Secure Degrees of Freedom for the MIMO Wire-tap Channel with a Multi-antenna Cooperative Jammer This paper was presented in part at the 2014 IEEE Information Theory Workshop, and the 2015 IEEE International Conference on Communications. This work was supported by NSF Grants CCF 09-64362, 13-19338 and CNS 13-14719.

Secure Degrees of Freedom for the MIMO Wire-tap Channel with a Multi-antenna Cooperative Jammer thanks: This paper was presented in part at the 2014 IEEE Information Theory Workshop, and the 2015 IEEE International Conference on Communications. This work was supported by NSF Grants CCF 09-64362, 13-19338 and CNS 13-14719.

Mohamed Nafea Wireless Communications and Networking Laboratory (WCAN)
Electrical Engineering Department
The Pennsylvania State University, University Park, PA 16802.
mnafea@psu.edu    yener@engr.psu.edu
Aylin Yener Wireless Communications and Networking Laboratory (WCAN)
Electrical Engineering Department
The Pennsylvania State University, University Park, PA 16802.
mnafea@psu.edu    yener@engr.psu.edu
Abstract

In this paper, a multiple antenna wire-tap channel in the presence of a multi-antenna cooperative jammer is studied. In particular, the secure degrees of freedom (s.d.o.f.) of this channel is established, with antennas at the transmitter, antennas at the legitimate receiver, and antennas at the eavesdropper, for all possible values of the number of antennas, , at the cooperative jammer. In establishing the result, several different ranges of need to be considered separately. The lower and upper bounds for these ranges of are derived, and are shown to be tight. The achievability techniques developed rely on a variety of signaling, beamforming, and alignment techniques which vary according to the (relative) number of antennas at each terminal and whether the s.d.o.f. is integer valued. Specifically, it is shown that, whenever the s.d.o.f. is integer valued, Gaussian signaling for both transmission and cooperative jamming, linear precoding at the transmitter and the cooperative jammer, and linear processing at the legitimate receiver, are sufficient for achieving the s.d.o.f. of the channel. By contrast, when the s.d.o.f. is not an integer, the achievable schemes need to rely on structured signaling at the transmitter and the cooperative jammer, and joint signal space and signal scale alignment. The converse is established by combining an upper bound which allows for full cooperation between the transmitter and the cooperative jammer, with another upper bound which exploits the secrecy and reliability constraints.

July 2, 2019

I Introduction

Information theoretically secure message transmission in noisy communication channels was first considered in the seminal work by Wyner [1]. Reference [2] subsequently identified the secrecy capacity of a general discrete memoryless wire-tap channel. Reference [3] studied the Gaussian wire-tap channel and its secrecy capacity. More recently, an extensive body of work was devoted to study a variety of network information theoretic models under secrecy constraint(s), see for example [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. The secrecy capacity region for most of multi-terminal models remain open despite significant progress on bounds and associated insights. Recent work thus includes efforts that concentrate on characterizing the more tractable high signal-to-noise ratio (SNR) scaling behavior of secrecy capacity region for Gaussian multi-terminal models [19, 20, 21, 22, 23, 24].

Among the multi-transmitter models studied, a recurrent theme in achievability is enlisting one or more terminals to transmit intentional interference with the specific goal of diminishing the reception capability of the eavesdropper, known as cooperative jamming [25]. For the Gaussian wire-tap channel, adding a cooperative jammer terminal transmitting Gaussian noise can improve the secrecy rate considerably [9], albeit not the scaling of the secrecy capacity with power at high SNR. Recently, reference [21] has shown that, for the Gaussian wire-tap channel, adding a cooperative jammer and utilizing structured codes for message transmission and cooperative jamming, provide an achievable secrecy rate scalable with power, i.e., a positive secure degrees of freedom (s.d.o.f.), an improvement from the zero degrees of freedom of the Gaussian wire-tap channel. More recently, reference [22] has proved that, for this channel, the s.d.o.f. , achievable by codebooks constructed from integer lattices along with real interference alignment, is tight. References [23, 24] have subsequently identified the s.d.o.f. region for multi-terminal Gaussian wire-tap channel models.

