Guaranteeing Positive Secrecy Capacity with Finite-Rate Feedback using Artificial Noise
aa r X i v : . [ c s . I T ] N ov Guaranteeing Positive Secrecy Capacity withFinite-Rate Feedback using Artificial Noise
Shuiyin Liu, Yi Hong, and Emanuele Viterbo
Abstract —While the impact of finite-rate feedback on thecapacity of fading channels has been extensively studied in theliterature, not much attention has been paid to this problemunder secrecy constraint. In this work, we study the ergodic secretcapacity of a multiple-input multiple-output multiple-antenna-eavesdropper (MIMOME) wiretap channel with quantized chan-nel state information (CSI) at the transmitter and perfect CSIat the legitimate receiver, under the assumption that only thestatistics of eavesdropper CSI is known at the transmitter. Werefine the analysis of the random vector quantization (RVQ) basedartificial noise (AN) scheme in [1], where a heuristic upper boundon the secrecy rate loss, when compared to the perfect CSIcase, was given. We propose a lower bound on the ergodicsecrecy capacity. We show that the lower bound and the secrecycapacity with perfect CSI coincide asymptotically as the numberof feedback bits and the AN power go to infinity. For practicalapplications, we propose a very efficient quantization codebookconstruction method for the two transmit antennas case.
Index Terms —artificial noise, secret capacity, physical layersecurity, wiretap channel.
I. I
NTRODUCTION
Complexity-based cryptographic technologies (e.g. AES[2]) have traditionally been used to provide a secure gatewayfor communications and data exchanges at the network layer.The security is achieved if an eavesdropper (Eve) withoutthe key cannot decipher the message in a reasonable amountof time. This premise becomes controversial with the rapiddevelopments of computing devices (e.g. quantum computer).In contrast, physical layer security (PLS) does not depend ona specific computational model and can provide security evenwhen Eve has unlimited computing power. Wyner [3] and laterCsisz´ar and K¨orner [4] proposed the wiretap channel modelas a basic framework for PLS. Wyner has shown that fordiscrete memoryless channels, if Eve intercepts a degradedversion of the intended receiver’s (Bob’s) signal, a prescribeddegree of data confidentiality could be simultaneously attainedby channel coding without any secret key. The associatednotion of secrecy capacity was introduced to characterize themaximum transmission rate from the transmitter (Alice) toBob, below which Eve is unable to obtain any information.Wyner’s wiretap channel model has been extended to fad-ing channel [5], Gaussian broadcast channel [6], multiple-
S. Liu, Y. Hong and E. Viterbo are with the Department of Electrical andComputer Systems Engineering, Monash University, Clayton, VIC 3800, Aus-tralia (e-mail: { shuiyin.liu, yi.hong, emanuele.viterbo } @monash.edu). Thiswork was performed at the Monash Software Defined TelecommunicationsLab and the authors were supported by the Australian Research CouncilDiscovery Project with ARC DP130100336. input single-output multiple-antenna-eavesdropper (MISOME)channel [7], and multiple-input multiple-output multiple-antenna-eavesdropper (MIMOME) channel [8]. All theseworks rely on the perfect knowledge of Bob’s channel stateinformation (CSI) at Alice to compute the secrecy capacityand enable secure encoding. In particular, Eve’s CSI is alsoassumed to be known at Alice in [5], [6], [8], although the CSIof a passive Eve is very hard to be unveiled at Alice. It is morereasonable to assume that Alice only knows the statistics ofEve’s channel. Even the assumption of perfect knowing Bob’sCSI is not realistic. In practice, Bob can only provide Alicewith a quantized version of his CSI via a rate constrainedfeedback channel (i.e., finite-rate feedback).In this work, we are interested in the secrecy capacity condi-tioned on the quantized CSI of Bob’s channel and the statisticsof Eve’s channel. While the impact of finite-rate feedback onthe capacity of fading channels has been extensively studied(see [9]–[13]), not much attention has been given to thisproblem under secrecy constraint. In [14], assuming that Aliceonly knows the statistics of Eve’s channel, the authors derivedlower and upper bounds on the ergodic secrecy capacityfor a single-input single-output single-antenna-eavesdropper(SISOSE) system with finite-rate feedback of Bob’s CSI. In theMIMOME scenario, the artificial noise (AN) scheme has beenshown to guarantee positive secrecy capacity without knowingEve’s CSI in [15]. Alice is assumed to