, Bustin et al. derived its closed-form expression under a power-covariance constraint [15]. Under a total power (trace) constraint, the secrecy capacity of the MIMO Gaussian wiretap channel is expressed as [19, Theorem  3]
where Open image in new window are the generalized eigen-values greater than one of the following pencil:
(In [15, 19] the authors consider the real matrices Open image in new window , Open image in new window . Nevertheless, it is conjectured that for complex matrices the following expression without Open image in new window in the prelog holds.) As explicitly characterized in [15, Theorem  2], the optimal input covariance achieving the above region is chosen such that the confidential message is sent over Open image in new window subchannels where receiver 1 observes stronger signals than receiver 2. Moreover, in the high SNR regime the optimal strategy converges to beamforming into the null subspace of Open image in new window [5, 11, 12, 14] as for the MISO case [14, 18]. In order to characterize the behavior of the secrecy capacity region in the high SNR regime, we define the d.o.f. region as

where Open image in new window denotes s.d.o.f. which corresponds precisely to the number Open image in new window of the generalized eigenvalues greater than one in the high SNR.

3. Vandermonde Precoding

For the frequency-selective BCC specified in Section 2, we wish to design a practical linear precoding scheme which fully exploits the d.o.f. offered by the frequency-selective channel. We remarked previously that for a special case when only the confidential message is sent to receiver 1 (without a common message), the optimal strategy consists of beamforming the confidential signal into the null subspace of receiver 2. By applying this intuitive result to the special Toeplitz MIMO channels Open image in new window , Open image in new window while including a common message, we propose a linear precoding strategy named Vandermonde precoding. Prior to the definition of the Vandermonde precoding, we provide some properties of a Vandermonde matrix [31].

Property 1.

Given a full-rank Toeplitz matrix Open image in new window , there exists a Vandermonde matrix Open image in new window for Open image in new window whose structure is given by
where Open image in new window are the Open image in new window roots of the polynomial Open image in new window with Open image in new window coefficients of the channel Open image in new window . Clearly Open image in new window satisfies the following orthogonal condition:

and Open image in new window if Open image in new window are all different.

It is well known that as the dimension of Open image in new window and Open image in new window increases, the Vandermonde matrix Open image in new window becomes ill-conditioned unless the roots are on the unit circle. In other words, the elements of each column either grow in energy or tend to zero [31]. Hence, instead of the brut Vandermonde matrix (14), we consider a unitary Vandermonde matrix obtained either by applying the Gram-Schmidt orthogonalization or singular value decomposition (SVD) on Open image in new window .

Definition 1.

We let Open image in new window be a unitary Vandermonde matrix obtained by orthogonalizing the columns of Open image in new window . We let Open image in new window be a unitary matrix in the null space of Open image in new window such that Open image in new window . The common message Open image in new window , the confidential message Open image in new window , is sent along Open image in new window , Open image in new window , respectively. We call Open image in new window Vandermonde precoder.

Further, the precoding matrix Open image in new window for the confidential message satisfies the following property.

Lemma 2.

Given two Toeplitz matrices Open image in new window , Open image in new window where Open image in new window , Open image in new window are linearly independent, there exists a unitary Vandermonde matrix Open image in new window for Open image in new window satisfying

Proof.

Appendix A.

In order to send the confidential message intended to receiver 1 as well as the common message to both receivers over the frequency-selective channel (1), we consider the Gaussian superposition coding based on the Vandermonde precoder of Definition 1. Namely, at block Open image in new window , we form the transmit vector as
where the common message vector Open image in new window and the confidential message vector Open image in new window are mutually independent Gaussian vectors with zero mean and covariance Open image in new window , Open image in new window , respectively. Under this condition, the input covariances subject to

satisfy the power constraint (2). We let Open image in new window denote the feasible set Open image in new window satisfying (18).

Theorem 3.

The Vandermonde precoding achieves the following secrecy rate region:

where Open image in new window denotes the convex hull and we let Open image in new window , Open image in new window , Open image in new window .

Proof.

Due to the orthogonal property (16) of the unitary Vandermonde matrix, receiver 2 only observes the common message, which yields the received signals given by
where we drop the block index. We examine the achievable rate region Open image in new window of the Vandermonde precoding. By letting the auxiliary variables Open image in new window and Open image in new window , we have

Plugging these expressions to (8), we obtain (19).