While the above development is for single-antenna terminals, multiple antennas have also been utilized to improve secrecy rates and s.d.o.f. for several channel models, see for example [26, 27, 28, 5, 6, 7, 19, 29, 30, 31]. The secrecy capacity of the multi-antenna (MIMO) wire-tap channel, identified in[26] scales with power only when the legitimate transmitter has an advantage over the eavesdropper in the number of antennas. It then follows naturally to utilize a cooperative jamming terminal to improve the secrecy rate and scaling for multi-antenna wire-tap channels as well which is the focus of this work.

In this paper, we study the multi-antenna wire-tap channel with a multi-antenna cooperative jammer. We characterize the high SNR scaling of the secrecy capacity, i.e., the s.d.o.f., of the channel with antennas at the cooperative jammer, antennas at transmitter, antennas at the receiver, and antennas at the eavesdropper. The achievability and converse techniques both are methodologically developed for ranges of the parameters, i.e., the number of antennas at each terminal. The upper and lower bounds for all parameter values are shown to match one another. The s.d.o.f. results in this paper match the achievability results derived in [32, 33], which are special cases for , , , and real channel gains. The s.d.o.f. for the cases and , for all possible values of , were reported in [34], [35], respectively.

The proposed achievable schemes for different ranges of the values for , , , and all involve linear precoding and linear receiver processing. The common goal to all these schemes is to perfectly align the cooperative jamming signals over the information signals observed at the eavesdropper while simultaneously enabling information and cooperative jamming signal separation at the legitimate receiver. We show that whenever the s.d.o.f. of the channel is integer valued, Gaussian signaling both at the transmitter and the cooperative jammer suffices to achieve the s.d.o.f. By contrast, non-integer s.d.o.f. requires structured signaling along with joint signal space and signal scale alignment in the complex plane [36, 37]. The necessity of structured signaling follows from the fact that fractional s.d.o.f. indicates sharing at least one spatial dimension between information and cooperative jamming signals at the receiver’s signal space. In this case, sharing the same spatial dimension between Gaussian information and jamming signals, which have similar power scaling, does not provide positive degrees of freedom, and we need for structured signals that can be separated over this single dimension at high SNR. The tools that enable the signal scale alignment are available in the field of transcendental number theory [37, 38, 39], which we utilize.

The paper is organized as follows. Section II introduces the channel model, and Section III provides the main results. For clarity of exposition, we first present the converse and achievability for the MIMO wire-tap channel with in Sections IV and V. Section VI then extends the converse and achievability proofs for the case . Section VII discusses the results of this work and Section VIII concludes the paper.

Overall, this study determines the value in jointly utilizing signal scale and spatial interference alignment techniques for secrecy and quantifies the impact of a multi-antenna helper for the MIMO wire-tap channel by settling the question of the secrecy prelog for the MIMO wire-tap channel in the presence of an -antenna cooperative jammer, for all possible values of . In contrast with the single antenna case, where integer lattice codes and real interference alignment suffice to achieve the s.d.o.f. of the channel, in the MIMO setting, one needs to utilize a variety of signaling, beam-forming, and alignment techniques, in order to coordinate the transmitted and received signals for different values of , and .

Ii Channel Model and Definitions

First, we remark the notation we use throughout the paper: Small letters denote scalars and capital letters denote random variables. Vectors are denoted by bold small letters, while matrices and random vectors are denoted by bold capital letters111The distinction between matrices and random vectors is clear from the context.. Sets are denoted using calligraphic fonts. All logarithms are taken to be base . The set of integers is denoted by . denotes an matrix of zeros, and denotes an identity matrix. For matrix , denotes its null space, denotes its determinant, and denotes its induced norm. For vector , denotes its Euclidean norm, and denotes the th to th components in . We use to denote the -letter extension of the random vector , i.e., . The operators , , and denote the transpose, Hermitian, and pseudo inverse operations. We use , , and , to denote the sets of real, complex, rational, and integer numbers, respectively. denotes the set of Gaussian (complex) integers. A circularly symmetric Gaussian random vector with zero mean and covariance matrix is denoted by .

