Secure Degrees of Freedom for the MIMO Wiretap ... - WCAN@PSU
Secure Degrees of Freedom for the MIMO Wiretap Channel with a Multiantenna Cooperative Jammer Mohamed Nafea and Aylin Yener Department of Electrical Engineering, The Pennsylvania State University, University Park, PA 16802
Abstract—A multiple antenna Gaussian wiretap channel with a multiantenna cooperative jammer (CJ) is considered and the secure degrees of freedom (s.d.o.f.), with N antennas at the sender, receiver, and eavesdropper, is derived for all possible values of the number of antennas at the cooperative jammer, K. In particular, the upper and lower bounds for the s.d.o.f. are provided for different ranges of K and shown to coincide. Gaussian signaling both for transmission and jamming is shown to be sufficient to achieve the s.d.o.f. of the channel, when the s.d.o.f. is integer-valued. By contrast, when the channel has a non-integer s.d.o.f., structured signaling and joint signal space and signal scale alignment are employed to achieve the s.d.o.f.
I. I NTRODUCTION Information theoretic secrecy [1] guarantees secure communication in the presence of an eavesdropper. Information theoretic secrecy of multiterminal and multiantenna channels has been studied extensively, e.g., [2]–[9]. In particular, the Gaussian wiretap channel (WTC) with a cooperative jammer (CJ) was studied in [2]–[8]. A CJ can improve the secrecy rate and even the prelog factor of the secrecy rate, i.e., the secure degrees of freedom (s.d.o.f.) [5]. Relying on cooperative jamming and structured signaling, [6], [7] identified the s.d.o.f. for several single-antenna WTC models. In this paper, we extend [6] to a multiantenna secure communication scenario. We focus on a multiantenna Gaussian WTC with a Kantenna CJ and N antennas at each of the transmitter, receiver, and wiretapper. For 0 ≤ K ≤ 2N , we characterize the s.d.o.f. arriving at the s.d.o.f. of N at K = 2N , concluding that increasing K over 2N cannot improve the s.d.o.f. We derive an upper bound on the s.d.o.f. which allows for cooperation between the transmitter and CJ, and show that this bound is tight for 0 ≤ K ≤ N2 . Next, for N ≤ K ≤ 2N , we derive another upper bound which incorporates both the secrecy and reliability constraints. We use this upper bound for K = N to N upper bound the s.d.o.f. N for all 2 < K < N . We show that increasing K from 2 to N does not increase the s.d.o.f. Next, we divide 0 ≤ K ≤ 2N into five different ranges and propose an achievable scheme which meets with the derived upper bound for each range. Whenever the s.d.o.f. is integervalued, we show that Gaussian signaling at the transmitter and the CJ is sufficient to achieve the s.d.o.f. of the channel. By contrast, when the s.d.o.f. is not an integer, structured signaling along with joint signal space and signal scale alignment are This work was supported by NSF Grants CCF 09-64362, 13-19338 and CNS 13-14719.
978-1-4799-5999-0/14/$31.00 ©2014 IEEE
Fig. 1.
MIMO Gaussian WTC with a multiantenna CJ.
needed. When K is larger than N , a linear precoder is utilized at the CJ so that some jamming signals are transmitted over directions invisible to the legitimate receiver. Overall, this study settles the secrecy capacity of the N × N × N MIMO wiretap channel in the high SNR, when a multiantenna CJ is available as a helper. As compared to the single antenna counterpart that uses real interference alignment with one dimensional lattices, here we need to use a variety of spatial and signal scale alignment techniques in concert to coordinate multiple antenna transmissions. Notation: For matrix A, ||A|| denotes its induced norm. For vector V, Vij = [Vi · · · Vj ]T , where 1 ≤ i < j ≤ n, and ||V|| denotes Euclidean norm. 0m×n denotes an m × n matrix of zeros. The set of integers {−Q, · · · , Q} is denoted by (−Q, Q)Z . Z[j] denotes the set of complex integers. II. C HANNEL M ODEL AND D EFINITIONS We consider a multiantenna Gaussian WTC composed of a transmitter, a receiver, an eavesdropper each with N antennas, and a K-antenna CJ, see Fig.1. The received signals at the receiver and eavesdropper at the nth channel use are given by Yr (n) = Ht Xt (n) + Hc Xc (n) + Zr (n)
(1)
Ye (n) = Gt Xt (n) + Gc Xc (n) + Ze (n),
(2)
where Xt (n), Xc (n) are the transmitted signals from the transmitter and CJ, respectively. Ht , Gt ∈ CN ×N are the transmitter’s channel matrices to the legitimate receiver and to the eavesdropper, and Hc , Gc ∈ CN ×K are the channel matrices from the CJ to the legitimate receiver and eavesdropper. The channel gains are static, and complex-valued. Zr (n) and Ze (n) denote the complex Gaussian noise at the nth channel use, i.e., Zr (n), Ze (n) ∼ CN (0, IN ), are independent from one another and both are independent and identically
627
1 distributed (i.i.d.) across n. The at the power Hconstraints transmitter and CJ are E XH X , E X X ≤ P . t c t c Let Xnt = [Xt (1) · · · Xt (n)]. Xnc , Yrn , Yen , Znr , Zne are defined similarly. The transmitter intends to send a secret message W ∈ W to the legitimate receiver in the presence of the eavesdropper. W is mapped into the transmitted signal Xnt ∈ Xtn by using a stochastic encoder f : W 7→ Xtn at the transmitter. The receiver forms an estimate of W , denoted by ˆ . Secrecy rate Rs is achievable if for any > 0, there exists W a channel code, 2nRs , n , such that n o ˆ 6= W ≤ ; 1 H(W |Yen ) ≥ 1 H(W ) − . (3) Pr W n n The achievable s.d.o.f. for a given secrecy rate, Rs , is
Rs Ds = lim . P →∞ log P
(4)
The CJ transmits Xnc ∈ Xcn . The jamming signal Xnc is not meant to convey a message. There is no common randomness between the transmitter and the CJ. III. M AIN R ESULT Theorem 1 The s.d.o.f. of the multiantenna Gaussian WTC with a K-antenna CJ and N antennas at each of its nodes is for 0 ≤ K ≤ N2 K, (5) Ds = N2 , for N2 < K ≤ N K , for N < K ≤ 2N. 2
The secrecy rate for the original channel, Rs , is upper bounded ¯ s . Using (4), the s.d.o.f., Ds , is upper bounded by K. by R B. N ≤ K ≤ 2N Here, we extend the converse proof in [6] to the multiantenna channel. Let φi , i = 1, · · · , 6, denote constants that do not depend on the power P . Rs can be upper bounded as nRs = H(W ) ≤ H(W |Yen ) + n − H(W |Yrn ) + nδ ≤ I(W ; Yrn |Yen ) + nφ1 ≤ h(Yrn |Yen ) − h(Yrn |W Yen Xnt Xnc ) + nφ1 =
A. 0 ≤ K ≤ N We allow for cooperation between the transmitter and CJ, obtain, in effect a multiantenna Gaussian WTC with N + Kantenna transmitter, N -antenna receiver, and N -antenna eavesdropper. The secrecy rate of this channel is bounded as [9] P ¯ ¯]¯H ¯ Rs ≤ log det IN + HG H + o(log P ), (6) K ¯ G ¯ ∈ CN ×N +K are the channel matrices from the where H, ¯] combined transmitter to the receiver and eavesdropper, and G ¯ is the projection matrix onto N (G). We also have [9] 0N −K×N −K 0N −K×K ]¯H ¯ ¯ HG H = Ψ ΨH , (7) 0K×N −K Ω where Ψ is a unitary matrix and Ω is a non-singular matrix. By substituting (7) in (6), it can be easily shown that ¯ s ≤ K log P + o(log P ). R 1 Throughout
−
h(Yen )
−
h(Znr )
(11)
+ nφ1 .