have perfect knowledgeof Bob’s eigenchannel vectors. This assumption allows her toalign artificial noise within the null space of a MIMO channelbetween Alice and Bob, so that only Eve’s equivocation isenhanced. In [1], the authors show that if only quantized CSIis available at Alice, the artificial noise will leak into Bob’schannel, causing a decrease in the achievable secrecy rate. A heuristic upper bound on the secrecy rate loss (compared tothe perfect CSI case) is proposed in [1, Eq. 34].The main contribution of this paper is to provide a lowerbound on the ergodic secrecy capacity for the AN schemewith quantized CSI, valid for any number of Alice/Bob/Eveantennas, as well as for any Bob/Eve SNR regimes. Followingthe work in [1], we use the random vector quantization (RVQ)scheme in [9]. Namely, given B feedback bits, Bob quantizeshis eigenchannel matrix to one of N = 2 B random unitarymatrices and feeds back the corresponding index. We firstshow that RVQ is asymptotically optimal for security purpose,i.e., the secrecy capacity/rate loss compared to the perfect CSIcase converges to as B → ∞ . This result implies that theheuristic bound in [1, Eq. 34] is not tight, since it reduces toa positive constant as B → ∞ . To refine the analysis in [1], we establish a tighter upper bound on the secrecy rate loss,which leads to an explicit lower bound on the ergodic secrecycapacity. We further show that the lower bound and the secrecycapacity with perfect CSI coincide asymptotically as B and theAN power go to infinity. This allows us to provide a sufficientcondition guaranteeing positive secrecy capacity.From a practical point of view, it is often desirable to usea deterministic quantization codebook rather than a randomone. The problem of derandomizing RVQ codebooks is relatedto discretizing the complex Grassmannian manifold [9], [10].Since the optimal constructions are possible only in veryspecial cases, deterministic codebooks are mostly generatedby computer search [16]. Interestingly, the case of codebookdesign with two transmit antennas is equivalent to quantizinga real sphere [13]. According to this fact, we propose avery efficient codebook construction method for the two-antenna case. Simulation results demonstrate that near-RVQperformance is achieved by a moderate number of feedbackbits.The paper is organized as follows: Section II presents thesystem model, followed by the analysis of secrecy capacitywith finite-rate feedback in Section III. Section IV providesthe deterministic quantization codebook construction methodfor the two-antenna case. Conclusions are drawn in Section V.Proofs of the theorems are given in Appendix. Notation:
Matrices and column vectors are denoted by upperand lowercase boldface letters, and the Hermitian transpose,inverse, pseudoinverse of a matrix B by B H , B − , and B † ,respectively. | B | denotes the determinant of B . Let the randomvariables { X n } and X be defined on the same probabilityspace. We write X n a.s. → X if X n converges to X almost surelyor with probability one. I n denotes the identity matrix of size n . An m × n null matrix is denoted by m × n . A circularlysymmetric complex Gaussian random variable x with variance σ is defined as x ∽ N C (0 , σ ) . The real, complex, integer andcomplex integer numbers are denoted by R , C , Z and Z [ i ] ,respectively. I ( x ; y ) represents the mutual information of tworandom variables x and y . We use the standard asymptoticnotation f ( x ) = O ( g ( x )) when lim sup x →∞ | f ( x ) /g ( x ) | < ∞ . ⌈ x ⌋ rounds to the closest integer, while ⌊ x ⌋ to the closestinteger smaller than or equal to x and ⌈ x ⌉ to the closest integerlarger than or equal to x . A central complex Wishart matrix A ∈ C m × m with n degrees of freedom and covariance matrix Σ , is defined as A ∽ W m ( n , Σ ) . Trace of a square matrix B is denoted by Tr ( B ) . We write , for equality in definition.II. S YSTEM M ODEL