The boundary of the achievable rate region of the Vandermonde precoding can be characterized by solving the weighted sum rate maximization. Any point Open image in new window on the boundary of the convex region Open image in new window is obtained by solving
for nonnegative weights Open image in new window satisfying Open image in new window . When the region Open image in new window , obtained without convex hull, is nonconvex, the set of the optimal covariances Open image in new window achieving the boundary point might not be unique. Figure 2 depicts an example in which the achievable rate region Open image in new window is obtained by the convex hull operation on the region Open image in new window , that is, replacing the non-convex subregion by the line segment Open image in new window , Open image in new window . For the weight ratio Open image in new window corresponding to the slope of the line segment Open image in new window , Open image in new window , there exist two optimal sets of the covariances yielding the points Open image in new window and Open image in new window (which clearly dominate the point Open image in new window ). These points are the solution to the weighted sum rate maximization (22). In summary, an optimal covariance set achieving (22) (might not be unique) is the solution of
where we let
Following [34, Section  II-C] (and also [7, Lemma  2]), we remark that the solution to the max-min problem (23) can be found by hypothesis testing of three cases, Open image in new window , Open image in new window , and Open image in new window . Formally, we have the following lemma.
Figure 2

Achievable rate region Open image in new window obtained by the convex hull on Open image in new window .

Lemma 4.

The optimal Open image in new window , solution of (23), is given by one of the three solutions.

Case 1.

and satisfies Open image in new window .

Case 2.

and satisfies Open image in new window .

Case 3.

and satisfies Open image in new window for some Open image in new window .

Before considering the weighted sum rate maximization (23), one applies SVD to Open image in new window , Open image in new window

where Open image in new window , Open image in new window , and Open image in new window are unitary, Open image in new window contain positive singular values Open image in new window , Open image in new window , respectively. Following [7, Theorem  3], one applies Lemma 4 to solve the weighted sum rate maximization.

Theorem 5.

The set of the optimal covariances Open image in new window , achieving the boundary of the achievable rate region Open image in new window of the Vandermonde precoding, corresponds to one of the following three solutions.

Case 1.

Open image in new window , if Open image in new window , solution of the following KKT conditions, satisfies Open image in new window

where Open image in new window with a positive semidefinite Open image in new window for Open image in new window , Open image in new window is determined such that Open image in new window , and we let Open image in new window .

Case 2.

Open image in new window if the following Open image in new window fulfills Open image in new window .

where Open image in new window is determined such that Open image in new window .

Case 3.

where Open image in new window with a positive semidefinite Open image in new window for Open image in new window ,  Open image in new window is determined such that Open image in new window .

Proof.

Appendix B.

Remark 6.

Due to the non-concavity of the underlying weighted sum rate functions, it is generally difficult to characterize the boundary of the achievable rate region Open image in new window except for some special cases. The special cases include the corner points, in particular, the secrecy rate for the case of sending only the confidential message ( Open image in new window ), as well as the maximum sum rate point for the equal weight case ( Open image in new window ). It is worth noticing that under equal weight the objective functions in three cases are all concave in Open image in new window , Open image in new window since Open image in new window is concave if Open image in new window and Open image in new window is concave if Open image in new window and Open image in new window .

The maximum sum rate point Open image in new window can be found by applying the following greedy search [7].

Greedy Search to Find the Maximum Sum Rate Point

( Open image in new window ) Find Open image in new window , Open image in new window maximizing Open image in new window and check Open image in new window . If yes stop. Otherwise go to (2).

( Open image in new window ) Find Open image in new window , Open image in new window maximizing Open image in new window and check Open image in new window . If yes stop. Otherwise go to (3).

( Open image in new window ) Find Open image in new window , Open image in new window maximizing Open image in new window and check Open image in new window for some Open image in new window .

For the special case of Open image in new window , Theorem 5 yields the achievable secrecy rate with the Vandermonde precoding.

Corollary 7.

The Vandermonde precoding achieves the secrecy rate

where the last equality is obtained by applying SVD to Open image in new window and plugging the power allocation of (30) with Open image in new window , Open image in new window , Open image in new window is determined such that Open image in new window .

Finally, by focusing the behavior of the achievable rate region in the high SNR regime, we characterize the achievable d.o.f. region of the frequency-selective BCC (1).