(12)
(9) follows from (3) and Fano’s inequality; φ1 = + δ. ˜ t = Xt + Z ˜ t, X ˜ c = Xc + Z ˜ c , where Z ˜t ∼ Define X ˜ CN (0, Kt ) and Zc ∼ CN (0, Kc ). Kt , Kc are chosen as Kt = ρ2 IN , Kc = β 2 IK , where 0 < ρ ≤ ||G1H || and t ˜ t is independent from Z ˜ c and both are in0 < β ≤ ||H1H || . Z c ˜ n, Z ˜ n are i.i.d. sequences dependent from {Xt , Xc , Zr , Ze }. Z t c ˜ ˜ ˜ ˜ t − Hc Z ˜ c + Zr and of Zt , Zc , respectively. Let Z1 = −Ht Z ˜ ˜ ˜ ˜ Z2 = −Gt Zt − Gc Zc + Ze , where Z1 ∼ CN (0, ΣZ˜ 1 ), ˜ 2 ∼ CN (0, Σ ˜ ), Σ ˜ = Ht Kt HH + Hc Kc HH + IN , Z t c Z2 Z1 H ˜n ˜n and ΣZ˜ 2 = Gt Kt GH t + Gc Kc Gc + IN . Z1 , Z2 are i.i.d. ˜ 1, Z ˜ 2 . Thus, using (12), we have sequences of Z ˜ nX ˜ n ) − h(X ˜ nX ˜ n |Yn Yn ) nRs ≤ h(Yrn Yen X t c t c r e − h(Yen ) + nφ2 (13) n n n n n n n n ˜ X ˜ ) + h(Y |X ˜ X ˜ ) + h(Y |X ˜ X ˜ ) ≤ h(X t c r t c e t c n n n n n n n ˜ X ˜ |Y Y X X ) − h(Y ) + nφ2 (14) − h(X t c r e t c e ˜ n ) + h(X ˜ n ) + h(Z ˜ n |X ˜ nX ˜ n ) + h(Z ˜ n |X ˜ nX ˜ n) ≤ h(X
Theorem 1 provides a complete characterization for the s.d.o.f. of the channel. The s.d.o.f. for K = 2N is equal to N , that is equal to the d.o.f. of the N -antenna Gaussian channel with no secrecy constraint. Thus, the s.d.o.f. can not be increased by increasing K over 2N . Interestingly, Theorem 1 shows that the s.d.o.f. N of the channel is not increased by increasing K from to N . In Sections IV and V, we provide the converse 2 and achievability proofs for Theorem 1. IV. C ONVERSE
h(Yrn Yen )
(9) (10)
t
c
1
t
c
2
n ˜ nZ ˜n − h(Z t c ) − h(Ye ) + nφ2 ˜ n ) + h(X ˜ n ) − h(Yn ) + nφ3 . ≤ h(X t c e
t
c
(15) (16)
Using infinite divisibility of Gaussian distribution, a ˜t + stochastically equivalent form of Ze is Z0e = Gt Z H ˜ ˜ Ze . Ze ∼ CN (0, IN − Gt Kt Gt ) is independent from ˜ t, Z ˜ c , Xt , Xc , Zr }. Thus, a stochastically equivalent form {Z n n ˜ n +Gc Xn + Z ˜ n . Since Gt X ˜ n , Gc Xn + of Ye is Ye0 = Gt X t c e t c n ˜ Ze are independent, we have n ˜ n) h(Yen ) = h(Ye0 ) ≥ h(Gt X t ˜ n ) + n log det(Gt ). = h(X t
(17) (18)
Substituting (18) in (16) gives us ˜ n ) + h(X ˜ n |X ˜ n ) + nφ4 , nRs ≤ h(X c2 c1 c2
(19)
˜ c = [X ˜ c = [X ˜c · · · X ˜ c ]T , X ˜c ˜ c ]T , and where X ···X 1 1 2 N N +1 K ˜ ˜ Xci = Xci + Zci . Let hck be the kth column vector of Hc = [Hc1 Hc2 ], Hc1 = [hc1 · · · hcN ] and Hc2 = [hcN +1 · · · hcK ]. In order to reliably transmit the message W , we must have
(8)
nRs ≤ I(Xnt ; Yrn ) = h(Yrn ) − h(Hc Xnc + Znr ) ˜ n + Hc X ˜n ) ≤ h(Yn ) − h(Hc X ≤
the paper, we omit index n whenever possible.
628
r h(Yrn )
c1 c2 2 ˜ n |Hc X ˜n ) h(Hc1 X c1 c2 2 1
−
(20) (21) (22)
˜n ) ˜ n |X ≤ h(Yrn ) − h(Hc1 X c1 c2 ˜ n ) − n log det(Hc ), ˜ n |X = h(Yrn ) − h(X c1 c2 1
(23) (24)
where (21) follows similar to (17), and (22) follows since for correlated X and Y, h(X + Y) ≥ h(X + Y|Y) = h(X|Y). Combining (19) and (24), we get n N K X 1X X ˜ c (i)) + nφ5 . nRs ≤ h(Yrk (i)) + h(X k 2 i=1 k=1
k=N +1
(25) Using Cauchy-Schwarz inequality and the power constraints, we can show that the variance of Yrk (i) is bounded as T Var{Yrk (i)} ≤ 1 + h2 P , where h2 = max (||hrtk ||2 + T ||hrck ||2 ),
k
K −N N log(1 + h2 P ) + log(β 2 + P ) + φ6 , (26) 2 2
and the s.d.o.f. is upper bounded as Ds ≤
K 2 .
C. Obtaining the upper bound for all K We use the upper bound obtained in Section IV-A for 0 ≤ K ≤ N2 , and the upper bound obtained in Section IV-B for N ≤ K ≤ 2N . By comparing the two, it is evident that the upper bound from Section IV-A is greater than N2 for N 2 < K ≤ N . Since we know, from Section IV-B, that at K = N , the upper bound is N2 , we can use N2 as the upper bound for N2 < K ≤ N . Combining these statements, we get (5). Next, we shall see the achievability of (5). V. ACHIEVABLE S CHEMES We divide the range 0 ≤ K ≤ 2N into five cases and propose an achievable scheme for each case. For all the achievable schemes, we have the n-letter signals, Xnt and Xnc , as i.i.d. sequences. Since Xnc is independent from Xnt , we have in effect a memoryless WTC, and the following secrecy rate is achievable by stochastic encoding [1]: Rs = [I(Xt ; Yr ) − I(Xt ; Ye )]+ . A. Case 1: 0 ≤ K ≤
(27)
N 2
The transmitter sends K independent Gaussian information streams and the CJ sends K independent Gaussian jamming streams. Since 2K ≤ N , the legitimate receiver can decode all the information and jamming streams at high SNR. The transmitter chooses a precoder, Pt , which aligns its information streams over the jamming streams at the eavesdropper. The signals transmitted by the transmitter and the CJ are Xt = Pt Ut ,
X c = Jc V c ,
T
+ Zr
(29)
(30) −1 We lower bound (27) as follows. First, Ht Gt Gc Hc is almost surely (a.s.) full column-rank. Thus, I(Xt ; Yr ) ≥ K log P + o(log P ).
(31)
Next, we upper bound the term I(Xt ; Ye ) as follows: I(Xt ; Ye ) = log
det(IK + 2P¯ GH c Gc ) ≤ K. H ¯ det(IK + P Gc Gc )
(32)
By substituting (31) and (32) in (27), we have Rs ≥ K log P + o(log P ) − K.