We consider secure communications over a three-terminalsystem, including a transmitter (Alice), the intended receiver(Bob), and an unauthorized receiver (Eve), equipped with N A , N B , and N E antennas, respectively. The signal vectors receivedby Bob and Eve are z = Hx + n B , (1) y = Gx + n E , (2)where x ∈ C N A is the transmit signal vector, H ∈ C N B × N A and G ∈ C N E × N A are the respective channel matrices between Alice to Bob and Alice to Eve, and n B , n E are AWGN vectorswith i.i.d. entries ∼ N C (0 , σ B ) and N C (0 , σ E ) . We assume thatthe entries of H and G are i.i.d. complex random variables ∼ N C (0 , .Without loss of generality, we normalize Bob’s channelnoise variance to one, i.e., σ B = 1 . (3)In this paper, we assume that Bob knows its own channelmatrix H instantaneously and Eve knows both its own channelmatrix G and the main channel H , instantaneously; whereasAlice is only aware of the statistics of H and G . There is alsoan error-free public feedback channel with limited capacityfrom Bob to Alice that can be tracked by Eve. In our setting,the feedback is exclusively used to send the index of thecodeword in a quantization codebook that describes the mainchannel state information H . The quantization codebook isassumed to be known a priori to Alice, Bob and Eve. A. Artificial Noise Scheme with Perfect CSI
The original AN scheme assumes N B < N A , in orderto ensure that H has a non-trivial null space with an or-thonormal basis Z = null ( H ) ∈ C N A × ( N A − N B ) (such that HZ = N B × ( N A − N B ) ) [15]. Let H = UΛV H be the singularvalue decomposition (SVD) of H , where U ∈ C N B × N B and V ∈ C N A × N A are unitary matrices. Then, we can write theunitary matrix V as V = [ ˜V , Z ] , (4)where the N B columns of ˜V ∈ C N A × N B span the orthogo-nal complement subspace to the null space spanned by thecolumns of Z ∈ C N A × ( N A − N B ) .With unlimited feedback (i.e., perfect CSI), Alice has per-fect knowledge of the precoding matrix V , and transmits x = ˜Vu + Zv = V " uv , (5)where u ∈ C N B is the information vector and v ∈ C ( N A − N B ) isthe “artificial noise”. For the purpose of evaluating the achiev-able secrecy rate, both u and v are assumed to be circularsymmetric Gaussian random vectors with i.i.d. complex entries ∼ N C (0 , σ u ) and N C (0 , σ v ) , respectively. In [17], we haveshown that Gaussian input alphabets asymptotically achievesthe secrecy capacity as σ v → ∞ .Equations (1) and (2) can then be rewritten as z = H ˜Vu + HZv + n B = H ˜Vu + n B , (6) y = G ˜Vu + GZv + n E , (7)to show that with unlimited feedback, the artificial noise onlydegrades Eve’s channel, resulting in increased secrecy capacity(compared to the non-AN case). B. Artificial Noise Scheme with Quantized CSI
In [1], the authors analyzed the impact of finite-rate feed-back on the secrecy rate achievable by the AN scheme. Toquantize the matrix ˜V in (4), the random vector quantization (RVQ) scheme in [9] is used. Given B feedback bits perfading channel, Bob specifies ˜V from a random quantizationcodebook V = n ˜V i , ≤ i ≤ B o , (8)where the entries are independent N A × N B random unitarymatrices, i.e., ˜V Hi ˜V i = I N B . The codebook V is known apriori to both Alice, Bob and Eve. Bob selects the ˜V j thatminimize the chordal distance between ˜V i and ˜V [11]: ˜V j = min ˜V i ∈V d (cid:16) ˜V i , ˜V (cid:17) , (9)where d (cid:16) ˜V i , ˜V (cid:17) = N B − Tr (cid:16) ˜V H ˜V i ˜V Hi ˜V (cid:17) . (10)Note that Tr ( A ) denotes the trace of the square matrix A . Andthen, Bob relays the corresponding index j back to Alice.Alice generates the precoding matrix from ˜V j as follows.Let ˜v , ... , ˜v N B be the columns of ˜V j , and e , ... , e N A − N B be the standard basis vectors. Alice applies the Gram-Schmidtalgorithm to the matrix [ ˜v , ... , ˜v N B , e , ... , e N A − N B ] to generate the remaining orthonormal basis vectors spanningthe orthogonal complement space to the one generated by thecolumns of ˜V j . This provides Alice with a unitary matrix ˆV = [ ˜V j , ˆZ ] ∈ C N A × N A , (11)that can be used to precode u and v as in (5).Since ˆZ = Z , the interference term HˆZv cannot be nulledat Bob. Therefore, equations (6) and (7) reduce to z = H ˜V j u + HˆZv + n B , (12) y = G ˜V j u + GˆZv + n E , (13)and show that with finite rate feedback (i.e., quantized CSI),some of the artificial noise will inevitably leak into the mainchannel from Alice to Bob, causing degradation in the secrecycapacity (compared to the unlimited feedback case). C. Assumptions and Notations
The analysis in [1], [15] are based on the assumption of N E < N A . Clearly, this assumption is not always realistic. Inthis work, we remove this assumption and evaluate the secrecycapacity for any number of Eve antennas.Since ˆV in (11) is a unitary matrix, the total transmissionpower can be written as || x || = " uv H ˆV H ˆV " uv = || u || + || v || . (14)Then the average transmit power constraint P is P = E ( || x || ) = P u + P v , (15)where P u = E ( || u || ) = σ u N B , P v = E ( || v || ) = σ v ( N A − N B ) , (16)are fixed by the power allocation scheme that selects the powerbalance between σ u and σ v .We define Bob’s and Eve’s SNRs as • SNR B , σ u /σ B • SNR E , σ u /σ E To simplify our notation, we define three system parameters: • α , σ u /σ E = SNR E • β , σ v /σ u (AN power allocation) • γ , σ E /σ B (Eve-to-Bob noise-power ratio)Note that SNR B = αγ . If γ > , we say Eve has a degraded channel. Since we have normalized σ B to one, we can rewrite(16) as • P u = αγN B • P v = αβγ ( N A − N B ) D. Instantaneous and Ergodic Secrecy Capacities