Theorem 8.

The d.o.f. region of the frequency-selective BCC (1) with Open image in new window Toeplitz matrices Open image in new window is given as a union of Open image in new window satisfying

where Open image in new window , Open image in new window denote non-negative integers. The Vandermonde precoding achieves the above d.o.f. region.

Proof.

The achievability follows rather trivially by applying Theorem 3. By considering equal power allocation over all Open image in new window streams such that Open image in new window , Open image in new window , we obtain the rate tuple Open image in new window where Open image in new window
We first notice that the prelog factor of Open image in new window as Open image in new window depends only on the rank of Open image in new window . From Lemma 2, we obtain
where (a) follows from orthogonality between Open image in new window and Open image in new window , (b) follows from the fact that Open image in new window is unitary satisfying Open image in new window . Notice that (36) yields Open image in new window . For the d.o.f. Open image in new window of the common message, (36) and (38) yield

which is dominated by the pre-log of Open image in new window in (37). This establishes the achievability.

The converse follows by noticing that the inequalities (33) and (34) correspond to trivial upper bounds. The first inequality (33) corresponds to the s.d.o.f. of the MIMO wiretap channel with the legitimate channel Open image in new window and the eavesdropper channel Open image in new window , which is bounded by Open image in new window . The second inequality (34) follows because the total number of streams for receiver 1 cannot be larger than the d.o.f. of Open image in new window , that is, Open image in new window .

Figure 3 illustrates the region Open image in new window of the frequency-selective BCC over Open image in new window dimensions. We notice that the s.d.o.f. constraint (33) yields the line segment Open image in new window , Open image in new window while the constraint (34) in terms of the total number of streams for receiver 1 yields the line segment Open image in new window , Open image in new window .
Figure 3

d. o.f. region Open image in new window of frequency-selective BCC.

4. Multiuser Secure Communications

In this section, we provide some applications of the Vandermonde precoding in the multi-user secure communication scenarios where the transmitter wishes to send confidential messages to more than one intended receivers. The scenarios that we address are: (a) a Open image in new window -user frequency-selective BCC with Open image in new window confidential messages and one common message, (b) a two-user frequency-selective BCC with two confidential messages and one common message. For each scenario, by focusing on the behavior in the high SNR regime, we characterize the achievable s.d.o.f. region and show the optimality of the Vandermonde precoding.

4.1. K + 1-User BCC with K Confidential Messages

As an extension of Section 3, we consider the Open image in new window -user frequency-selective BCC where the transmitter sends Open image in new window confidential messages Open image in new window to the first Open image in new window receivers as well as one common message Open image in new window to all receivers. Each of the confidential messages must be kept secret to receiver Open image in new window . Notice that this model, called multireceiver wiretap channel, has been studied in the literature ([20, 22, 23, 24, 25, 26] and reference therein). In particular, the secrecy capacity region of the Gaussian MIMO multireceiver wiretap channel has been characterized in [24, 26] for Open image in new window , an arbitrary Open image in new window , respectively, where the optimality of the S-DPC is proved.

The received signal Open image in new window of receiver Open image in new window and the received signal Open image in new window of receiver Open image in new window at any block are given by

where Open image in new window is the transmit vector satisfying the total power constraint and Open image in new window , Open image in new window are mutually independent AWGN with covariance Open image in new window . We assume that the Open image in new window vectors Open image in new window , Open image in new window of length Open image in new window are linearly independent and perfectly known to all the terminals. As an extension of the frequency-selective BCC in Section 2, we say that the rate tuple Open image in new window is achievable if for any Open image in new window there exists a sequence of codes Open image in new window such that
where we denote Open image in new window and define

An achievable secrecy rate region Open image in new window for the case of Open image in new window , when the transmitter sends two confidential messages in the presence of an external eavesdropper, is provided in [25, Theorem 1]. This theorem can be extended to an arbitrary Open image in new window while including the common message. Formally we state the following lemma.

Lemma 9.

An achievable rate region of the Open image in new window +1-user BCC, where the transmitter sends Open image in new window confidential messages intended to the first Open image in new window receivers as well as a common message to all users, is given as a union of all non-negative rate-tuple satisfying

Proof.

Appendix C.