(33)
Hence, the achievable s.d.o.f. is lower bounded as Ds ≥ K. B. Case 2:
N 2
< K ≤ N and N is even
The s.d.o.f. is upper bounded by N2 for all N2 < K ≤ N . When N is even, the achievable scheme for K = N2 can be used to achieve the s.d.o.f. of the channel for all N2 < K ≤ N . The transmitted signals are given by (28), with Jc = ¯ ). [I N 0 N ×K− N ]T , Pt = G−1 t Gc Jc , Ut , Vc ∼ CN (0, P I N 2 2 2 2 Using the same analysis as in the previous case, the achievable N N s.d.o.f. is 2 for any 2 < K ≤ N , where N is even. C. Case 3:
N 2
< K ≤ N and N is odd
For this case, we utilize structured signaling both at the transmitter and the CJ. In particular, we propose to use joint signal space alignment and the complex field equivalent of real interference alignment [10], [11]. The transmitter and CJ send N2+1 streams each. The transmitter aligns its information streams over the jamming at the eavesdropper. The legitimate receiver projects its received signal over a direction orthogonal to all but one information and one jamming streams, decodes these two streams from the projection, and subtracts their effect from its received signal, leaving N − 1 spatial dimensions for the other streams. For notational simplicity, let d = N2+1 . The transmitted signals are given by (28) with Jc = T [Id 0d×K−d ]T , Pt = G−1 t Gc Jc , Ut = [U1 · · · Ud ] , Vc = T [V1 · · · Vd ] , Ui = UiRe + jUiIm , Vi = ViRe + jViIm , for i = 2, · · · , d, and U1 , V1 , {UiRe }di=2 , {UiIm }di=2 , {ViRe }di=2 , {ViIm }di=2 are i.i.d. uniform over the set {a(−Q, Q)Z }. The values for a and the integer Q are chosen as 1−
Q = P 2+ − ν,
3
a = γP 2(2+) ,
(34)
where > 0 can be arbitrarily small, and ν, γ are constants. ˜ c = Gc Jc . The received signal at the eavesdropper is Let G ˜ c (Ut + Vc ) + Ze . Ye = G
(28)
where Ut , Vc ∼ CN (0, P¯ IK ). Ut and Vc are the information and jamming streams, respectively. Pt = G−1 t Gc = [pt1 · · · ptK ] , and Jc = IK . P¯ = α1 P , where α = PK max{K, i=1 ||pti ||2 } to satisfy the power constraints. The
VcT
Ye = Gc (Ut + Vc ) + Ze .
hrtk , hrck
and are the kth row vectors of Ht and Hc . Thus, h(Yrk (i)) ≤ log 2πe(1 + h2 P ). Similarly, we can ˜ c (i)) ≤ log 2πe(β 2 + P ). Thus, show that h(X k Rs ≤
received signals are expressed as Yr = Ht G−1 Hc UTt t Gc
(35)
We upper bound I(Xt ; Ye ) as follows:
629
I(Xt ; Ye ) ≤ I(Xt ; Ye , Ze ) = I(Xt ; Ye |Ze ) ˜ c (Ut + Vc ) − H G ˜ c Vc =H G
(36) (37)
= H(Ut + Vc ) − H(Vc ) 4Q + 1 ≤ N, ≤ (2d − 1) log 2Q + 1
(38) (39)
˜ c (Ut + where (38) follows since the mappings Ut + Vc 7→ G ˜ ˜ c are Vc ), Vc 7→ Gc Vc are invertible, since the entries of G 2 rationally independent . Next, we derive a lower bound for I(Xt ; Yr ). The received signal at the legitimate receiver is ¯ c Vc + Zr , Yr = AUt + H
(40)
¯ where A = Ht G−1 t Gc Jc and Hc = Hc Jc . Let ai and ¯ c , respectively. hci be the ith column vectors of A and H N The legitimate receiver chooses b ∈ C such that b ⊥ span {a2 , · · · , ad , hc2 , · · · , hcd } and multiplies its received signal by the decoding matrix, bH D= , (41) 0N −1×1 IN −1
inequality. Since the mapping (U1 , V1 ) 7→ f1 U1 + f2 V1 is invertible, the only source for error is the Gaussian noise Z 0 . n o ˆ1 , Vˆ1 ) 6= (U1 , V1 ) Pe1 ≤ Pr (U (50) 2 −dmin dmin , (51) = exp ≤ Pr |Z 0 | ≥ 2 4||b||2 √ ) and dmin is the minimum where |Z 0 | ∼ Rayleigh( ||b|| 2 distance between points in the constellation R1 , which can be lower bounded using the following lemma [10], [11].
Lemma 1 For almost all z ∈ Cn and for all > 0, −( n−1+ ) 2 |p + z.q| > max qi , i
holds for all q ∈ Zn , p ∈ Z except for finitely many of them. Thus, for almost all channel gains, dmin is dmin =
e r = DYr = [Yer (Y e N )T ]T , where to obtain Y r2 1 Yer1 = f1 U1 + f2 V1 + Z 0 e N = AU e t+H e c Vc + ZN , Y r2 r2
(42)
e r) I(Xt ; Yr ) ≥ I(Ut ; Y e N |U1 Yer ), ≥ I(U1 ; Yer1 ) + I(Udt2 ; Y r2 1
≥
(43)
e = [˜ ˜d ], f1 = bH a1 , f2 = bH hc1 , Z 0 = bH Zr , A a1 · · · a N N ˜c · · · h ˜ c ], a ˜ e c = [h ˜ H = a , and h = h . i ci ci2 1 d i2 The legitimate receiver uses Yer1 to decode U1 and V1 . Since f1 , f2 are a.s. rationally independent, the mapping (U1 , V1 ) 7→ f1 U1 + f2 V1 is bijective (i.e., invertible) [10]. The legitimate receiver employs a hard decision decoder which maps Yer1 ∈ Y˜r1 to the nearest point in the constellation R1 = f1 U1 +f2 V1 , where U1 , V1 = {a(−Q, Q)Z }. Then, the legitimate receiver passes the output of the hard decision decoder through the bijective map f1 U1 + f2 V1 7→ (U1 , V1 ) to decode both U1 , V1 , e N to obtain and subtracts their effect from Y r2 h iT ¯ r = B UdT VdT Y + ZN (44) r2 . t2 c2 h i ˜c · · · h ˜ c ∈ CN −1×N −1 is a.s. full rank ˜2 · · · a ˜d h B= a 2 d due to the random generation assumption on the channel gains. ¯ r. Finally, by zero forcing, the receiver obtains Udt2 from Y The term I(Xt ; Yr ) is lower bounded as follows: (45)
= (1 − Pe1 ) log(2Q + 1) − 1,
(49)
ˆ1 6= U1 }, and (48) follows from Fano’s where Pe1 = Pr{U
a|f1 | − 2 P 2. ≥ γ|f1 |2 (2Q) 2
(53) (54)
(55)
e N |U1 Yer ). Define Next, we lower bound the term I(Udt2 ; Y r2 1 h iT ˜ c bH ; Y ¯ 0 = B UdT VdT e = [0N −1×1 IN −1 ] − 1 h B + r t2 c2 1 f2 i h T T e r , and P d = ¯ 0 = Ud VdT + B−1 BZ e r; Y b 0 = B−1 Y BZ c2 e2 t2 r r d d ˆ 6= U }. Thus, we have Pr{U t2 t2 e N |U1 Yer ) = I Ud ; Y e N U1 , f2 V1 + Z 0 I(Udt2 ; Y (56) r2 t2 r2 1 ¯ 0 |f2 V1 + Z) = I(Udt2 ; Y (57) r d ¯0 d b0 ≥ I(Ut2 ; Yr ) ≥ I(Ut2 ; Yr ) (58) d N −1 ≥ 1 − Pe2 log(2Q + 1) − 1. (59) b r = ΞZr = [Zˆr · · · Zˆr ]T , where Ξ = B−1 B. e Thus, Let Z 2 N H b ˆ Zr ∼ CN (0, ΞΞ ) and |Zri | ∼ Rayleigh(σi ), where σi2 = ΞΞH (i, i). Using the union bound, we have Ped2 ≤ ≤
(47) (48)
|f1 U1 + f2 V1 |
1− I(U1 ; Yer1 ) ≥ log P + o(log P ). 2+
(46)
≥ H(U1 ) − 1 − Pe1 log |U1 |
inf
U1 ,V1 ∈{a(−2Q,2Q)Z }
Substituting (54) in (51) gives Pe1 ≤ exp(−µP ), where µ = γ 2 |f1 |2 2− 4||b||2 . Using (34) and (49), we have
e r forms a where (45) follows since Ut → Xt → Yr → Y Markov chain. The term I(U1 ; Yer1 ) can be bounded as I(U1 ; Yer1 ) = H(U1 ) − H(U1 |Yer1 )
(52)
d X
d n o X n ao ˆi 6= Ui ≤ Pr U Pr |Zˆri | ≥ 2 i=2 i=2
(60)
0 N −1 exp(−µ0 P ), 2
(61)
where µ0 =
γ2 , 2 8σmax
σmax = max σi , and 0 = i
3 2+ .