We recall from [8] the definition of instantaneous secrecycapacity for MIMOME channel: C S , max p ( u ) { I ( u ; z ) − I ( u ; y ) } . (17)where the maximum is taken over all possible input distri-butions p ( u ) . We remark that C S is a function of H and G ,which are embedded in z and y . To average out the channelrandomness, we further define the ergodic secrecy capacity, asin [15] E( C S ) , max p ( u ) { I ( u ; z | H ) − I ( u ; y | H , G ) } , (18)where I ( X ; Y | Z ) , E Z [ I ( X ; Y ) | Z ] , following the notation in[18].Since closed form expressions for C S and E( C S ) are notalways available, we often consider the corresponding secrecyrates, given by R S , I ( u ; z ) − I ( u ; y ) , (19) E( R S ) , I ( u ; z | H ) − I ( u ; y | H , G ) , (20)assuming Gaussian input alphabets, i.e., v and u are mutuallyindependent Gaussian vectors with i.i.d. complex entries N C (0 , σ v ) and N C (0 , σ u ) , respectively.From (6), (7) and (19), the achievable secrecy rate withperfect CSI can be written as R S = log (cid:12)(cid:12)(cid:12) I N B + αγ HH H (cid:12)(cid:12)(cid:12) + log (cid:12)(cid:12)(cid:12) I N E + αβ ( GZ )( GZ ) H (cid:12)(cid:12)(cid:12) − log (cid:12)(cid:12)(cid:12) I N E + α ( G ˜V )( G ˜V ) H + αβ ( GZ )( GZ ) H (cid:12)(cid:12)(cid:12) . (21)The closed-form expression of E( R S ) can be found in [17, Th.1]. It is shown that E( R S ) → E( C S ) as the AN power P v → ∞ in [17, Th. 3].From (12), (13) and (19), the achievable secrecy rate withquantized CSI can be written as R S,Q = log (cid:12)(cid:12)(cid:12) I N B + αγ ( H ˜V j )( H ˜V j ) H + αβγ ( HˆZ )( HˆZ ) H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I N B + αβγ ( HˆZ )( HˆZ ) H (cid:12)(cid:12)(cid:12) − log (cid:12)(cid:12)(cid:12) I N E + α ( GV ,j )( G ˜V j ) H + αβ ( GˆZ )( GˆZ ) H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I N E + αβ ( GˆZ )( GˆZ ) H (cid:12)(cid:12)(cid:12) .(22) E. Open Problems and Motivations
Using [19, Eq. 2, pp. 56], it is simple to show that E( R S ) ≥ E( R S,Q ) . (23)In [1], the ergodic secrecy rate loss is defined by: E(∆ R S ) , E( R S ) − E( R S,Q ) . (24)A heuristic upper bound was proposed in [1, Eq. 34]: E(∆ R S ) / N B log N B + αβγN A D (cid:16) N A , N B , B (cid:17) N B − D (cid:0) N A , N B , B (cid:1) + N B log (cid:18) αγ ( N A − N B ) (cid:19) , UB heuristic . (25)where D (cid:16) N A , N B , B (cid:17) = E (cid:16) d (cid:16) ˜V j , ˜V (cid:17)(cid:17) , (26)and d ( · , · ) is give in (10).However, (25) is insufficient to characterize the impactof quantized CSI on the secrecy rate achievable by the ANscheme. To see this, we first provide the following theorem. Theorem 1:
For the RVQ-based AN scheme, as B → ∞ , ˆV → V , (27)where V is given in (4) and ˆV is given in (11). Proof:
See Appendix A.Theorem 1 shows the RVQ scheme is asymptotically opti-mal for large B , i.e., the secrecy capacity/rate loss (comparedto the perfect CSI case) converges to zero. In contrast, as B → ∞ , UB heuristic in (25) reduces to a positive constant:UB heuristic → N B log (cid:18) αγ ( N A − N B ) (cid:19) , (28)since D (cid:16) N A , N B , B (cid:17) → as B → ∞ [11]. Hence, theheuristic bound in (25) is not tight. Remark 1:
The ergodic secrecy capacity with quantizedCSI, denoted by E( C S,Q ) , is lower bounded by E( C S,Q ) ≥ E( R S,Q ) = E( R S ) − E(∆ R S ) . (29)Using the closed-form expression of E( R S ) given in [17, Th.1], we are motivated to establish a tighter upper bound on E(∆ R S ) , which allows us to obtain a lower bound on E( C S,Q ) .III. S ECRECY C APACITY WITH Q UANTIZED