Notice that the second term of the last equation in (44) can be also expressed by

It can be easily seen that without the secrecy constraint the above region reduces to the Marton's achievable region for the general Open image in new window -user broadcast channel [35].

In order to focus on the behavior of the region in the high SNR regime, we define the s.d.o.f. region as

where Open image in new window denotes the d.o.f. of the common message and Open image in new window denotes the s.d.o.f. of confidential message Open image in new window . As an extension of Theorem 8, we have the following s.d.o.f. region result.

Theorem 10.

The s.d.o.f. region of the Open image in new window -user frequency-selective BCC (40) is a union of Open image in new window satisfying

where Open image in new window are non-negative integers. The Vandermonde precoding achieves this region.

Proof.

Appendix D.

Figure 4 illustrates the region Open image in new window for the case of Open image in new window confidential messages. It can be easily seen that the constraint (49) in terms of the total number of streams for the virtual receiver yields the subspace Open image in new window , Open image in new window , Open image in new window while the s.d.o.f. constraint (48) for the virtual receiver yields the subspace Open image in new window , Open image in new window , Open image in new window , Open image in new window . We remark that for the special case of one confidential message and one common message ( Open image in new window ), the region reduces to Figure 3.
Figure 4

s. d.o.f. region Open image in new window over Open image in new window dimensions of three-user frequency-selective BCC.

Remark 11.

When only the Open image in new window confidential messages are transmitted to the Open image in new window intended receivers in the presence of the eavesdropper, the s.d.o.f. region has the equivalent MIMO interpretation [36]. More specifically, the frequency-selective BCC (40) is equivalent to the MIMO-BCC where the transmitter with Open image in new window dimensions (antennas) sends messages to Open image in new window receivers with Open image in new window antennas each in the presence of the eavesdropper with Open image in new window antennas. The secrecy constraint (orthogonal constraint) consumes Open image in new window dimensions of the channel seen by the virtual receiver and lets the number of effective transmit antennas be Open image in new window . The resulting channel is the MIMO-BC without secrecy constraint with Open image in new window transmit antennas and Open image in new window receivers with Open image in new window antennas each, whose multiplexing gain is Open image in new window (we assume Open image in new window ). Figure 5 illustrates the example with Open image in new window , Open image in new window , Open image in new window .
Figure 5

Equivalent MIMO interpretation for three-user frequency-selective BCC with two confidential messages.

4.2. Two-User BCC with Two Confidential Messages

We consider the two-user BCC where the transmitter sends two confidential messages Open image in new window , Open image in new window as well as one common message Open image in new window . Each of the confidential messages must be kept secret to the unintended receiver. This model has been studied in [17, 18, 19] for the case of two confidential messages and in [20] for the case of two confidential messages and a common message. In [19], the secrecy capacity region of the MIMO Gaussian BCC was characterized. The received signal at receivers 1, 2 at any block is given, respectively, by

where Open image in new window is the input vector satisfying the total power constraint and Open image in new window , Open image in new window are mutually independent AWGN with covariance Open image in new window . We assume the channel vectors Open image in new window , Open image in new window are linearly independent.

We say that the rate tuple Open image in new window is achievable if for any Open image in new window there exists a sequence of codes Open image in new window such that
where we define the average error probability as
where Open image in new window is the output of decoders 1, 2, respectively. A secrecy achievable rate region of the two-user BCC with two confidential messages and a common message is given by [20, Theorem  1]
where the random variables satisfy the Markov chain

We extend Theorem 8 to the two-user frequency-selective BCC (50) and obtain the following s.d.o.f. result.

Theorem 12.

The s.d.o.f. region of the two-user frequency-selective BCC (50) is a union of Open image in new window satisfying

where Open image in new window are non-negative integers. The Vandermonde precoding achieves the region.

Proof.

Appendix F.

Figure 6 represents the s.d.o.f. region Open image in new window over Open image in new window dimensions of the two-user frequency-selective BCC. The per-receiver s.d.o.f. constraints (55) yield the subspace Open image in new window , Open image in new window , Open image in new window , Open image in new window for user 1 and the subspace Open image in new window , Open image in new window , Open image in new window , Open image in new window for user 2. The constraints (56) in terms of the total number of streams per receiver yield the subregion Open image in new window , Open image in new window , Open image in new window for user 1 and the subregion Open image in new window , Open image in new window , Open image in new window for user 2. For the special case of one confidential message and one common message, the region reduces to Figure 3.
Figure 6

s. d.o.f. region Open image in new window over Open image in new window dimensions of Open image in new window -user frequency-selective BCC.