Thus,
e N |U1 Yer ) ≥ 1 − (N − 1) log P + o(log P ). (62) I(Udt2 ; Y r2 1 2+ Thus, substituting (55) and (62) in (46) gives us
2 A set of complex numbers {c , · · · , c } are rationally independent if 1 L there is no setPof rational numbers {r1 , · · · , rL }, rather than the all zeros L set, such that i=1 ri ci = 0 [10].
630
I(Xt ; Yr ) ≥
1− N log P + o(log P ), 2+
(63)
and using (4), (27), (39), and (63) results in Ds ≥ 1− 2+ N . Since > 0 is arbitrarily small, we can achieve s.d.o.f. of N2 . D. Case 4: N < K ≤ 2N and K is even The achievable scheme for this case involves transmitting K 2 Gaussian information and K 2 Gaussian jamming streams. The CJ sends K−N out of its K 2 streams over the null space of Hc , N (Hc ), leaving only N − K 2 streams visible to the legitimate receiver. At high SNR, the legitimate receiver can decode K the K 2 information and the N − 2 jamming streams. For K notational simplicity, let g = N − 2 . The transmitted signals are given by (28), with Pt = G−1 t Gc Jc , Jc = [JI Jn ], JI = [Ig 0g×K−g ]T , Jn = [n1 · · · nK−N ], where {n1 , · · · , nK−N } span N (Hc ), Ut , Vc ∼ CN (0, P¯ I K ), P¯ = α10 P , and α0 is 2 chosen to satisfy the power constraints. −1 Let A = Ht Gt Gc . The received signals are given by h iT ¯ UT V g T Yr = B + Zr (64) t c1 Ye = Gc Jc (Ut + Vc ) + Ze , (65) ¯ = AJI AJn Hc JI ∈ CN ×N can be written as where B JI Jn 0K×g ¯ B = A Hc . (66) 0K×g 0K×K−N JI By extending the proof in the Appendix in [8], we can show ¯ is a.s. full rank. Thus, we have I(Xt ; Yr ) = K log P + that B 2 o(log P ). Similar to (32), we have I(Xt ; Ye ) ≤ K 2 . Thus, the s.d.o.f. is lower bounded as Ds ≥ K 2 . E. Case 5: N < K ≤ 2N and K is odd The transmitter sends K+1 structured information and the 2 K+1 CJ sends 2 structured jamming streams. The CJ sends K − N out of its streams over N (Hc ). The legitimate receiver projects its received signal over a direction orthogonal to all but one information and one jamming streams, decodes these two streams from the projection, and subtracts their effect from its received signal to decode the other information streams. Let m = K+1 and l = N − K−1 2 2 . The transmitted signals are given by (28), with Pt = G−1 t Gc Jc , Jc = [JI Jn ], JI = [Il 0l×K−l ]T , Jn is defined as in the previous subsection, and Ut , Vc ∈ Zm [j], a, Q are defined as in Section V-C. Similar to going from (36) to (39), we have I(Xt ; Ye ) ≤ K. The received signal at the legitimate receiver is given by iT h ¯ c UT V l T Yr = A H + Zr , (67) t c1 ¯ where A = Ht G−1 t Gc Jc , Hc = Hc JI . The receiver chooses b ⊥ span {a2 , · · · , am , hc2 , · · · , hcl } and multiplies its ree r = [Yer (Y e N )T ]T . ceived signal by D in (41) to obtain Y r2 1 Using similar analysis as in Section V-C, we have eN e I(Xt ; Yr ) ≥ I(U1 ; Yer1 ) + I(Um t2 ; Yr2 |U1 Yr1 ) 1− I(U1 ; Yer1 ) ≥ log P + o(log P ) 2+ m ¯ 00 eN e I(Um t2 ; Yr2 |U1 Yr1 ) ≥ I(Ut2 ; Yr ),
(68) (69) (70)
¯ 00 , in (70), is given by where > 0 is arbitrarily small. Y r h iT T T l ¯ 00 = B b Um e r, Y + BZ (71) V r t2 c2 ˜c · · · h ˜ c ]. Using the argument in b = [˜ ˜m h where B a2 · · · a 2 l b b 00 = B b −1 Y ¯ 00 . Thus, Section V-D, B is a.s. full rank. Define Y r r m b 00 eN e I(Um (72) t2 ; Yr2 |U1 Yr1 ) ≥ I(Ut2 ; Yr ) 1− (K − 1) log P + o(log P ), (73) ≥ 2+
Thus, Ds ≥ 1− 2+ K. Since is arbitrarily small, the s.d.o.f. of K is achievable, which completes the proof for Theorem 1. 2 VI. C ONCLUSION We have characterized the s.d.o.f. for the multiantenna Gaussian wiretap channel with a K-antenna cooperative jammer (CJ) and N antennas at each of its nodes for all possible values of K. We have shown that when the s.d.o.f. of the channel is integer-valued, the s.d.o.f. can be achieved by a scheme which involves linear precoding, Gaussian signaling both for transmission and jamming, and linear receiver processing. In contrast, we have proved that, when the s.d.o.f. is non-integer, a scheme which employs structured signaling along with joint signal space and signal scale alignment achieves the s.d.o.f. of the channel. The converse was proved by allowing for cooperation between the transmitter and CJ for a certain range of K, and by incorporating both the secrecy and reliability constraints, for the other values of K. Although we identified its prelog factor, the secrecy capacity of this model remains open and deserves further attention. R EFERENCES [1] A. D. Wyner, “The wire-tap channel,” Bell System Technical Journal, vol. 54, no. 8, pp. 1355–1387, 1975. [2] E. Tekin and A. Yener, “Achievable rates for the general Gaussian multiple access wire-tap channel with collective secrecy,” in 44th Annual Allerton Conf. On Communication, Control, and Computing, Sep. 2006. [3] ——, “The general Gaussian multiple-access and two-way wiretap channels: Achievable rates and cooperative jamming,” IEEE Trans. Info. Theory, vol. 54, no. 6, pp. 2735–2751, 2008. [4] X. Tang, R. Liu, P. Spasojevic, and H. V. Poor, “Interference assisted secret communication,” IEEE Trans. Info. Theory, vol. 57, no. 5, pp. 3153–3167, 2011. [5] X. He and A. Yener, “Providing secrecy with structured codes: Twouser Gaussian channels,” IEEE Trans. Info. Theory, vol. 60, no. 4, pp. 2121–2138, 2014. [6] J. Xie and S. Ulukus, “Secure degrees of freedom of one-hop wireless networks,” IEEE Trans. Info. Theory, vol. 60, no. 6, pp. 3359–3378, 2014. [7] ——, “Secure degrees of freedom of K-user Gaussian interference channels: A unified view,” Submitted to IEEE Trans. Info. Theory, 2013, arXiv preprint arXiv:1305.7214. [8] M. Nafea and A. Yener, “How many antennas does a cooperative jammer need for achieving the degrees of freedom of multiple antenna Gaussian channels in the presence of an eavesdropper,” in 51st Annual Allerton Conference on Communication, Control, and Computing, Oct. 2013. [9] A. Khisti and G. W. Wornell, “Secure transmission with multiple antennas-Part II: The MIMOME wiretap channel,” IEEE Trans. Info. Theory, vol. 56, no. 11, pp. 5515–5532, 2010. [10] M. A. Maddah-Ali, “On the degrees of freedom of the compound MIMO broadcast channels with finite states,” 2009, arXiv preprint arXiv:0909.5006. [11] D. Kleinbock, “Baker-sprindzhuk conjectures for complex analytic manifolds,” 2002, arXiv preprint math/0210369.
