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

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., [4‐14] 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 [8‐13, 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 [17‐20]. 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 [21‐26] (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 paths such that the channel remains fixed during an entire transmission of blocks for an arbitrary large . 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 [28‐30]) where a guard interval of symbols is inserted at the beginning of each block to avoid the interblock interference and both receivers discard these 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 symbols followed by a guard interval of symbols discarded at both receivers, the frequency-selective channel can be modeled as an 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, where 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 where 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 streams of the confidential message and streams of the common message for simultaneously over a block of dimensions. Interestingly, the proposed Vandermonde precoding can be applied to multiuser secure communication scenarios: (a) a -user frequency-selective BCC with 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, will denote transpose operator, conjugation, and hermitian transpose. , represent the identity matrix, zero matrix. denote a determinant, rank, trace of a matrix , respectively. denotes the sequence . , , , , , denote the realization of the random variables , , , , , . Finally, " '' denotes less or equal to in the positive semidefinite ordering between positive semidefinite matrices, that is, we have if is positive semidefinite.

We consider the quasistatic frequency-selective fading BCC illustrated in Figure 1. The received signal of receivers 1, 2 at block is given by

where denote an Toeplitz matrix with the -path channel vector of user 1, of user 2, respectively, denotes the transmit vector, and finally are mutually independent additive white Gaussian noise (AWGN). The input vector is subject to the power constraint given by

where we let . The structure of is given by

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

The transmitter wishes to send a common message to two receivers and a confidential message to receiver 1. A code consists of the following: (1) two message sets and with the messages uniformly distributed over the sets , , respectively; (2) a stochastic encoder that maps each message pair to a codeword ; (3) one decoder at receiver 1 that maps a received sequence to a message pair and another at receiver 2 that maps a received sequence to a message . The average error probability of a code is defined as

where denotes the error probability when the message pair is sent defined by

The secrecy level of the confidential message at receiver 2 is measured by the equivocation rate 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 is said to be achievable if for any there exists a sequence of codes such that we have

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

where the union is over all possible distribution , , satisfying [20, Lemma  1]

where might be a deterministic function of . Recently, the secrecy capacity region of the two-user MIMO-BCC (1) was characterized in [16] and is given by all possible rate tuples satisfying

for some with denotes the input covariance satisfying and , 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 [10‐12, 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 are the generalized eigen-values greater than one of the following pencil:

(In [15, 19] the authors consider the real matrices , . Nevertheless, it is conjectured that for complex matrices the following expression without 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 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 [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 denotes s.d.o.f. which corresponds precisely to the number of the generalized eigenvalues greater than one in the high SNR.

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 , 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 , there exists a Vandermonde matrix for whose structure is given by

where are the roots of the polynomial with coefficients of the channel . Clearly satisfies the following orthogonal condition:

and if are all different.

It is well known that as the dimension of and increases, the Vandermonde matrix 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 .

Definition 1.

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

Further, the precoding matrix for the confidential message satisfies the following property.

Lemma 2.

Given two Toeplitz matrices , where , are linearly independent, there exists a unitary Vandermonde matrix for 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 , we form the transmit vector as

where the common message vector and the confidential message vector are mutually independent Gaussian vectors with zero mean and covariance , , respectively. Under this condition, the input covariances subject to

satisfy the power constraint (2). We let denote the feasible set satisfying (18).

Theorem 3.

The Vandermonde precoding achieves the following secrecy rate region:

where denotes the convex hull and we let , , .

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 of the Vandermonde precoding. By letting the auxiliary variables and , 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 on the boundary of the convex region is obtained by solving

for nonnegative weights satisfying . When the region , obtained without convex hull, is nonconvex, the set of the optimal covariances achieving the boundary point might not be unique. Figure 2 depicts an example in which the achievable rate region is obtained by the convex hull operation on the region , that is, replacing the non-convex subregion by the line segment , . For the weight ratio corresponding to the slope of the line segment , , there exist two optimal sets of the covariances yielding the points and (which clearly dominate the point ). 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, , , and . Formally, we have the following lemma.

Lemma 4.

The optimal , solution of (23), is given by one of the three solutions.

Case 1.

maximizes

and satisfies .

Case 2.

maximizes

and satisfies .

Case 3.

maximizes

and satisfies for some .

Before considering the weighted sum rate maximization (23), one applies SVD to ,

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

Theorem 5.

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

Case 1.

, if , solution of the following KKT conditions, satisfies

where with a positive semidefinite for , is determined such that , and we let .

Case 2.

if the following fulfills .

We let and where , are diagonal with the th element given by

where is determined such that .

Case 3.

, if , solution of the following KKT conditions, satisfies for some

where with a positive semidefinite for ,  is determined such that .

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 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 ( ), as well as the maximum sum rate point for the equal weight case ( ). It is worth noticing that under equal weight the objective functions in three cases are all concave in , since is concave if and is concave if and .

The maximum sum rate point can be found by applying the following greedy search [7].

Greedy Search to Find the Maximum Sum Rate Point

( ) Find , maximizing and check . If yes stop. Otherwise go to (2).

( ) Find , maximizing and check . If yes stop. Otherwise go to (3).

( ) Find , maximizing and check for some .

For the special case of , 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 and plugging the power allocation of (30) with , , is determined such that .

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 Toeplitz matrices is given as a union of satisfying

where , 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 streams such that , , we obtain the rate tuple where

We first notice that the prelog factor of as depends only on the rank of . From Lemma 2, we obtain

where (a) follows from orthogonality between and , (b) follows from the fact that is unitary satisfying . Notice that (36) yields . For the d.o.f. of the common message, (36) and (38) yield

which is dominated by the pre-log of 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 and the eavesdropper channel , which is bounded by . The second inequality (34) follows because the total number of streams for receiver 1 cannot be larger than the d.o.f. of , that is, .

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

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 -user frequency-selective BCC with 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.

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

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 symbols followed by a guard interval of symbols discarded at both receivers, the frequency-selective channel can be modeled as an Toeplitz matrix. For this special type of MIMO channels, we proposed a practical yet order-optimal Vandermonde precoding which enables to send streams of the confidential messages and streams of the common messages simultaneously over a block of 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.