Remark 13.

Comparing Theorems 10, 12 as well as Figures 4, 6 for Open image in new window , it clearly appears that the s.d.o.f. of Open image in new window -user BCC with Open image in new window confidential messages is dominated by the s.d.o.f. of Open image in new window -user BCC with Open image in new window confidential messages. In other words, the s.d.o.f. region critically depends on the assumption on the eavesdropper(s) to whom each confidential message must be kept secret.

Remark 14.

When only two confidential messages are transmitted in the two-user frequency-selective BCC, the set of the s.d.o.f. has the equivalent MIMO interpretation [36]. More specifically, the frequency-selective BCC (40) is equivalent to the MIMO-BCC where the transmitter with Open image in new window dimensions (antennas) sends two confidential messages to two receivers with Open image in new window antennas. The secrecy constraint consumes Open image in new window dimensions for each MIMO link and lets the number of effective transmit antennas be Open image in new window for each user. The resulting channel is a two parallel Open image in new window point-to-point MIMO channel without eavesdropper. Notice that the same parallel MIMO links can be obtained by applying the block diagonalization on the MIMO-BC without secrecy constraint [36]. In other words, the secrecy constraint in the BCC with inner eavesdroppers is equivalent to the orthogonal constraint in the classical MIMO-BC. Figure 7 shows the example with Open image in new window , Open image in new window and Open image in new window confidential messages.
Figure 7

Equivalent MIMO interpretation for the two-user frequency-selective BCC with two confidential messages.

5. Numerical Examples

In order to examine the performance of the proposed Vandermonde precoding, this section provides some numerical results in different settings.

5.1. Secrecy Rate versus SNR

We evaluate the achievable secrecy rate Open image in new window in (32) when the transmitter sends only a confidential message to receiver 1 (without a common message) in the presence of receiver 2 (eavesdropper) over the frequency-selective BCC studied in Section 3.

5.1.1. MISO Wiretap Channel

For the sake of comparison (albeit unrealistic), we consider the special case of the frequency-selective wiretap channel when receiver 1 has a scalar observation and the eavesdropper has Open image in new window observations. This is equivalent to the MISO wiretap channel with the receiver 1 channel Open image in new window and the eavesdropper channel Open image in new window . Without loss of generality, we assume that the observation at receiver 1 is the first row of Open image in new window . We consider that all entries of Open image in new window , Open image in new window are i.i.d. Open image in new window and average the secrecy rate over a large number of randomly generated channels with Open image in new window , Open image in new window . In Figure 8, we compare the optimal beamforming strategy [10, 13, 14] and the Vandermonde precoding as a function of SNR Open image in new window . Since only one stream is sent to receiver 1, the s.d.o.f. is Open image in new window . In fact, the MISO secrecy capacity in the high SNR regime is given by
where Open image in new window is the beamforming vector. The Vandermonde precoding achieves
where Open image in new window denotes the Open image in new window th column of Open image in new window orthogonal to Open image in new window . Clearly, there exists a constant gap between (57) and (58) due to the suboptimal choice of the beamforming vector.
Figure 8

Achievable secrecy rate with one observation at receiver 1 and Open image in new window , Open image in new window (MISO wiretap channel).

5.1.2. MIMO Wiretap Channel

We consider the frequency-selective wiretap channel with Open image in new window , Open image in new window . Although there exists a closed-form expression under a power-covariance constraint [15], the secrecy capacity under a total power constraint in (11) is still difficult to compute (especially for a large dimension of Open image in new window and Open image in new window ) because it requires a search over all possible power covariances constraints. Therefore, in Figure 9, we compare the averaged secrecy rate achieved by the generalized SVD scheme [5] and the Vandermonde precoding. We assume that all entries of Open image in new window are i.i.d. Open image in new window . For the Vandermonde precoding, we show the achievable rate with waterfilling power allocation (32) and equal power allocation (36) by allocating Open image in new window to Open image in new window streams. As observed, these two suboptimal schemes achieve the same s.d.o.f. of Open image in new window although the generalized SVD incurs a substantial power loss. The result agrees well with Theorem 8. We remark also that the optimal waterfilling power allocation yields a negligible gain.
Figure 9

Achievable secrecy rate with Open image in new window , Open image in new window (MIMO wiretap channel).

