Secret Communcaton usng Artfcal Nose Roht Neg, Satashu Goel C Department, Carnege Mellon Unversty, PA 151, USA {neg,satashug}@ece.cmu.edu Abstract The problem of secret communcaton between two nodes over a wreless lnk s consdered, where a passve eavesdropper may overhear the communcaton. It s desred that the eavesdropper be unable to decode the message. We show that secrecy can be acheved by addng artfcally generated nose to the nformaton bearng sgnal such that t does not degrade the ntended recever s channel. We consder two dfferent scenaros; one n whch the transmtter has multple transmt antennas and the other n whch the transmtter has a sngle antenna but helper nodes are avalable. In the multple antenna scenaro, the degrees of freedom provded by the multple antennas s used to generate nose ntellgently so that t degrades only the eavesdropper s channel. In the multple helper scenaro, even though the transmtter does not have multple antennas, the helper nodes smulate the effect of multple antennas and allow the transmtter to generate artfcal nose as n the prevous case. Keywords: Prvacy, secrecy capacty, wreless I. INTRODUCTION The broadcast nature of wreless medum gves rse to a number of securty ssues. In partcular, t makes t hard to lmt access to wreless networks and makes t easy to eavesdrop, even from a dstance. Secrecy problems nvolve three nodes; transmtter, recever and an eavesdropper. The transmtter wants to communcate wth the ntended recever wthout the eavesdropper beng able to decode the secret message. We consder the problem of secret communcaton n the wreless envronment, where a passve eavesdropper may overhear the communcaton. Snce the eavesdropper s assumed to be passve, n general ts locaton and even ts presence wll not be known to the transmtter. Hence, any secrecy scheme for ths stuaton must not assume knowledge about the eavesdropper s locaton or ts presence. Our approach s nspred by the result n [1], whch showed that secret communcaton s possble f the eavesdropper s channel s worse than the recever s channel. [1] also defned a noton of secrecy capacty, whch essentally s the maxmum rate at whch the ntended recever s decodng error probablty tends to zero, whle the eavesdropper s error probablty tends to one. However, n general, there s no guarantee that the recever wll have better channel than the eavesdropper (e.g., consder the case where eavesdropper s closer to the transmtter than s the recever). In ths paper, we show an nterestng applcaton of multple antennas, where ths condton can be satsfed by producng artfcal nose. Ths nose s Ths work was supported n part by Cylab, CMU under grant DAAD19- -1-89 from the Army Research Offce. created such that t degrades the eavesdropper s channel but does not affect the channel of the ntended recever, thus allowng perfectly secure communcaton. In a smple case, t s possble to create such artfcal nose f the transmtter has multple antennas. However, even f the transmtter has a sngle antenna but there are helper nodes avalable, such nose can stll be produced and perfect secrecy can be acheved. Note that ths paper consders nformaton theoretc securty whch s provable as opposed to computatonal securty, whch s generally unprovable []. A technque for secret communcaton usng channel state nformaton (CSI) as the secret key was descrbed n [5] and was generalzed for the mult-antenna scenaro by [4]. An abstract characterzaton of secrecy capacty of the knd dscussed by [5] was obtaned by []. In contrast, our paper assumes that the CSI s publcly avalable, so that t cannot be used to obtan a secret key. Secton II formally ntroduces the problem of secret communcaton n the wreless envronment. Secton III develops the concept of artfcal nose. Two scenaros are consdered; one where the transmtter has multple antennas and the other when the transmtter does not have multple antennas but helper nodes are avalable. Smulaton results are presented n Secton IV. Fnally, Secton V concludes the paper. II. PROLM SCNARIO Vectors are denoted by bold font. For convenence, we measure nformaton n nats nstead of bts (.e. log e ( ) s assumed for entropy). Wherever applcable, we assume that transmssons of all nodes are synchronzed (whch s clearly an dealstc assumpton). We consder two dfferent scenaros. Scenaro 1 (Multple antennas): Fg. 1(a) shows a transmtter A wth N antennas and a recever and eavesdropper wth only one antenna each. We assume that multple eavesdroppers (f they exst) cannot collude to form a multple antenna recever. An example of such a scenaro s a wreless LAN wth the base staton as the transmtter. The channel gan vectors for channels from A to and, at tme k, are gven by h k and g k respectvely. We assume that both channels are flat fadng and the recever can estmate h k perfectly and feed t back to the transmtter wthout errors. Also, we assume that the transmtter can authentcate the fed back h k (perhaps usng a shared ntal key). We assume that the eavesdropper s passve (.e., lstens, but does not transmt). Thus, the transmtter does not know the eavesdropper s channel gan g k. However, the eavesdropper may know both h k and g k. It s