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Abstract—A multiple antenna Gaussian wiretap channel with a multiantenna cooperative jammer (CJ) is considered and the secure degrees of freedom (s.d.o.f.), with N antennas at the sender, receiver, and eavesdropper, is derived for all possible values of the number of antennas at the cooperative jammer, K. In particular, the upper and lower bounds for the s.d.o.f. are provided for different ranges of K and shown to coincide. Gaussian signaling both for transmission and jamming is shown to be sufficient to achieve the s.d.o.f. of the channel, when the s.d.o.f. is integer-valued. By contrast, when the channel has a non-integer s.d.o.f., structured signaling and joint signal space and signal scale alignment are employed to achieve the s.d.o.f.
I. I NTRODUCTION Information theoretic secrecy [1] guarantees secure communication in the presence of an eavesdropper. Information theoretic secrecy of multiterminal and multiantenna channels has been studied extensively, e.g., [2]–[9]. In particular, the Gaussian wiretap channel (WTC) with a cooperative jammer (CJ) was studied in [2]–[8]. A CJ can improve the secrecy rate and even the prelog factor of the secrecy rate, i.e., the secure degrees of freedom (s.d.o.f.) [5]. Relying on cooperative jamming and structured signaling, [6], [7] identified the s.d.o.f. for several single-antenna WTC models. In this paper, we extend [6] to a multiantenna secure communication scenario. We focus on a multiantenna Gaussian WTC with a Kantenna CJ and N antennas at each of the transmitter, receiver, and wiretapper. For 0 ≤ K ≤ 2N , we characterize the s.d.o.f. arriving at the s.d.o.f. of N at K = 2N , concluding that increasing K over 2N cannot improve the s.d.o.f. We derive an upper bound on the s.d.o.f. which allows for cooperation between the transmitter and CJ, and show that this bound is tight for 0 ≤ K ≤ N2 . Next, for N ≤ K ≤ 2N , we derive another upper bound which incorporates both the secrecy and reliability constraints. We use this upper bound for K = N to N upper bound the s.d.o.f. N for all 2 < K < N . We show that increasing K from 2 to N does not increase the s.d.o.f. Next, we divide 0 ≤ K ≤ 2N into five different ranges and propose an achievable scheme which meets with the derived upper bound for each range. Whenever the s.d.o.f. is integervalued, we show that Gaussian signaling at the transmitter and the CJ is sufficient to achieve the s.d.o.f. of the channel. By contrast, when the s.d.o.f. is not an integer, structured signaling along with joint signal space and signal scale alignment are This work was supported by NSF Grants CCF 09-64362, 13-19338 and CNS 13-14719.
978-1-4799-5999-0/14/$31.00 ©2014 IEEE
Fig. 1.
MIMO Gaussian WTC with a multiantenna CJ.
needed. When K is larger than N , a linear precoder is utilized at the CJ so that some jamming signals are transmitted over directions invisible to the legitimate receiver. Overall, this study settles the secrecy capacity of the N × N × N MIMO wiretap channel in the high SNR, when a multiantenna CJ is available as a helper. As compared to the single antenna counterpart that uses real interference alignment with one dimensional lattices, here we need to use a variety of spatial and signal scale alignment techniques in concert to coordinate multiple antenna transmissions. Notation: For matrix A, ||A|| denotes its induced norm. For vector V, Vij = [Vi · · · Vj ]T , where 1 ≤ i < j ≤ n, and ||V|| denotes Euclidean norm. 0m×n denotes an m × n matrix of zeros. The set of integers {−Q, · · · , Q} is denoted by (−Q, Q)Z . Z[j] denotes the set of complex integers. II. C HANNEL M ODEL AND D EFINITIONS We consider a multiantenna Gaussian WTC composed of a transmitter, a receiver, an eavesdropper each with N antennas, and a K-antenna CJ, see Fig.1. The received signals at the receiver and eavesdropper at the nth channel use are given by Yr (n) = Ht Xt (n) + Hc Xc (n) + Zr (n)
(1)
Ye (n) = Gt Xt (n) + Gc Xc (n) + Ze (n),
(2)
where Xt (n), Xc (n) are the transmitted signals from the transmitter and CJ, respectively. Ht , Gt ∈ CN ×N are the transmitter’s channel matrices to the legitimate receiver and to the eavesdropper, and Hc , Gc ∈ CN ×K are the channel matrices from the CJ to the legitimate receiver and eavesdropper. The channel gains are static, and complex-valued. Zr (n) and Ze (n) denote the complex Gaussian noise at the nth channel use, i.e., Zr (n), Ze (n) ∼ CN (0, IN ), are independent from one another and both are independent and identically
627
1 distributed (i.i.d.) across n. The at the power Hconstraints transmitter and CJ are E XH X , E X X ≤ P . t c t c Let Xnt = [Xt (1) · · · Xt (n)]. Xnc , Yrn , Yen , Znr , Zne are defined similarly. The transmitter intends to send a secret message W ∈ W to the legitimate receiver in the presence of the eavesdropper. W is mapped into the transmitted signal Xnt ∈ Xtn by using a stochastic encoder f : W 7→ Xtn at the transmitter. The receiver forms an estimate of W , denoted by ˆ . Secrecy rate Rs is achievable if for any > 0, there exists W a channel code, 2nRs , n , such that n o ˆ 6= W ≤ ; 1 H(W |Yen ) ≥ 1 H(W ) − . (3) Pr W n n The achievable s.d.o.f. for a given secrecy rate, Rs , is
Rs Ds = lim . P →∞ log P
(4)
The CJ transmits Xnc ∈ Xcn . The jamming signal Xnc is not meant to convey a message. There is no common randomness between the transmitter and the CJ. III. M AIN R ESULT Theorem 1 The s.d.o.f. of the multiantenna Gaussian WTC with a K-antenna CJ and N antennas at each of its nodes is for 0 ≤ K ≤ N2 K, (5) Ds = N2 , for N2 < K ≤ N K , for N < K ≤ 2N. 2
The secrecy rate for the original channel, Rs , is upper bounded ¯ s . Using (4), the s.d.o.f., Ds , is upper bounded by K. by R B. N ≤ K ≤ 2N Here, we extend the converse proof in [6] to the multiantenna channel. Let φi , i = 1, · · · , 6, denote constants that do not depend on the power P . Rs can be upper bounded as nRs = H(W ) ≤ H(W |Yen ) + n − H(W |Yrn ) + nδ ≤ I(W ; Yrn |Yen ) + nφ1 ≤ h(Yrn |Yen ) − h(Yrn |W Yen Xnt Xnc ) + nφ1 =
A. 0 ≤ K ≤ N We allow for cooperation between the transmitter and CJ, obtain, in effect a multiantenna Gaussian WTC with N + Kantenna transmitter, N -antenna receiver, and N -antenna eavesdropper. The secrecy rate of this channel is bounded as [9] P ¯ ¯]¯H ¯ Rs ≤ log det IN + HG H + o(log P ), (6) K ¯ G ¯ ∈ CN ×N +K are the channel matrices from the where H, ¯] combined transmitter to the receiver and eavesdropper, and G ¯ is the projection matrix onto N (G). We also have [9] 0N −K×N −K 0N −K×K ]¯H ¯ ¯ HG H = Ψ ΨH , (7) 0K×N −K Ω where Ψ is a unitary matrix and Ω is a non-singular matrix. By substituting (7) in (6), it can be easily shown that ¯ s ≤ K log P + o(log P ). R 1 Throughout
−
h(Yen )
−
h(Znr )
(11)
+ nφ1 .