CSIIn this section, we wish to determine the secrecy capacitywith RVQ scheme. A tight upper bound on the ergodic secrecyrate loss and a lower bound on the ergodic secrecy capacityare provided in Theorem 2 and Theorem 4, respectively. InTheorem 5, we show that the lower bound and the secrecycapacity with perfect/quantized CSI coincide asymptoticallyas B and P v go to infinity. This provides a sufficient conditionguaranteeing positive secrecy capacity. To describe our result, we first recall the following functionfrom [20]: Θ( m, n, x ) , e − /x m − X k =0 k X l =0 2 l X i =0 ( ( − i (2 l )!( n − m + i )!2 k − i l ! i !( n − m + l )! · k − l ) k − l ! · l + n − m )2 l − i ! · n − m + i X j =0 x − j Γ( − j, /x ) ) ,(30)where (cid:16) ab (cid:17) = a ! / (( a − b )! b !) is the binomial coefficient, n ≥ m are positive integers, and Γ( a, b ) is the incomplete Gammafunction Γ( a, b ) = Z ∞ b x a − e − x dx . (31)We further define N min , min { N E , N A − N B } , (32) N max , max { N E , N A − N B } , (33) ˆ N min , min { N E , N A } , (34) ˆ N max , max { N E , N A } . (35)Finally, we define a set of N A power ratios { θ i } N A , where θ i , ( α ≤ i ≤ N B αβ N B + 1 ≤ i ≤ N A (36)We recall from [11, Th. 4] that µ (cid:16) N A , N B , B (cid:17) ≤ D (cid:16) N A , N B , B (cid:17) ≤ η (cid:16) N A , N B , B (cid:17) , (37)where D ( · , · , · ) is given in (26) and η ( n, p, K ) = Γ (cid:18) p ( n − p ) (cid:19) p ( n − p ) ( Kc ( n, p )) − p ( n − p )+ p exp( − ( Kc ( n, p )) − ζ ) , (38) µ ( n, p, K ) = p ( n − p ) p ( n − p ) + 1 ( Kc ( n, p )) − p ( n − p ) , (39) c ( n, p ) = p ( n − p ) + 1) p Q i =1 Γ( n − i + 1)Γ( p − i + 1) , n ≥ p p ( n − p ) + 1) n − p Q i =1 Γ( n − i + 1)Γ( n − p − i + 1) , n ≤ p (40)for any < ζ < . Note that Γ( a ) is the Gamma function. A. Bounds on Ergodic/Instantaneous Secrecy Rate Loss
We first consider the ergodic secrecy rate loss
E(∆ R S ) . Theorem 2:
Let θ min = min { αγ, αβγ } . We have E(∆ R S ) ≤ Θ( N B , N A , αγ ) − Θ( N B , N A , θ min )+ Θ N B , N A , αβγ η (cid:16) N A , N B , B (cid:17) N B , UB, (41)where Θ( · , · , · ) is given in (30) and η ( · , · , · ) is given in (38). Proof:
See Appendix B. B E ( ∆ R S ) n a t Heuristic UB in (25)Error Floor in (28)Proposed UB in (41)E(∆ R S ) Fig. 1. E (∆ R S ) vs. B with β = 1 , αγ = 1 , N A = 4 and N B = 2 . Theorem 2 gives a tight upper bound on
E(∆ R S ) , for anynumber of Alice/Bob/Eve antennas, as well as for any Bob/EveSNR regimes. Different from (28), if β ≥ , as B → ∞ ,UB → , (42)which is consistent with Theorem 1. Example 1:
Let us apply Theorem 2 to the analysis of aRVQ-based AN scheme with β = 1 , αγ = 1 , N A = 4 and N B = 2 . The numerical result in Fig. 1 shows that the proposedupper bound in (41) is much tighter than the heuristic one in(25), and captures the behavior of E(∆ R S ) .We then study the distribution of instantaneous secrecy rateloss, defined by ∆ R S , R S − R S,Q . (43)Here, we consider the large system limit as N A and B → ∞ with fixed ratio B/N A . An interesting case that leads to aclosed-form bound can be found when N B = N E = 1 . Theorem 3: If N B = N E = 1 , as N A and B → ∞ with B/N A → ¯ B , ∆ R S a.s. → log (1 + P/β ) + log (cid:16) − ¯ B P (cid:17) − log (cid:18) P + 1 − ββ (1 − − ¯ B ) P (cid:19) . (44) Proof:
See Appendix C.Theorem 3 provides a closed-form asymptotic expressionfor ∆ R S when N B = N E = 1 . Hence, the ergodic secrecyrate loss also converges to the same constant, as stated in thefollowing corollary. Corollary 1:
Under the same assumptions of Theorem 3,
E(∆ R S ) → log (1 + P/β ) + log (cid:16) − ¯ B P (cid:17) − log (cid:18) P + 1 − ββ (1 − − ¯ B ) P (cid:19) . (45) Proof:
The proof is straightforward.
Example 2:
The numerical result in Fig. 2 shows that (45)is very accurate even for finite N A and B . B E ( ∆ R S ) n a t Estimation in (45)Simulation
Fig. 2. E (∆ R S ) vs. B with β = 1 , P = 1 , N A = 10 and N B = 1 . B. A Lower Bound on Ergodic Secrecy Capacity
A lower bound on E( C S,Q ) can be derived using the resultsfrom (29), Theorem 2 and [17, Th. 1]. Theorem 4: E( C S,Q ) ≥ Θ( N min , N max , αβ ) − Ω + Θ( N B , N A , θ min ) − Θ N B , N A , αβγ η (cid:16) N A , N B , B (cid:17) N B , ¯ C LB,Q , (46)where Θ( · , · , · ) is given in (30), η ( · , · , · ) is given in (38) and Ω = K ˆ N min P k =1 det (cid:16) R ( k ) (cid:17) , β = 1Θ( ˆ N min , ˆ N max , α ) , β = 1 (47) K = ( − N E ( N A − ˆ N min ) Γ ˆ N min ( N E ) Q i =1 µ m i N E i Q i =1 Γ m i ( m i ) Q i
50 100 150 200 250 300 350 4002.22.32.42.52.62.72.82.93 P v ¯ C L B , Q n a t ¯ C Bob ¯ C LB , Q : B = 70¯ C LB , Q : B = 50 Fig. 3. ¯ C LB,Q vs. P v with N A = 4 , N B = N E = 2 , and α = γ = 1 . ϕ ( i, j ) = N E − ˆ N min + j − d i . Proof:
See Appendix D.Theorem 4 gives a lower bound on E( C S,Q ) , for any numberof Alice/Bob/Eve antennas, as well as for any Bob/Eve SNRregimes. The lower bound in (46) is an increasing function ofthe number of feedback bits B . To guarantee a positive secrecycapacity, Alice just needs to increase B and checks whether ¯ C LB,Q > . C. Positive Secrecy Capacity with Quantized CSI
To characterize the achievability of positive secrecy capacitywith quantized CSI, we start by analyzing the tightness of (46).