As the channel model, we consider the MIMO wire-tap channel with an -antenna transmitter, -antenna receiver, -antenna eavesdropper, and an -antenna cooperative jammer as depicted in Fig. 1.

Fig. 1: multiple antenna wire-tap channel with an -antenna cooperative jammer.

The received signals at the receiver and eavesdropper, at the th channel use, are given by

(1)
(2)

where and are the transmitted signals from the transmitter and the cooperative jammer at the th channel use. , are the channel gain matrices from the transmitter and the cooperative jammer to the receiver, while , are the channel gain matrices from the transmitter and the cooperative jammer to the eavesdropper. It is assumed that the channel gains are static, independently drawn from a complex-valued continuous distribution, and known at all terminals. and are the complex Gaussian noise at the receiver and eavesdropper at the th channel use, where and for all . is independent from and both are independent and identically distributed (i.i.d.) across the time index222Throughout the paper, we omit the index whenever possible. . The power constraints on the transmitted signals at the transmitter and the cooperative jammer are .

The transmitter aims to send a message to the receiver, and keep it secret from the external eavesdropper. A stochastic encoder, which maps the message to the transmitted signal , is used at the transmitter. The receiver uses its observation, , to obtain an estimate of the transmitted message. Secrecy rate is achievable if for any , there is a channel code satisfying333We consider weak secrecy throughout this paper.

(3)
(4)

The secrecy capacity of a channel, , is defined as the closure of all its achievable secrecy rates. For a channel with complex-valued coefficients, the achievable secure degrees of freedom (s.d.o.f.), for a given secrecy rate , is defined as

(5)

The cooperative jammer transmits the signal in order to reduce the reception capability of the eavesdropper. However, this transmission affects the receiver as well, as interference. The jamming signal, , does not carry any information. Additionally, there is no shared secret between the transmitter and the cooperative jammer.

Iii Main Result

We first state the s.d.o.f. results for .

Theorem 1

The s.d.o.f. of the MIMO wire-tap channel with an -antenna cooperative jammer, antennas at each of the transmitter and receiver, and antennas at the eavesdropper is given by

(6)

Proof:  The proof for Theorem 1 is provided in Sections IV and V.

Next, in Theorem 2 below, we generalize the result in Theorem 1 to .

Theorem 2

The s.d.o.f. of the MIMO wire-tap channel with an -antenna cooperative jammer, -antenna transmitter, -antenna receiver, and -antenna eavesdropper is given by

(7)

where,

Proof:  The proof for Theorem 2 is provided in Section VI.

Remark 1 Theorem 2 provides a complete characterization for the s.d.o.f. of the channel. The s.d.o.f. at is equal to , which is equal to the d.o.f of the point-to-point MIMO Gaussian channel. Thus, increasing the number of antenna at the cooperative jammer, , over can not increase the s.d.o.f. over .

Remark 2 For , the s.d.o.f. of the channel is equal to at , i.e., the maximum s.d.o.f. of the channel is achieved without the help of the cooperative jammer.

Remark 3 The converse proof for Theorem 2 involves combining two upper bounds for the s.d.o.f. derived for two different ranges of . These two bounds are a straight forward generalization of those derived for the symmetric case in Theorem 1. However, combining them is more tedious since more cases of the number of antennas at the different terminals should be handled carefully. Achievability for Theorem 2 utilizes similar techniques to those used for Theorem 1 as well, where handling more cases is required. For clarity of exposition, we derive the s.d.o.f. for the symmetric case first in order to present the main ideas, and then utilize these ideas and generalize the result to the asymmetric case of Theorem 2.

For illustration purposes, the s.d.o.f. for , and varies from to , is depicted in Fig. 2. We provide the discussion of the results of this work in Section VII.