(12)
(9) follows from (3) and Fano’s inequality; φ1 = + δ. ˜ t = Xt + Z ˜ t, X ˜ c = Xc + Z ˜ c , where Z ˜t ∼ Define X ˜ CN (0, Kt ) and Zc ∼ CN (0, Kc ). Kt , Kc are chosen as Kt = ρ2 IN , Kc = β 2 IK , where 0 < ρ ≤ ||G1H || and t ˜ t is independent from Z ˜ c and both are in0 < β ≤ ||H1H || . Z c ˜ n, Z ˜ n are i.i.d. sequences dependent from {Xt , Xc , Zr , Ze }. Z t c ˜ ˜ ˜ ˜ t − Hc Z ˜ c + Zr and of Zt , Zc , respectively. Let Z1 = −Ht Z ˜ ˜ ˜ ˜ Z2 = −Gt Zt − Gc Zc + Ze , where Z1 ∼ CN (0, ΣZ˜ 1 ), ˜ 2 ∼ CN (0, Σ ˜ ), Σ ˜ = Ht Kt HH + Hc Kc HH + IN , Z t c Z2 Z1 H ˜n ˜n and ΣZ˜ 2 = Gt Kt GH t + Gc Kc Gc + IN . Z1 , Z2 are i.i.d. ˜ 1, Z ˜ 2 . Thus, using (12), we have sequences of Z ˜ nX ˜ n ) − h(X ˜ nX ˜ n |Yn Yn ) nRs ≤ h(Yrn Yen X t c t c r e − h(Yen ) + nφ2 (13) n n n n n n n n ˜ X ˜ ) + h(Y |X ˜ X ˜ ) + h(Y |X ˜ X ˜ ) ≤ h(X t c r t c e t c n n n n n n n ˜ X ˜ |Y Y X X ) − h(Y ) + nφ2 (14) − h(X t c r e t c e ˜ n ) + h(X ˜ n ) + h(Z ˜ n |X ˜ nX ˜ n ) + h(Z ˜ n |X ˜ nX ˜ n) ≤ h(X
Theorem 1 provides a complete characterization for the s.d.o.f. of the channel. The s.d.o.f. for K = 2N is equal to N , that is equal to the d.o.f. of the N -antenna Gaussian channel with no secrecy constraint. Thus, the s.d.o.f. can not be increased by increasing K over 2N . Interestingly, Theorem 1 shows that the s.d.o.f. N of the channel is not increased by increasing K from to N . In Sections IV and V, we provide the converse 2 and achievability proofs for Theorem 1. IV. C ONVERSE
h(Yrn Yen )
(9) (10)
t
c
1
t
c
2
n ˜ nZ ˜n − h(Z t c ) − h(Ye ) + nφ2 ˜ n ) + h(X ˜ n ) − h(Yn ) + nφ3 . ≤ h(X t c e
t
c
(15) (16)
Using infinite divisibility of Gaussian distribution, a ˜t + stochastically equivalent form of Ze is Z0e = Gt Z H ˜ ˜ Ze . Ze ∼ CN (0, IN − Gt Kt Gt ) is independent from ˜ t, Z ˜ c , Xt , Xc , Zr }. Thus, a stochastically equivalent form {Z n n ˜ n +Gc Xn + Z ˜ n . Since Gt X ˜ n , Gc Xn + of Ye is Ye0 = Gt X t c e t c n ˜ Ze are independent, we have n ˜ n) h(Yen ) = h(Ye0 ) ≥ h(Gt X t ˜ n ) + n log det(Gt ). = h(X t
(17) (18)
Substituting (18) in (16) gives us ˜ n ) + h(X ˜ n |X ˜ n ) + nφ4 , nRs ≤ h(X c2 c1 c2
(19)
˜ c = [X ˜ c = [X ˜c · · · X ˜ c ]T , X ˜c ˜ c ]T , and where X ···X 1 1 2 N N +1 K ˜ ˜ Xci = Xci + Zci . Let hck be the kth column vector of Hc = [Hc1 Hc2 ], Hc1 = [hc1 · · · hcN ] and Hc2 = [hcN +1 · · · hcK ]. In order to reliably transmit the message W , we must have
(8)
nRs ≤ I(Xnt ; Yrn ) = h(Yrn ) − h(Hc Xnc + Znr ) ˜ n + Hc X ˜n ) ≤ h(Yn ) − h(Hc X ≤
the paper, we omit index n whenever possible.
628
r h(Yrn )
c1 c2 2 ˜ n |Hc X ˜n ) h(Hc1 X c1 c2 2 1
−
(20) (21) (22)
˜n ) ˜ n |X ≤ h(Yrn ) − h(Hc1 X c1 c2 ˜ n ) − n log det(Hc ), ˜ n |X = h(Yrn ) − h(X c1 c2 1
(23) (24)
where (21) follows similar to (17), and (22) follows since for correlated X and Y, h(X + Y) ≥ h(X + Y|Y) = h(X|Y). Combining (19) and (24), we get n N K X 1X X ˜ c (i)) + nφ5 . nRs ≤ h(Yrk (i)) + h(X k 2 i=1 k=1
k=N +1
(25) Using Cauchy-Schwarz inequality and the power constraints, we can show that the variance of Yrk (i) is bounded as T Var{Yrk (i)} ≤ 1 + h2 P , where h2 = max (||hrtk ||2 + T ||hrck ||2 ),
k
K −N N log(1 + h2 P ) + log(β 2 + P ) + φ6 , (26) 2 2
and the s.d.o.f. is upper bounded as Ds ≤
K 2 .
C. Obtaining the upper bound for all K We use the upper bound obtained in Section IV-A for 0 ≤ K ≤ N2 , and the upper bound obtained in Section IV-B for N ≤ K ≤ 2N . By comparing the two, it is evident that the upper bound from Section IV-A is greater than N2 for N 2 < K ≤ N . Since we know, from Section IV-B, that at K = N , the upper bound is N2 , we can use N2 as the upper bound for N2 < K ≤ N . Combining these statements, we get (5). Next, we shall see the achievability of (5). V. ACHIEVABLE S CHEMES We divide the range 0 ≤ K ≤ 2N into five cases and propose an achievable scheme for each case. For all the achievable schemes, we have the n-letter signals, Xnt and Xnc , as i.i.d. sequences. Since Xnc is independent from Xnt , we have in effect a memoryless WTC, and the following secrecy rate is achievable by stochastic encoding [1]: Rs = [I(Xt ; Yr ) − I(Xt ; Ye )]+ . A. Case 1: 0 ≤ K ≤
(27)
N 2
The transmitter sends K independent Gaussian information streams and the CJ sends K independent Gaussian jamming streams. Since 2K ≤ N , the legitimate receiver can decode all the information and jamming streams at high SNR. The transmitter chooses a precoder, Pt , which aligns its information streams over the jamming streams at the eavesdropper. The signals transmitted by the transmitter and the CJ are Xt = Pt Ut ,
X c = Jc V c ,
T
+ Zr
(29)
(30) −1 We lower bound (27) as follows. First, Ht Gt Gc Hc is almost surely (a.s.) full column-rank. Thus, I(Xt ; Yr ) ≥ K log P + o(log P ).
(31)
Next, we upper bound the term I(Xt ; Ye ) as follows: I(Xt ; Ye ) = log
det(IK + 2P¯ GH c Gc ) ≤ K. H ¯ det(IK + P Gc Gc )
(32)
By substituting (31) and (32) in (27), we have Rs ≥ K log P + o(log P ) − K.