Theorem 5: If N E ≤ N A − N B and β ≥ , as αβ , B → ∞ , ¯ C LB,Q = E( C S,Q ) = E( C S ) = ¯ C Bob , (50)where ¯ C Bob represents Bob’s ergodic channel capacity.
Proof:
See Appendix E.We have shown that ¯ C LB,Q , E( C S ) and ¯ C Bob coincide asymp-totically as B and P v = αβγ ( N A − N B ) go to infinity. Weremark that according to (18), a universal upper bound on theergodic secrecy capacity is given by E( C S ) ≤ max p ( u ) { I ( u ; z | H ) } = ¯ C Bob . (51)
Remark 2:
Note that ¯ C Bob > and ¯ C LB,Q is derived basedon Gaussian input alphabets. From Theorem 5, we statethat a positive secrecy capacity for MIMOME channel withquantized CSI is always achieved by using RVQ-based ANtransmission scheme and Gaussian input alphabets for large B and P v , if N E ≤ N A − N B . Example 3:
Fig. 3 compares ¯ C LB,Q and ¯ C Bob as a functionof AN power P v , with N A = 4 , N B = N E = 2 , and α = γ = 1 .Since P u = αγN B and P v = αβγ ( N A − N B ) , we have P u = 2 and P v = 2 β . The simulation result shows that ¯ C LB,Q approachesto ¯ C Bob as P v increases, for sufficiently large B .IV. I MPLEMENTATION U SING A D ETERMINISTIC C ODEBOOK
In the previous section, random quantization codebookshave been used to prove new results on secrecy capacity with quantized CSI. The methods of constructing random unitarymatrices ˜V i in (8) can be found in [21]. In practice, it isoften desirable that the quantization codebook is deterministic.The problem of derandomizing RVQ codebooks is typicallyreferred to as Grassmannian subspace packing [9], [10].Despite of a few special cases (e.g., B ≤ [13]), analyticalcodebook design in general remains an intricate task. In thissection, we propose a very efficient quantization codebookconstruction method for the case of N A = 2 and N B = 1 .According to [13, Eq. (20)], the codeword ˜V i can beexpressed as ˜V i ( ω, φ ) = " cos ωe jφ sin ω , (52)which fully describes the complex Grassmannian manifold G , by setting ≤ φ ≤ π and ≤ ω ≤ π/ . Let (ˆ ω, ˆ φ ) be spherical coordinates parameterizing the unit sphere S ,where ≤ ˆ φ ≤ π and ≤ ˆ ω ≤ π . In [13, Lemma 1], theauthors further show that the map S → G , (ˆ ω, ˆ φ ) ˜V i (ˆ ω/ , ˆ φ ) (53)is an isomorphism. In other words, the sampling problem on G , can analogically be addressed on the real sphere S .The method of sampling points uniformly from S isprovided in [22]. In details, one can parameterize ( x, y, z ) ∈ S using spherical coordinates (ˆ ω, ˆ φ ) : x = sin ˆ ω cos ˆ φ , y = sin ˆ ω sin ˆ φ , z = cos ˆ ω . (54)The area element of S is given by d S = sin ˆ ω d ˆ ω d ˆ φ = − d (cos ˆ ω ) d ˆ φ . (55)Hence, to obtain a uniform distribution over S , one has topick ˆ φ ∈ [0 , π ] and t ∈ [ − , uniformly and compute ˆ ω by: ˆ ω = arccos t . (56)In this way cos ˆ ω = t will be uniformly distributed in [ − , .Based on above analysis, we give a straightforward methodfor codebook construction: ˆ V = ( ˆV ,i = " cos(0 . t i ) e jφ i sin(0 . t i ) i = 1 , ... , B ) , (57)where t i = − l i/ ⌈ B/ ⌉ m − ⌊ B/ ⌋ , (58) φ i = 2 π (cid:16) i mod 2 ⌈ B/ ⌉ (cid:17) ⌈ B/ ⌉ . (59)Note that ⌊ x ⌋ rounds to the closest integer smaller than or equalto x , while ⌈ x ⌉ to the closest integer larger than or equal to x .Using the deterministic codebook in (57) can save storagespace on Alice, since she can generate the target codeword ˆV ,j directly without the knowledge of the whole codebook ˆ V . We remark that the proposed codebook construction is valid B E ( R S , Q ) n a t Optimal Deterministic Codebook in [13]Proposed Deterministic Codebook in (57)Random Codebook in (8)
