Artificial Noise Revisited: When Eve Has more Antennas than Alice

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1 Artificial Noise Reisited: When e Has more Antennas than Alice Shiyin Li Yi Hong and manele Viterbo CS Department Monash Uniersity Melborne VIC 3800 Astralia mail: shiyin.li yi.hong emanele.iterbo@monash.ed Abstract We consider secre commnications oer MIMO wiretap channels in the presence of a passie eaesdropper with an nlimited nmber of antennas. In this scenario we characterize the performance of the artificial noise scheme proposed by Goel et al. and show that non-zero secrecy capacity is aailable een when the eaesdropper has more antennas than the transmitter. Or reslts are deried based on the fraction of the transmission power sed for artificial noise and the ratio of the channel noise ariances of the eaesdropper and the intended receier. Finally we inestigate the attack option for the eaesdropper to drie the secrecy rate to zero by increasing the nmber of antennas. I. INTRODUCTION The isses of establishing a priate and secre link at the physical layer hae known a growing interest in the past few years. Physical layer secrity lays its fondation on the wiretap channel [1] where the pre-exchanged secret key in Shannon s model [2] is replaced by the channel noise to proide randomness. For this model the notion of secrecy capacity is frther deeloped to characterize the maximm transmission rate at which the eaesdropper (e) is nable to obtain any information [3]. For qasi-static fading channel the aerage secrecy rate is deried in [4]. For the ergodic fading channel [5] proides a detailed analysis of secrecy capacity. In the literatre the achieable aerage secrecy rate has been adopted as a metric of secrity [4 6]. The artificial noise (AN) recently emerged as a promising method to increase secrecy rate [6] where the transmitter (Alice) aligns a jamming signal named artificial noise within the nll space between itself and the legitimate receier (Bob) ths AN only degrades e s channel. In [6] assming that both signal and AN follow a mltiariate Gassian distribtion non-zero aerage secrecy rate is obsered in simlation when the nmber of e s antennas N is strictly smaller than the nmber of Alice s antennas N A i.e. N < N A. Seeral channel models hae been considered to generalize the idea of AN. When the nmber of Bob s antennas = 1 [7] proides an asymptotic analysis of the secrecy capacity. When all channel matrices (inclding e) are fixed and known to all the terminals [8] proides a detailed characterization in terms of the secrecy capacity. More recently the notion of practical This work is spported by ARC nder Grant Discoery Project No. DP This research is also spported by the 2013 Monash Faclty of ngineering Seed Fnding Scheme. secrecy is proposed for the AN-based systems that make se of a finite alphabet (e.g. M-QAM) [9]. Instead of increasing secrecy rate e s error probability is maximized by the randomly distribted AN (e.g. not necessarily Gassian). In this work we inestigate the comprehensie relationship between secrecy capacity mltiple antennas and signal-tonoise ratio (SNR) within the framework of the original Gassian distribted AN scheme [6]. The contribtions of the paper are as follows. We show that the aerage secrecy capacity is achieed with Gassian inpt alphabets when N N A. We proide pper and lower bonds on both aerage and instantaneos secrecy rates with Gassian inpt alphabets that acconts for arbitrary e s SNR and N. From Bob s perspectie gien N we derie the ales of N A and to ensre positie aerage secrecy rate. From e s perspectie gien N A and we derie the ale of N to nll the instantaneos secrecy rate. The paper is organized as follows: Section II presents the system model followed