Fig. 2: Secure degrees of freedom for a MIMO wire-tap channel, with antennas at each of its nodes, and a cooperative jammer with antennas, where varies from to .

Iv Converse for

In Section IV-A, we derive the upper bound for the s.d.o.f. for . In Section IV-B, we derive the upper bound for . The two bounds are combined in Section IV-C to provide the desired upper bound in (6).

Iv-A

Allow for full cooperation between the transmitter and the cooperative jammer. This cooperation can not decrease the s.d.o.f. of the channel, and yields a MIMO wire-tap channel with -antenna transmitter, -antenna receiver, and -antenna eavesdropper. It has been shown in [26] that, at high SNR, i.e., , the secrecy capacity of this channel, , takes the asymptotic form

(8)

where , and are the channel gains from the combined transmitter to the receiver and eavesdropper, and is the projection matrix onto the null space of , . , where is the space orthogonal to the null space of . Due to the randomly generated channel gains, if a vector , then almost surely (a.s.), for all . Thus, .

can be decomposed as

(9)

where is a unitary matrix and is a non-singular matrix [26]. Let , where and . Substituting (9) in (8) yields

(10)
(11)
(12)
(13)

where (11) follows from Sylvester’s determinant identity and (12) follows from being unitary.

The achievable secrecy rate of the original channel, , is upper bounded by . Thus, the s.d.o.f. of the original channel, for , is upper bounded as

(14)
(15)

Iv-B

The upper bound we derive here is inspired by the converse of the single antenna Gaussian wire-tap channel with a single antenna cooperative jammer derived in [22], though as we will see shortly, the vector channel extension resulting from multiple antennas does require care. Let , for , denote constants which do not depend on the power .

The secrecy rate can be upper bounded as follows

(16)
(17)
(18)
(19)
(20)
(21)
(22)

where (18) follows since by the secrecy constraint in (4), by Fano’s inequality, and by the fact that conditioning does not increase entropy, (22) follows since is independent from , and .

Let and , where and . Note that and are noisy versions of the transmitted signals and , respectively. is independent from and both are independent from . and are i.i.d. sequences of the random vectors and . In addition, let and . Note that and , where and . and are i.i.d. sequences of and , since each of is i.i.d. across time. The covariance matrices, and , are chosen as and , where . This choice of and guarantees the finiteness , and as shown in Appendix A. Starting from (22), we have

(23)
(24)
(25)
(26)
(27)
(28)
(29)

where (26) follows since and are independent from , , , and . We now consider the following two cases.

Case 1:

We first lower bound in (29) as follows. Using the infinite divisibility of Gaussian distribution, we can express a stochastically equivalent form of , denoted by , as

(30)

where444The choice of guarantees that is a valid covariance matrix. is independent from . is an i.i.d. sequence of the random vectors . Using (30), a stochastically equivalent form of is

(31)

Let , , and , where , , and , . In addition, let , where and . Using (31), we have

(32)
(33)
(34)
(35)

where the inequality in (33) follows since and are independent, as for two independent random vectors and , we have .

Substituting (35) in (29) results in

(36)
(37)

where .

We now exploit the reliability constraint in (3) to derive another upper bound for , which we combine with the bound in (37) in order to obtain the desired bound for the s.d.o.f. when and . The reliability constraint in (3) can be achieved only if [40]

(38)
(39)

Similar to (30), a stochastically equivalent form of is given by

(40)

where555The choice of guarantees that is a valid covariance matrix. is independent from . is an i.i.d. sequence of the random vectors .

Let , , and , where , , and , . In addition, let , where and . Using (40), we have

(41)
(42)
(43)
(44)

Substituting (44) in (39) yields

(45)

Let . Summing (37) and (45) results in

(46)
(47)

where .

In Appendix B, we show, for , , and , that

(48)
(49)

where ; and denote the transpose of the th row vectors of and , respectively. Using (47), (48), and (49), we have