5.2. The Maximum Sum Rate Point (R0, R1) versus SNR

We consider the frequency-selective BCC with one confidential message to receiver 1 and one common message to two receivers. In particular, we characterize the maximum sum rate-tuple corresponding to Open image in new window on the boundary of the achievable rate region Open image in new window . Figure 10 shows the averaged maximum sum rate-tuple Open image in new window of the Vandermonde precoding both with optimal input covariance computed by the greedy algorithm and with equal power allocation. We remark that there is essentially no loss with the equal power allocation.
Figure 10

Achievable secrecy/common rates Open image in new window in the frequency-selective BCC.

5.3. Two-User Secrecy Rate Region in the Frequency-Selective BCC

We consider the two-user frequency-selective BCC where the transmitter sends two confidential messages (no common message) of Section 4.2. For the sake of comparison (albeit unrealistic), we consider the special case of one observation Open image in new window at each receiver. Notice that the two-user frequency-selective BCC is equivalent to the two-user MISO BCC with Open image in new window whose secrecy capacity region is achieved by the S-DPC scheme [18]. The proposed Vandermonde precoding achieves the secrecy rate region given by all possible rate-tuples Open image in new window
satisfying Open image in new window where Open image in new window denotes the Open image in new window th column of Open image in new window orthogonal to Open image in new window , Open image in new window orthogonal to Open image in new window , respectively. Figure 11 compares the averaged secrecy rate region of the Vandermonde precoding, zero-forcing beamforming, and the optimal S-DPC scheme for Open image in new window where all entries of Open image in new window are i.i.d. Open image in new window . As observed, the Vandermonde precoding achieves the near-optimal rate region. As the number of paths Open image in new window increases, the gap with respect to the S-DPC becomes smaller since the Vandermonde precoding tends to choose the optimal beamformer matched to the channels.
Figure 11

Achievable secrecy rate region Open image in new window (MISO-BCC).

6. Conclusions

We considered the secured communication over the frequency-selective channel by focusing on the frequency-selective BCC. In the case of a block transmission of Open image in new window symbols followed by a guard interval of Open image in new window symbols discarded at both receivers, the frequency-selective channel can be modeled as an Open image in new window Toeplitz matrix. For this special type of MIMO channels, we proposed a practical yet order-optimal Vandermonde precoding which enables to send Open image in new window streams of the confidential messages and Open image in new window streams of the common messages simultaneously over a block of Open image in new window dimensions. The key idea here consists of exploiting the frequency dimension to "hide" confidential information in the zeros of the channel seen by the unintended receiver similarly to the spatial beamforming. We also provided some application of the Vandermonde precoding in the multiuser secured communication scenarios and proved the optimality of the proposed scheme in terms of the achievable s.d.o.f. region.

We conclude this paper by noticing that there exists a simple approach to establish secured communications. More specifically, perfect secrecy can be built in two separated blocks: (1) a precoding that cancels the channel seen by the eavesdropper to fulfill the equivocation requirement, (2) the powerful off-the-shelf encoding techniques to achieve the secrecy rate. Since the practical implementation of secrecy encoding techniques (double binning) remains a formidable challenge, such design is of great interest for the future secrecy systems.

Notes

Acknowledgments

The work is supported by the European Commission in the framework of the FP7 Network of Excellence in Wireless COMmunications NEWCOM++. The work of M. Debbah is supported by Alcatel-Lucent within the Alcatel-Lucent Chair on Flexible Radio at Supelec. The authors wish to thank Yingbin Liang for helpful discussions, and the anonymous reviewers for constructive comments. The material in this paper was partially presented at IEEE 19th International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC), Cannes, France, September 2008.

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Copyright information

© Mari Kobayashi et al. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  • Mari Kobayashi
    • 1
  • Mérouane Debbah
    • 2
  • Shlomo Shamai (Shitz)
    • 3
  1. 1.Department of TelecommunicationsSUPELECGif-sur-YvetteFrance
  2. 2.Alcatel-Lucent Chair on Flexible RadioSUPELECGif-sur-YvetteFrance
  3. 3.Department of Electrical EngineeringTechnion-Israel Institute of TechnologyHaifaIsrael

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