by the analysis of secrecy capacity in Section III. Section IV discsses the attack to drie the secrecy rate to zero. Conclsions are drawn in Section V. Proofs of the theorems are gien in Appendix. Notation: Matrices and colmn ectors are denoted by pper and lowercase boldface letters and the Hermitian transpose inerse psedoinerse of a matrix B by B H B 1 and B respectiely. B denotes the determinant of B. Let {X n X} be defined on the same probability space. We write X n X if X n conerges to X almost srely or with probability one. The identity matrix of size n is denoted by I n. An m n nll matrix is denoted by 0 m n. We write for eqality in definition. A circlarly symmetric complex Gassian random ariable x with ariance σ 2 is defined as x N C (0 σ 2 ). The real complex integer and complex integer nmbers are denoted by R C Z and Z [i] respectiely. I(x; y) represents the mtal information of two random ariables x and y. We se the standard asymptotic notation f (x) = O (g (x)) when lim sp f(x)/g(x) <. x ronds to the closest integer. x II. SYSTM MODL We consider a MIMO wiretap system inclding a transmitter (Alice) an intended receier (Bob) and a passie eaesdropper (e) with N A and N antennas respectiely /14/$ I

2 We assme that the matrices H C NB NA and G C N NA represent the channels from Alice to Bob and Alice to e. H and G are assmed to be independent (i.e. all terminals are not co-located) and hae i.i.d. entries N C (0 1). Assming < N A H then has a non-triial nll space Z = nll(h) i.e. HZ = 0 NB (N A ). Let H = UΛV H be the singlar ale decomposition (SVD) of H. The relationship between matrices Z and V is gien by V = [V 1 Z] (1) where V 1 represents the matrix with the first colmns of V. Using the AN scheme [6] Alice transmits x = V 1 + Z = V (2) where is the secret data ector and is the artificial noise. Gassian inpt alphabets are assmed i.e. both C NB 1 and C (NA NB) 1 are mtally independent Gassian ector with i.i.d. entries N C (0 σ 2 ) and N C (0 σ 2 ) respectiely. The signals receied by Bob and e are gien by z = HV 1 + HZ + n B = HV 1 + n B (3) y = GV 1 + GZ + n. (4) where n B and n are additie white Gassian noise ectors at Bob and e respectiely with i.i.d. entries N C (0 σ 2 B ) and N C (0 σ 2 ). From qations (3) and (4) we see that only increases eaesdropper s ncertainty abot the secret message bt does not affect Bob. Since V is nitary the total transmission power x 2 is x 2 = H V H V = (5) We set the aerage transmit power constraints P and P : P = 2 = σ 2 P = 2 = σ 2 (N A ). (6) We define Bob s and e s SNRs as SNR B σ 2 /σ 2 B SNR σ 2 /σ 2. (7) Throghot the paper we assme the worst-case scenario for Alice and Bob described in [6]: Alice only knows H. e knows H G Z and V 1. Different from [6] we assme no pper bond on N. To simplify or analysis we define three system parameters: α σ 2 /σ 2 (SNR ) β σ 2 /σ 2 (AN power fraction) γ σ 2 /σ2 B (e-to-bob noise-power ratio) If γ > 1 we say e has a degraded channel. Note that SNR B = αγ. For conenience we fix σ 2 B = 1 ths P = αγ. To ealate the asymptotic secrecy rate we assme N A /N β 1 N A / β 2 /N β 3 III. NON-ZRO SCRCY CAPACITY In this section we reisit the problem of garanteeing non-zero instantaneos and aerage secrecy capacities sing artificial noise. To present or reslt we first define some sefl fnctions. A. Definitions We recall the definition of instantaneos secrecy capacity: C S max {I(û; z) I(û; y)} (8) which is a special case of the definition in [10]. The maximm is taken oer all possible inpt distribtions p (û). Note that C S is a random ariable depending on Gassian random matrices H and G which are embedded in z and y. We frther define aerage secrecy capacity as in [6] C S max {I(û; z H) I(û; y H G)}. (9) where I (X; Y Z) Z [I (X; Y ) Z] [11]. Since closed form expressions for C S and C S are not always aailable (except for Theorem 1) we often resort to lower bonds gien by C S I(; z) I(; y) R S (10) C S I(; z H) I(; y H G) R S (11) assming Gassian inpt alphabets i.e. and are mtally independent Gassian ector with i.i.d. entries N C (0 σ 2 ) and N C (0 σ 2 ) respectiely. We then define the following fnction as in [12] m 1 1/x Θ(m n x) e 2(k l) k l where a b k 2l k=0 l=0 i=0 2(l + n m) 2l i ( 1) i (2l)!