(33)
Hence, the achievable s.d.o.f. is lower bounded as Ds ≥ K. B. Case 2:
N 2
< K ≤ N and N is even
The s.d.o.f. is upper bounded by N2 for all N2 < K ≤ N . When N is even, the achievable scheme for K = N2 can be used to achieve the s.d.o.f. of the channel for all N2 < K ≤ N . The transmitted signals are given by (28), with Jc = ¯ ). [I N 0 N ×K− N ]T , Pt = G−1 t Gc Jc , Ut , Vc ∼ CN (0, P I N 2 2 2 2 Using the same analysis as in the previous case, the achievable N N s.d.o.f. is 2 for any 2 < K ≤ N , where N is even. C. Case 3:
N 2
< K ≤ N and N is odd
For this case, we utilize structured signaling both at the transmitter and the CJ. In particular, we propose to use joint signal space alignment and the complex field equivalent of real interference alignment [10], [11]. The transmitter and CJ send N2+1 streams each. The transmitter aligns its information streams over the jamming at the eavesdropper. The legitimate receiver projects its received signal over a direction orthogonal to all but one information and one jamming streams, decodes these two streams from the projection, and subtracts their effect from its received signal, leaving N − 1 spatial dimensions for the other streams. For notational simplicity, let d = N2+1 . The transmitted signals are given by (28) with Jc = T [Id 0d×K−d ]T , Pt = G−1 t Gc Jc , Ut = [U1 · · · Ud ] , Vc = T [V1 · · · Vd ] , Ui = UiRe + jUiIm , Vi = ViRe + jViIm , for i = 2, · · · , d, and U1 , V1 , {UiRe }di=2 , {UiIm }di=2 , {ViRe }di=2 , {ViIm }di=2 are i.i.d. uniform over the set {a(−Q, Q)Z }. The values for a and the integer Q are chosen as 1−
Q = P 2+ − ν,
3
a = γP 2(2+) ,
(34)
where > 0 can be arbitrarily small, and ν, γ are constants. ˜ c = Gc Jc . The received signal at the eavesdropper is Let G ˜ c (Ut + Vc ) + Ze . Ye = G
(28)
where Ut , Vc ∼ CN (0, P¯ IK ). Ut and Vc are the information and jamming streams, respectively. Pt = G−1 t Gc = [pt1 · · · ptK ] , and Jc = IK . P¯ = α1 P , where α = PK max{K, i=1 ||pti ||2 } to satisfy the power constraints. The
VcT
Ye = Gc (Ut + Vc ) + Ze .
hrtk , hrck
and are the kth row vectors of Ht and Hc . Thus, h(Yrk (i)) ≤ log 2πe(1 + h2 P ). Similarly, we can ˜ c (i)) ≤ log 2πe(β 2 + P ). Thus, show that h(X k Rs ≤
received signals are expressed as Yr = Ht G−1 Hc UTt t Gc
(35)
We upper bound I(Xt ; Ye ) as follows:
629
I(Xt ; Ye ) ≤ I(Xt ; Ye , Ze ) = I(Xt ; Ye |Ze ) ˜ c (Ut + Vc ) − H G ˜ c Vc =H G
(36) (37)
= H(Ut + Vc ) − H(Vc ) 4Q + 1 ≤ N, ≤ (2d − 1) log 2Q + 1
(38) (39)
˜ c (Ut + where (38) follows since the mappings Ut + Vc 7→ G ˜ ˜ c are Vc ), Vc 7→ Gc Vc are invertible, since the entries of G 2 rationally independent . Next, we derive a lower bound for I(Xt ; Yr ). The received signal at the legitimate receiver is ¯ c Vc + Zr , Yr = AUt + H
(40)
¯ where A = Ht G−1 t Gc Jc and Hc = Hc Jc . Let ai and ¯ c , respectively. hci be the ith column vectors of A and H N The legitimate receiver chooses b ∈ C such that b ⊥ span {a2 , · · · , ad , hc2 , · · · , hcd } and multiplies its received signal by the decoding matrix, bH D= , (41) 0N −1×1 IN −1
inequality. Since the mapping (U1 , V1 ) 7→ f1 U1 + f2 V1 is invertible, the only source for error is the Gaussian noise Z 0 . n o ˆ1 , Vˆ1 ) 6= (U1 , V1 ) Pe1 ≤ Pr (U (50) 2 −dmin dmin , (51) = exp ≤ Pr |Z 0 | ≥ 2 4||b||2 √ ) and dmin is the minimum where |Z 0 | ∼ Rayleigh( ||b|| 2 distance between points in the constellation R1 , which can be lower bounded using the following lemma [10], [11].
Lemma 1 For almost all z ∈ Cn and for all > 0, −( n−1+ ) 2 |p + z.q| > max qi , i
holds for all q ∈ Zn , p ∈ Z except for finitely many of them. Thus, for almost all channel gains, dmin is dmin =
e r = DYr = [Yer (Y e N )T ]T , where to obtain Y r2 1 Yer1 = f1 U1 + f2 V1 + Z 0 e N = AU e t+H e c Vc + ZN , Y r2 r2
(42)
e r) I(Xt ; Yr ) ≥ I(Ut ; Y e N |U1 Yer ), ≥ I(U1 ; Yer1 ) + I(Udt2 ; Y r2 1
≥
(43)
e = [˜ ˜d ], f1 = bH a1 , f2 = bH hc1 , Z 0 = bH Zr , A a1 · · · a N N ˜c · · · h ˜ c ], a ˜ e c = [h ˜ H = a , and h = h . i ci ci2 1 d i2 The legitimate receiver uses Yer1 to decode U1 and V1 . Since f1 , f2 are a.s. rationally independent, the mapping (U1 , V1 ) 7→ f1 U1 + f2 V1 is bijective (i.e., invertible) [10]. The legitimate receiver employs a hard decision decoder which maps Yer1 ∈ Y˜r1 to the nearest point in the constellation R1 = f1 U1 +f2 V1 , where U1 , V1 = {a(−Q, Q)Z }. Then, the legitimate receiver passes the output of the hard decision decoder through the bijective map f1 U1 + f2 V1 7→ (U1 , V1 ) to decode both U1 , V1 , e N to obtain and subtracts their effect from Y r2 h iT ¯ r = B UdT VdT Y + ZN (44) r2 . t2 c2 h i ˜c · · · h ˜ c ∈ CN −1×N −1 is a.s. full rank ˜2 · · · a ˜d h B= a 2 d due to the random generation assumption on the channel gains. ¯ r. Finally, by zero forcing, the receiver obtains Udt2 from Y The term I(Xt ; Yr ) is lower bounded as follows: (45)
= (1 − Pe1 ) log(2Q + 1) − 1,
(49)
ˆ1 6= U1 }, and (48) follows from Fano’s where Pe1 = Pr{U
a|f1 | − 2 P 2. ≥ γ|f1 |2 (2Q) 2
(53) (54)
(55)
e N |U1 Yer ). Define Next, we lower bound the term I(Udt2 ; Y r2 1 h iT ˜ c bH ; Y ¯ 0 = B UdT VdT e = [0N −1×1 IN −1 ] − 1 h B + r t2 c2 1 f2 i h T T e r , and P d = ¯ 0 = Ud VdT + B−1 BZ e r; Y b 0 = B−1 Y BZ c2 e2 t2 r r d d ˆ 6= U }. Thus, we have Pr{U t2 t2 e N |U1 Yer ) = I Ud ; Y e N U1 , f2 V1 + Z 0 I(Udt2 ; Y (56) r2 t2 r2 1 ¯ 0 |f2 V1 + Z) = I(Udt2 ; Y (57) r d ¯0 d b0 ≥ I(Ut2 ; Yr ) ≥ I(Ut2 ; Yr ) (58) d N −1 ≥ 1 − Pe2 log(2Q + 1) − 1. (59) b r = ΞZr = [Zˆr · · · Zˆr ]T , where Ξ = B−1 B. e Thus, Let Z 2 N H b ˆ Zr ∼ CN (0, ΞΞ ) and |Zri | ∼ Rayleigh(σi ), where σi2 = ΞΞH (i, i). Using the union bound, we have Ped2 ≤ ≤
(47) (48)
|f1 U1 + f2 V1 |
1− I(U1 ; Yer1 ) ≥ log P + o(log P ). 2+
(46)
≥ H(U1 ) − 1 − Pe1 log |U1 |
inf
U1 ,V1 ∈{a(−2Q,2Q)Z }
Substituting (54) in (51) gives Pe1 ≤ exp(−µP ), where µ = γ 2 |f1 |2 2− 4||b||2 . Using (34) and (49), we have
e r forms a where (45) follows since Ut → Xt → Yr → Y Markov chain. The term I(U1 ; Yer1 ) can be bounded as I(U1 ; Yer1 ) = H(U1 ) − H(U1 |Yer1 )
(52)
d X
d n o X n ao ˆi 6= Ui ≤ Pr U Pr |Zˆri | ≥ 2 i=2 i=2
(60)
0 N −1 exp(−µ0 P ), 2
(61)
where µ0 =
γ2 , 2 8σmax
σmax = max σi , and 0 = i
3 2+ .