Fig. 4. E ( R S,Q ) vs. B with β = 2 , γ = 1 , P = 10 , N A = 2 , and N B = N E = 1 . for any B . This is different from the construction scheme in[13, Sec. VI-A], which is only possible for the case of B ≤ . Example 4:
Fig. 4 examines the performance of the pro-posed codebook construction with β = 2 , γ = 1 , P = 10 ,and N E = 1 . When B ≤ , it is seen that the performanceof codebook ˆ V in (57) is indistinguishable from the optimalone in [13, Sec. VI-A]. When B ≥ , the proposed codebookprovides the same performance as the random one in (8).V. C ONCLUSIONS
In this work, we have discussed the problem of guaran-teeing positive secrecy capacity for MIMOME channel withthe quantized CSI of Bob’s channel and the statistics ofEve’s channel. We analyzed the RVQ-based AN scheme andprovided a lower bound on the ergodic secrecy capacity. Weproved that a positive secrecy capacity is always achievableby Gaussian input alphabets when N E ≤ N A − N B , and thenumber of feedback bits B and the artificial noise power P v arelarge enough. We also proposed an efficient implementationof discretizing the RVQ codebook which exhibits similarperformance to that of random codebook.A PPENDIX
A. Proof of Theorem 1
According to [12], as B → ∞ , the RVQ operation in (9)can guarantee ˜V j → ˜V . (60)We then check the matrix ˆZ generated by Alice. The SVDdecomposition of H can be written as H = UΛ ˜V H . (61)From (60) and (61), as B → ∞ , we have HˆZ = UΛ ˜V H ˆZ → UΛ ˜V Hj ˆZ = N A × ( N A − N B ) , (62)which means ˆZ → null( H ) . (63)From (11), (60) and (63), we have ˆV → V as B → ∞ . (cid:4) B. Proof of Theorem 2
Using [19, Eq. 12, pp. 55], we have (cid:12)(cid:12)(cid:12) I N B + αγ ( H ˜V j )( H ˜V j ) H + αβγ ( HˆZ )( HˆZ ) H (cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12) I N B + θ min ( H ˜V j )( H ˜V j ) H + θ min ( HˆZ )( HˆZ ) H (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) I N B + θ min HH H (cid:12)(cid:12)(cid:12) , (64)where θ min = min { αγ, αβγ } .Since the unitary matrix ˆV = [ ˜V j , ˆZ ] is independent of G and its realization is known to Alice, G ˜V j ∈ C N E × N B and GˆZ ∈ C N E × ( N A − N B ) are mutually independent complexGaussian random matrices with i.i.d. entries [23, Th. 1]. Wecan write E log (cid:12)(cid:12)(cid:12) I N E + α ( G ˜V j )( G ˜V j ) H + αβ ( GˆZ )( GˆZ ) H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I N E + αβ ( GˆZ )( GˆZ ) H (cid:12)(cid:12)(cid:12) (65)as the average of a function of N E × N A i.i.d complex Gaussianrandom variables ∼ N C (0 , .Similarly, with unlimited feedback, we have E log (cid:12)(cid:12)(cid:12) I N E + α ( G ˜V )( G ˜V ) H + αβ ( GZ )( GZ ) H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I N E + αβ ( GZ )( GZ ) H (cid:12)(cid:12)(cid:12) (66)as the average of a function of N E × N A i.i.d complex Gaussianrandom variables ∼ N C (0 , .From (65) and (66), we have E log (cid:12)(cid:12)(cid:12) I N E + α ( G ˜V j )( G ˜V j ) H + αβ ( GˆZ )( GˆZ ) H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I N E + αβ ( GˆZ )( GˆZ ) H (cid:12)(cid:12)(cid:12) = E log (cid:12)(cid:12)(cid:12) I N E + α ( G ˜V )( G ˜V ) H + αβ ( GZ )( GZ ) H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I N E + αβ ( GZ )( GZ ) H (cid:12)(cid:12)(cid:12) .