(n m + i)! 2 2k i l!i!(n m + l)! n m+i j=0 x j Γ( j 1/x) (12) = a!/((a b)!b!) is the binomial coefficient n m are positie integers and Γ( ) is the incomplete Gamma fnction. Finally we define N max = max {N N A } N min = min {N N A }. (13) B. Non-zero Aerage Secrecy Capacity We proide some analytical insights relating R S and C S to N A N α β and γ. We first derie an pper bond on I(; y H G). Lemma 1: I(; y H G) N log(1 + α ) Θ(N min N max αβ) +Θ(N min N max αβ/(1 + α )) 1 = (N N min ) log α + O α Proof: See Appendix A. + O 1. β

3 Lemma 1 reeals the following relation between C S and R S. Theorem 1: If N N A as α β then C S = R S. (14) Proof: See Appendix B. In Theorem 1 we demonstrated the achieability of the aerage secrecy capacity of the AN scheme sing Gassian inpt alphabets nder the aboe condition. This reslt was not gien in [6] where only R S > 0 was obsered by simlation. The assmption that N is bonded can be n-natral in practice. In the following theorem we derie a lower bond on the achieable rate R S withot any limitation on N. Theorem 2: R S Θ( N A αγ) + Θ(N min N max αβ) N log(1 + α ) Θ(N min N max αβ/(1 + α )) R LB (15) Proof: See Appendix C. To gain frther intition we proide the following corollary giing a sfficient condition for positie aerage secrecy rate as. Corollary 1: If β 2 1 lim R S / > 0 when N < N A + (log γ 1 + Υ(P )) (16) log P log γ where 1 + 4P 1 Υ(P ) = + 2 tanh 1 1. (17) 2P 1 + 4P Proof: See Appendix D. Corollary 1 gies an example of application of Theorem 2 and shows that positie secrecy capacity is in fact achieable for any N ths remoes the restriction N < N A in [6] since the second term in (16) can be increased by γ. xample 1: Fig. 1 compares the ales of R S and R LB in (15) as fnctions of N with α = β = 3 db γ = 6 db = 3 and N A = 4. In this case (16) redces to N < 6. We obsere that R LB > 0 when N < 6. This example shows the sability of Corollary 1 for finite nmbers of antennas. Althogh the gap between R S and R LB increases with increasing N the cre of R LB can still confirm the fact that positie aerage secrecy rate is aailable een when N N A. C. Non-zero Instantaneos Secrecy Capacity We now analyze the instantaneos secrecy rate R S as a random ariable depending on the random matrices G and H. An interesting case that leads to a closed form bond can be fond when β = 1 (or 0 db). Theorem 3: If β = 1 as N A and N with fixed β 1 β 2 and β 3 R S Φ(P β N 2 ) Φ(P/() β 1 ) + Φ(P/() β 1 β 3 ) B β 3 β 3 where Φ (x y) y log 1 + x 14 F (x y) F (x y) 4x + log 1 + xy 14 F (x y) (18) nat N Fig. 1. RS and R LB s. N with α = β = 3 db γ = 6 db = 3 and N A = 4. Fig. 2. γ = 3. RS (nat) F (x y) 5 0 N =20 N =40 R S R LB RS s. and N with N A = 20 α = 10 db β = 0 db and x (1 + y) x (1 2 y) (19) Proof: Using [13 q. 1.14] the proof is straightforward. Corollary 2: Under the same assmptions of Theorem 3 if N N A (i.e. β 1 1) as P the maximm R S is achieed when = min max N A N 1 N A 1. (20) 1 + γ Proof: Aailable in the jornal ersion. Corollary 2 reeals the relationship between the maximm R S and γ for gien N A and N : the larger the channel degradation γ is the smaller the nll space size N A is reqired. xample 2: Fig. 2 shows the ariation of R S by simlation as a fnction of and N with N A = 20 α = 10 db β = 0 db and γ = 3. We obsere that for N = 20 and 40 RS is maximized when = N A 1+γ N = 15 and 10. Once exceeds the optimm ale RS starts to decrease. Ths Corollary 2 is ery accrate een for a finite system model. The reslts for general β and β 1 will be reported in the jornal ersion. IV. ATTACK ON AN In the preios section we took the side of Alice and Bob and we established the conditions that ensre non-zero secrecy rate for gien α and N. This proided a design strategy of