A h g A α A (a) Scenaro 1: Multple Antennas Fg. 1. α AHN α AH 1 αh 1 H 1 H N α H N α H 1 α H1 α HN α A αh N α (b) Scenaro : Multple Helpers Framework for Secrecy Capacty assumed that the components of h k and g k are..d. Raylegh dstrbuted. Scenaro (Multple helpers): Fg. 1(b) shows a transmtter A, ntended recever and eavesdropper wth only a sngle antenna each. ut several helper nodes (H 1, H,..., H N ) exst to ad secret communcaton from A to. As opposed to Scenaro 1, the helper node antennas are not under drect control of the transmtter. The channel gan from X to Y s denoted α XY. Note that the channels are not assumed to be recprocal,.e. n general α XY α Y X. It s assumed that the channel gans between A, and the helper nodes are known to all nodes (possbly, even to the eavesdropper). The secrecy of our communcaton scheme does not depend on secrecy of channel gans. The channel gans are assumed to be..d. Raylegh dstrbuted. III. SCRT COMMUNICATION USING ARTIFICIAL NOIS The key dea n ths paper s that a transmtter, perhaps n cooperaton wth the helper nodes, can generate nose artfcally to conceal the secret message that t s transmttng. The nose s generated such that only the eavesdropper s affected but not the ntended recever. We frst consder the smpler Scenaro 1. Scenaro 1 (Multple antennas): Here, the transmtter can use ts multple antennas to transmt artfcal nose n the null space of the ntended recever s channel, and thus, not affect the recever, whle degradng the eavesdropper s channel. Let the transmtter transmt x k at tme k. The sgnals receved by the legtmate recever () and the eavesdropper () are, respectvely, z k = h H k x k + n k, (1) y k = g H k x k + e k () where n k and e k are AWGN nose samples. The transmtter chooses x k to be the sum of p k u k and w k, where u k s the Gaussan dstrbuted nformaton bearng sgnal and w k s a statstcally ndependent, Gaussan dstrbuted artfcal nose vector, so that x k = p k u k + w k. () Here, w k s chosen such that h H k w k =,.e. w k les n the null space of h H k. p k s chosen such that h H k p k and p k s normalzed so that p k = 1. Then, the sgnals receved by and are gven by z k = h H k p k u k + n k, (4) y k = g H k p ku k + g H k w k + e k. (5) Note how the artfcal nose les n the null-space of h H k, so that t does not affect the ntended recever, whle the eavesdropper s channel s degraded wth hgh probablty. If w k s chosen to have a fxed drecton, the secrecy capacty wll be small f g H k w k s small. To avod ths, {w k } are chosen to be..d. Gaussan random vectors n the null space of h H k, w k = Γ k v k, (6) where Γ k s a null space matrx of h H k. It s assumed that Γ k s an orthogonal bass of the null space so that Γ H k Γ k = I holds. The components of v k are chosen to be..d. Gaussan wth zero mean and varance σv. Then, the secrecy capacty s gven by [1] C a sec I(Z; U) I(Y ; U) (7) = log(1+ hh k p k σ u σ n ) log(1+ gh k p k σu gk Hw ), (8) k + σe where gk Hw k = (g k Γ k Γ H k gh k )σ v. Thus, secrecy capacty s the dfference of the mutual nformaton between the transmtter and the ntended recever versus the eavesdropper. For a passve eavesdropper, g k s not known to the transmtter, so the secrecy capacty s maxmzed by choosng p k = h k / h k. Clearly, wth ths choce of p k, p k u k les n the range space of h k. Hence, the nformaton bearng sgnal s transmtted n the range space of h k whereas the artfcal nose s transmtted n the null space of h H k. Thus, the two spaces are well separated. Scenaro (Multple helpers): When multple antennas are not avalable at the transmtter, coordnaton wth the helper nodes can, hopefully, allow the prevous method to work. However, the helper nodes are not n drect control of the transmtter. How can they then coordnate n transmttng the artfcal nose (whch, by defnton, s random and cannot be known to the helpers)? A novel -stage protocol acheves ths, as descrbed below. In the frst stage, the transmtter and the ntended recever both transmt ndependent artfcal nose sgnals to the helper nodes. The helper nodes and the eavesdropper receve dfferent weghted versons of these two sgnals. In the second stage, the helper nodes smply replay a weghted verson of the receved sgnal, usng a publcly avalable sequence of weghts. At the same tme, the transmtter transmts ts secret message, whle also cancellng the artfcal nose at the ntended recever. Formally, Stage 1 In the frst stage, A and transmt α A x and y respectvely. The sgnals receved by H and are r H and r,1 respectvely. It s assumed that a node cannot transmt and receve