(67)From (21), (22), (64) and (67), E(∆ R S ) can be upperbounded by E(∆ R S ) ≤ E (cid:16) log (cid:12)(cid:12)(cid:12) I N B + αγ HH H (cid:12)(cid:12)(cid:12)(cid:17) − E (cid:16) log (cid:12)(cid:12)(cid:12) I N B + θ min HH H (cid:12)(cid:12)(cid:12)(cid:17) + E (cid:16) log (cid:12)(cid:12)(cid:12) I N B + αβγ ( HˆZ )( HˆZ ) H (cid:12)(cid:12)(cid:12)(cid:17) . (68)We then estimate the third term in (68). Let λ , ... , λ N B bethe eigenvalues of HH H . We have H H H = ˜VΛ ˜V H and Λ = diag ([ λ , ..., λ N B ]) . (69)Recalling the fact that for a Wishart matrix, its eigenvaluesand eigenvectors are independent. Therefore ˜V and Λ areindependent. This allows us to bound the third term in (68) by E H (cid:16) log (cid:12)(cid:12)(cid:12) I N B + αβγ ( HˆZ )( HˆZ ) H (cid:12)(cid:12)(cid:12)(cid:17) ( a ) = E H (cid:16) log (cid:12)(cid:12)(cid:12) I N A − N B + αβγ ( HˆZ ) H ( HˆZ ) (cid:12)(cid:12)(cid:12)(cid:17) = E Λ (cid:16) E ˜V (cid:16) log (cid:12)(cid:12)(cid:12) I N A − N B + αβγ ˆZ H ˜VΛ ˜V H ˆZ (cid:12)(cid:12)(cid:12)(cid:17)(cid:17) ( b ) = E Λ (cid:16) E ˜V (cid:16) log (cid:12)(cid:12)(cid:12) I N B + αβγ ˜V H ˆZˆZ H ˜VΛ (cid:12)(cid:12)(cid:12)(cid:17)(cid:17) ( c ) ≤ E Λ (cid:16) E ˜V (cid:16) log (cid:12)(cid:12)(cid:12) I N B + αβγ E ˜V (cid:16) ˜V H ˆZˆZ H ˜V (cid:17) Λ (cid:12)(cid:12)(cid:12)(cid:17)(cid:17) ( d ) = E Λ log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I N B + αβγD (cid:16) N A , N B , B (cid:17) N B Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( e ) ≤ E Λ log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I N B + αβγη (cid:16) N A , N B , B (cid:17) N B Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (70)where ( a ) and ( b ) hold because | I + AB | = | I + BA | , ( c ) follows from the concavity of log determinant function, ( d ) follows from [24, Lemma 1] [1, Lemma 2], ( e ) holds becauseof (37).Applying the fact [20, Th. 1] E (cid:16) log (cid:12)(cid:12)(cid:12) I N B + ρ HH H (cid:12)(cid:12)(cid:12)(cid:17) = E (log | I N B + ρ Λ | ) = Θ( N B , N A , ρ ) ,(71)to (68) and (70), we can simply obtain (41). (cid:4) C. Proof of Theorem 3
Recalling the fact that ZZ H = I N A − ˜V ˜V H and ˆZˆZ H = I N A − ˜V j ˜V Hj . From (21) and (22), if N B = N E = 1 , we canwrite ∆ R S as ∆ R S = log (cid:16) αγ HH H (cid:17) − log 1 + α GG H + α ( β − GZ )( GZ ) H αβ ( GZ )( GZ ) H − log 1 + αβγ HH H + αγ (1 − β ) ( H ˜V j )( H ˜V j ) H αβγ HH H − αβγ ( H ˜V j )( H ˜V j ) H + log 1 + α GG H + α ( β − GˆZ )( GˆZ ) H αβ ( GˆZ )( GˆZ ) H . (72)As N A and B → ∞ with B/N A → ¯ B , according to [12, Th.1], we have ( H ˜V j )( H ˜V j ) H HH H a.s. → (1 − − ¯ B ) . (73)Since β (AN power allocation), γ (Eve-to-Bob noise-powerratio), and P = αγ + αβγ ( N A − (average transmit powerconstraint) are fixed, the central limit theorem tells us that αγ HH H a.s. → P/β , α GG H a.s. → P/βγ , α ( GZ )( GZ ) H a.s. → P/βγ , α ( GˆZ )( GˆZ ) H a.s. → P/βγ . (74)Note that GZ (or GˆZ ) is a complex Gaussian random vectorwith i.i.d. entries [23, Th. 1].By substituting (73) and (74) into (72), we obtain (44). (cid:4)
D. Proof of Theorem 4
According to [17, Th. 1], we have E( R S ) = Θ( N B , N A , αγ ) + Θ( N min , N max , αβ ) − Ω , (75)where Θ( · , · , · ) is given in (30) and Ω is given in (47). Bysubstituting (41) and (75) into (29), we can obtain (46). (cid:4) E. Proof of Theorem 5 If β ≥ , then θ min = αγ . From (46) and (75), as B → ∞ , ¯ C LB,Q = Θ( N B , N A , αγ ) + Θ( N min , N max , αβ ) − Ω = E( R S ) .(76)According to [17, Th. 3], if N E ≤ N A − N B , as αβ → ∞ , E( R S ) = E( C S ) = ¯ C Bob , (77)where ¯ C Bob represents Bob’s average channel capacity.Meanwhile, it always holds that ¯ C LB,Q ≤ E( C S,Q ) ≤ ¯ C Bob . (78)From (76), (77) and (78), if N E ≤ N A − N B and β ≥ , as αβ , B → ∞ , we have ¯ C LB,Q = E( C S,Q ) = E( C S ) = ¯ C Bob . (79) (cid:4) R EFERENCES[1] S.-C. Lin, T.-H. Chang, Y.-L. Liang, Y. P. Hong, and C.-Y. Chi, “Onthe impact of quantized channel feedback in guaranteeing secrecy withartificial noise: The noise leakage problem,”
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