4 defence against eaesdropping. In this section we play the part of e. We want to estimate how many antennas e needs to drie the instantaneos secrecy rate R S to zero assming the worst case scenario that N A α β and γ are known to e. We define e s actiity as the attack on the AN scheme. In smmary attack and defence are related to pper and lower bonds on the secrecy rate respectiely. Let ŷ be a processed form of y. Thanks to the data processing ineqality we hae I(; y) I(; ŷ). (21) Let ˆR S I(; z) I(; ŷ). From (10) and (21) we obtain R S ˆR S. (22) In other words data processing cannot decrease R S bt may be sed to obtain an easily comptable pper bond. We se zero-forcing (ZF) processing since it can remoe the artificial noise and hence the dependency on parameter β. In particlar if N N A G has a left inerse denoted by G then the interference term GZ can be remoed by mltiplying y by W = HG i.e. Wy =HV 1 + Wn ŷ ZF. (23) Note that if N < N A ZF processing is not applicable. Theorem 4: Let ˆR S ZF I(; z) I(; ŷ ZF ). As N A and N with fixed β 1 β 2 and β 3 ˆR S ZF < Φ (P β N 2 ) log P(1 (24) B almost srely where Φ (x y) is gien in Theorem 3. Proof: See Appendix. From (22) for large scale system models we hae R S / < ˆR S ZF /. By setting ˆR S ZF / eqal to zero R S / is forced to be zero as stated in the following corollary. Corollary 3: Under the same assmptions of Theorem 4 R S / 0 if N = exp {Φ (P β 2 )} + 2 N A. (25) α Proof: The proof is straightforward. Corollary 3 proides e an analytical expression for choosing N to attack the AN scheme. V. CONCLUSIONS This paper characterizes the trade-off between the mltiple antennas and the secrecy rate achieed by the artificial noise scheme. By taking all the system parameters into accont we deeloped explicit lower and pper bonds on the secrecy rate with Gassian inpt alphabets. We hae shown that the lower bond tells Alice how many antennas she needs to ensre non-zero secrecy rate while the pper bond proides the minimm nmber of antennas for e to drie the secrecy rate to zero. Based on or analysis we describe the antenna nmber race between Alice and e as a defence erss attack battle oer physical layer secrity. APPNDIX A. Proof of Lemma 1 Since all entries in H and G are mtally independent I(; y) can be expressed as a fnction of these independent random entries. This allows s to take two steps to compte the expected ale of I(; y): we first compte I(; y G) gien H then compte H [I(; y G) H]. The adantage is that for gien H V = [V 1 Z] is a fixed nitary matrix. Then sing [14 Th. 1] GV 1 and GZ are mtally independent complex Gassian random matrices with i.i.d. entries N C (0 1). Let G 1 = GV 1 G 2 = GZ W 1 = G 1 G H 1 and W 2 = G 2 G H 2. According to [11] we hae I(; y G) I N σ 2 = G1G 2 + σ2 W 1 +σ 2 W 2 I N σ 2 +σ2 W 2 a I N σ 2 G2 + σ2 G1 (W 1 ) +σ 2 W 2 I N σ 2 +σ2 W 2 σ I N + 2 W σ = G NBσ2 I N + σ2 + N log σ2 + σ 2 W σ 2 2 σ 2 (26) where (a) holds de to the concaity of log-determinant fnction (Jensen s Ineqality). Note that I + G 2 G H 2 = I + G H 2 G 2 and define G 2 G H 2 W = G H 2 G 2 if N N A if N > N A Then W is a Wishart matrix W Nmin (N max I Nmin ) where N min and N max are gien in (13). Recalling the definitions of α and β in Sec.II. Based on aboe analysis and [12 Th. 1] the first term of (26) can be written as I Nmin + αβ 1+α W G2 I Nmin + αβw = Θ(N min N max αβ/(1 + α )) Θ(N min N max αβ) (27) where Θ(x y z) is gien in (12). From (26) and (27) we hae I(; y H G) = H [I(; y G) H] N log(1 + α ) Θ(N min N max αβ) +Θ(N min N max αβ/(1 + α )) 1 1 = (N N min ) log α + O + O. α β B. Proof of Theorem 1 If N N A from Lemma 1 and (10) as α and β R S = I(; z H). (28)