Thus,
e N |U1 Yer ) ≥ 1 − (N − 1) log P + o(log P ). (62) I(Udt2 ; Y r2 1 2+ Thus, substituting (55) and (62) in (46) gives us
2 A set of complex numbers {c , · · · , c } are rationally independent if 1 L there is no setPof rational numbers {r1 , · · · , rL }, rather than the all zeros L set, such that i=1 ri ci = 0 [10].
630
I(Xt ; Yr ) ≥
1− N log P + o(log P ), 2+
(63)
and using (4), (27), (39), and (63) results in Ds ≥ 1− 2+ N . Since > 0 is arbitrarily small, we can achieve s.d.o.f. of N2 . D. Case 4: N < K ≤ 2N and K is even The achievable scheme for this case involves transmitting K 2 Gaussian information and K 2 Gaussian jamming streams. The CJ sends K−N out of its K 2 streams over the null space of Hc , N (Hc ), leaving only N − K 2 streams visible to the legitimate receiver. At high SNR, the legitimate receiver can decode K the K 2 information and the N − 2 jamming streams. For K notational simplicity, let g = N − 2 . The transmitted signals are given by (28), with Pt = G−1 t Gc Jc , Jc = [JI Jn ], JI = [Ig 0g×K−g ]T , Jn = [n1 · · · nK−N ], where {n1 , · · · , nK−N } span N (Hc ), Ut , Vc ∼ CN (0, P¯ I K ), P¯ = α10 P , and α0 is 2 chosen to satisfy the power constraints. −1 Let A = Ht Gt Gc . The received signals are given by h iT ¯ UT V g T Yr = B + Zr (64) t c1 Ye = Gc Jc (Ut + Vc ) + Ze , (65) ¯ = AJI AJn Hc JI ∈ CN ×N can be written as where B JI Jn 0K×g ¯ B = A Hc . (66) 0K×g 0K×K−N JI By extending the proof in the Appendix in [8], we can show ¯ is a.s. full rank. Thus, we have I(Xt ; Yr ) = K log P + that B 2 o(log P ). Similar to (32), we have I(Xt ; Ye ) ≤ K 2 . Thus, the s.d.o.f. is lower bounded as Ds ≥ K 2 . E. Case 5: N < K ≤ 2N and K is odd The transmitter sends K+1 structured information and the 2 K+1 CJ sends 2 structured jamming streams. The CJ sends K − N out of its streams over N (Hc ). The legitimate receiver projects its received signal over a direction orthogonal to all but one information and one jamming streams, decodes these two streams from the projection, and subtracts their effect from its received signal to decode the other information streams. Let m = K+1 and l = N − K−1 2 2 . The transmitted signals are given by (28), with Pt = G−1 t Gc Jc , Jc = [JI Jn ], JI = [Il 0l×K−l ]T , Jn is defined as in the previous subsection, and Ut , Vc ∈ Zm [j], a, Q are defined as in Section V-C. Similar to going from (36) to (39), we have I(Xt ; Ye ) ≤ K. The received signal at the legitimate receiver is given by iT h ¯ c UT V l T Yr = A H + Zr , (67) t c1 ¯ where A = Ht G−1 t Gc Jc , Hc = Hc JI . The receiver chooses b ⊥ span {a2 , · · · , am , hc2 , · · · , hcl } and multiplies its ree r = [Yer (Y e N )T ]T . ceived signal by D in (41) to obtain Y r2 1 Using similar analysis as in Section V-C, we have eN e I(Xt ; Yr ) ≥ I(U1 ; Yer1 ) + I(Um t2 ; Yr2 |U1 Yr1 ) 1− I(U1 ; Yer1 ) ≥ log P + o(log P ) 2+ m ¯ 00 eN e I(Um t2 ; Yr2 |U1 Yr1 ) ≥ I(Ut2 ; Yr ),
(68) (69) (70)
¯ 00 , in (70), is given by where > 0 is arbitrarily small. Y r h iT T T l ¯ 00 = B b Um e r, Y + BZ (71) V r t2 c2 ˜c · · · h ˜ c ]. Using the argument in b = [˜ ˜m h where B a2 · · · a 2 l b b 00 = B b −1 Y ¯ 00 . Thus, Section V-D, B is a.s. full rank. Define Y r r m b 00 eN e I(Um (72) t2 ; Yr2 |U1 Yr1 ) ≥ I(Ut2 ; Yr ) 1− (K − 1) log P + o(log P ), (73) ≥ 2+
Thus, Ds ≥ 1− 2+ K. Since is arbitrarily small, the s.d.o.f. of K is achievable, which completes the proof for Theorem 1. 2 VI. C ONCLUSION We have characterized the s.d.o.f. for the multiantenna Gaussian wiretap channel with a K-antenna cooperative jammer (CJ) and N antennas at each of its nodes for all possible values of K. We have shown that when the s.d.o.f. of the channel is integer-valued, the s.d.o.f. can be achieved by a scheme which involves linear precoding, Gaussian signaling both for transmission and jamming, and linear receiver processing. In contrast, we have proved that, when the s.d.o.f. is non-integer, a scheme which employs structured signaling along with joint signal space and signal scale alignment achieves the s.d.o.f. of the channel. The converse was proved by allowing for cooperation between the transmitter and CJ for a certain range of K, and by incorporating both the secrecy and reliability constraints, for the other values of K. Although we identified its prelog factor, the secrecy capacity of this model remains open and deserves further attention. R EFERENCES [1] A. D. Wyner, “The wire-tap channel,” Bell System Technical Journal, vol. 54, no. 8, pp. 1355–1387, 1975. [2] E. Tekin and A. Yener, “Achievable rates for the general Gaussian multiple access wire-tap channel with collective secrecy,” in 44th Annual Allerton Conf. On Communication, Control, and Computing, Sep. 2006. [3] ——, “The general Gaussian multiple-access and two-way wiretap channels: Achievable rates and cooperative jamming,” IEEE Trans. Info. Theory, vol. 54, no. 6, pp. 2735–2751, 2008. [4] X. Tang, R. Liu, P. Spasojevic, and H. V. Poor, “Interference assisted secret communication,” IEEE Trans. Info. Theory, vol. 57, no. 5, pp. 3153–3167, 2011. [5] X. He and A. Yener, “Providing secrecy with structured codes: Twouser Gaussian channels,” IEEE Trans. Info. Theory, vol. 60, no. 4, pp. 2121–2138, 2014. [6] J. Xie and S. Ulukus, “Secure degrees of freedom of one-hop wireless networks,” IEEE Trans. Info. Theory, vol. 60, no. 6, pp. 3359–3378, 2014. [7] ——, “Secure degrees of freedom of K-user Gaussian interference channels: A unified view,” Submitted to IEEE Trans. Info. Theory, 2013, arXiv preprint arXiv:1305.7214. [8] M. Nafea and A. Yener, “How many antennas does a cooperative jammer need for achieving the degrees of freedom of multiple antenna Gaussian channels in the presence of an eavesdropper,” in 51st Annual Allerton Conference on Communication, Control, and Computing, Oct. 2013. [9] A. Khisti and G. W. Wornell, “Secure transmission with multiple antennas-Part II: The MIMOME wiretap channel,” IEEE Trans. Info. Theory, vol. 56, no. 11, pp. 5515–5532, 2010. [10] M. A. Maddah-Ali, “On the degrees of freedom of the compound MIMO broadcast channels with finite states,” 2009, arXiv preprint arXiv:0909.5006. [11] D. Kleinbock, “Baker-sprindzhuk conjectures for complex analytic manifolds,” 2002, arXiv preprint math/0210369.
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