5 Moreoer we hae C S max {I(û; z H)} = I(; z H). (29) The last eqation holds since the inpt is a circlarly symmetric complex Gassian random ector [11 Th. 1]. Based on (28) and (29) as α and β we hae C. Proof of Theorem 2 R S = C S. Since (HV 1 ) (HV 1 ) H = HH H sing [11 Th. 2] and [12 Th. 1] we hae I(; z H) = H log I NB + αγhh H = Θ( N A αγ). (30) where Θ(x y z) is gien in (12). From Lemma 1 and (30) we obtain R S Θ( N A αγ) + Θ(N min N max αβ) N log(1 + α ) Θ(N min N max αβ/(1 + α )) (31) where N min and N max are gien in (13). D. Proof of Corollary 1 If β 2 1 according to [11] we hae I(; z H) lim = log P 1 + Υ(P ) (32) where Υ(P ) is gien in (17). From Lemma 1 we hae I(; y H G) N lim lim N min log P γ. (33) Moreoer if β 2 1 we hae N min = N A. (34) Based on (32) (33) and (34) lim R S / > 0 if N < N A + (log γ 1 + Υ(P )). log P log γ (35). Proof of Theorem 4 We recall that ˆR S ZF = I(; z) I(; ŷ ZF ) (36) According to [13 q. 1.14] we hae I(; z) lim Φ (P β N 2 ) (37) B where Φ(x y) is gien in (18). Moreoer since W = HG we hae I(; ŷ ZF ) σ 2 (HV 1 ) (HV 1 ) H + σ 2 WWH = log σ 2 WWH 1 I σ 2 + σ 2 VH 1 G G H V1 = log σ 2 VH 1 G H G 1 V1 NB = log (1 + α/λ i ) (38) where λ i 1 i represent the eigenales of 1 V1 H G G H V1. From (38) we hae NB I(; ŷ ZF ) > log α/λ i. (39) Let δ min be the smallest eigenale of 1 N G H G. Then 1/δ min is the largest eigenale of 1 N G H G 1. According to [15 Th. 8 pp. 69] we hae Based on (39) and (40) we hae λ i N < δ NB min. (40) I(; ŷ ZF ) > log αn δ min. (41) As N A and N according to Marčenko-Pastr law δ min (1 β 1 ) 2 [13 q. 1.10] (41) redces to I(; ŷ ZF ) > log P(1 (42) almost srely. By sbstitting (37) and (42) into (36) as N A and N with fixed β 1 β 2 and β 3 ˆR S ZF < Φ (P β 2 ) log P(1 almost srely. RFRNCS [1] A. D. Wyner The wire-tap channel Bell Syst. Tech. J. ol. 54 no. 8 pp Oct [2] C.. Shannon Commnication theory of secrecy systems Confidential report [3] S. K. Leng-Yan-Cheong and M.. Hellman The Gassian wire-tap channel I Trans. Inf. Theory ol. 24 no. 4 pp Jl [4] M. Bloch J. Barros M. Rodriges and S. W. McLaghlin Wireless information-theoretic secrity I Trans. Inf. Theory ol. 54 no. 6 pp Jn [5] Y. Liang H. V. Poor and S. Shamai Secre commnication oer fading channels I Trans. Inf. Theory ol. 54 no. 6 pp Jn [6] S. Goel and R. Negi Garanteeing secrecy sing artificial noise I Trans. Wireless Commn. ol. 7 pp Jn [7] A. Khisti and G. W. Wornell Secre transmission with mltiple antennas I: The MISOM wiretap channel I Trans. Inf. Theory ol. 56 no. 7 pp [8] Secre transmission with mltiple antennas Part II: The MI- MOM wiretap channel I Trans. Inf. Theory ol. 56 no. 11 pp [9] S. Li Y. Hong and. Viterbo Practical secrecy sing artificial noise I Commnications Letters ol. 17 no. 7 pp [10] I. Csiszár and J. Körner Broadcast channels with confidential messages I Trans. Inf. Theory ol. 24 no. 3 pp May [11]. Telatar Capacity of mlti-antenna Gassian channels ropean Transactions on Telecommnications ol. 10 no. 6 pp [12] H. Shin and J. H. Lee Closed-form formlas for ergodic capacity of MIMO Rayleigh fading channels in Proc. I Int. Conf. Commn. (ICC 03) Anchorage US May 2003 pp [13] A. M. Tlino and S. Verdú Random Matrix Theory and Wireless Commnications. North America: Now Pblishers Inc [14]. Lkacs and. P. King A property of the normal distribtion Ann. Math. Statist. ol. 25 no. 2 pp [15] H. Lütkepohl Handbook of matrices. John Wiley and Sons 1996.