at the same tme. r H = α AH α A x + α H y + n (9) r,1 = α A α A x + α y + e 1 (1) Stage In the second stage, A and H transmt β α AH α H x + z and β r H respectvely. The sgnals receved by and are r and r, respectvely. r = α A z + β α H α H y + β α H n + e (11) r, = α A z + [α A β α AH α H α A β α AH α H] x + β α H α H y + β α H n + e (1) Here, e 1, e, e, {n } N =1 are AWGN nose samples. β are the random weghts used by the helper nodes (known publcly). z s the nformaton bearng sgnal whle x and y are transmtted to conceal the transmsson of z. Note how A s transmsson of β α AH α Hx cancels out the transmsson of the helper nodes precsely, only at the ntended recever, but not at the eavesdropper, thus causng artfcal nose n the latter. The equvalent channel from A to s gven by r = α A z + β α Hn + e. (1) Note that y s known to and hence s not ncluded n the equvalent channel. Varyng β s performs the same functon as varyng the drecton of w k n Scenaro 1, and thus, reduces the probablty of the artfcal nose beng nulled at the eavesdropper. The channel between A and can be wrtten as ( ) x r = h z z + H xy + n, (14) y ( ) ( ) h z =, n = e 1 α A β, (15) α H n + e ( ) αa α H xy = A α γ β, (16) α H α H where, γ = α A β α AH α H α A β α AH α H. quaton (14) represents a SIMO (Sngle Input Multple Output) channel whch s degraded by both AWGN nose and nterference. The capacty for ths channel s gven by [7], C = log h z h H z σ z + K log K, (17) ( ) σ K = H x xy σy H H xy + ( ) σ e1 α H σβ σn + σe. (18) Thus, the secrecy capacty s gven by C sec h I(Z; R ) I(Z; R,1, R, ) (19) = log(1 + α A σz σn ) log h zh H z σ z + K, () K where, σn = α H σβ σn + σe. The secrecy capacty obtaned n (8) and () s a functon of random channel gans. Therefore, the expected secrecy capacty, as well as the outage probablty (cumulatve dstrbuton functon) of the secrecy capacty can be computed for the two scenaros, usng Monte Carlo smulatons. In both the scenaros, we assume that the total power transmtted per transmsson s constraned to P. The total transmtted power n the multple antenna scenaro s gven by f 1 (σu, σ v ) = σ u + ( 1)σv. (1) The combnaton of powers (σ u, σ v) s chosen such that t maxmzes the lower bound on the expected secrecy capacty. Thus, Csec a max f 1(σu,σ v ) P [log(1+ hh k p k σu ) hk,g k σ n log(1+ gh k p k σu gk Hw )]. () k + σe The total transmtted power n the multple helper case s gven by f (σx, σ y, σ z, ξ) = (1+ ξ)σx +(1+ ξ)σy +σ z + ξ, () where we choose σ β = ξ for smplfcaton. In ths case, the combnaton of powers (σx, σ y, σ z, ξ) s chosen to maxmze the lower bound on the expected secrecy capacty and hence, Csec h max f (σx,σ y,σ z,ξ) P [log(1 + α A σz σn ) log(1 + α A σ z σ n )], (4) where the expectaton s over all the channel gans. For computng the outage probabltes for a gven number of transmt antennas (or number of helper nodes) and total transmt power P, the combnaton of powers used s the one found by performng the optmzaton n () (or (4)). IV. SIMULATION RSULTS We compute the expected secrecy capacty for the two scenaros, after optmzng the transmt powers, subject to the total power constrant of P. We compare t wth the capacty of the transmtter-recever lnk, for the same power constrant. The dfference of the two provdes an upper bound on the loss n capacty because of the secrecy requrement. We study the varaton of the expected value and the outage probablty of secrecy capacty wth the number of transmt antennas and the total avalable power P. In the multple antenna scenaro, t s assumed that P has been normalzed by the power of AWGN
xpected Secrecy Capacty (nats/symbol) Multple Antennas 8 N = T 7 N = T 6 N T N T 5 4 1 1 15 5 Power P (d) Outage Probablty 1 1 1 1 1 Multple Antennas = = 5 1 4.6.8 1 1. 1.4 Capacty (nats/symbol) Fg.. xpected Secrecy Capacty: Multple antenna scenaro Fg. 4. Outage Probablty: Multple antenna scenaro xpected Secrecy Capacty (nats/symbol) Multple Helpers 8 = 7 = 6 5 4 1 5 1 15 5 Power P (d) Outage Probablty 1 Multple Helpers = = 5 1.5.1.15. Capacty (nats/symbol) Fg.. xpected Secrecy Capacty: Multple helpers scenaro Fg. 5. Outage Probablty: Multple helper scenaro nose power sample n k, σn. = [ n k ]. Further, t s assumed that the AWGN nose samples n k and e k have the same power,.e. σe. = [ n k ] = σn. Smlar normalzaton s assumed n the multple helper scenaro, along wth the assumpton that all AWGN nose samples have equal power. The expected secrecy capacty and the outage probablty for secrecy capacty are computed usng Monte Carlo smulatons. In the multple antenna scenaro, for each h H k, ts null space matrx Γ k s computed. Then, gven h k and g k, secrecy capacty s computed usng (8). The expected secrecy capacty s obtaned by averagng over the varous realzatons of h k and g k. The combnaton of powers (σu, σ v ) s chosen to maxmze the expected secrecy capacty, accordng to (). Ths s done by performng a brute force search over a dscretzed space. Smlarly, n the multple helper scenaro, gven a set of channel gans, secrecy capacty s computed usng (). The combnaton of powers (σx, σ y, σ z, ξ) s chosen accordng to (4), usng a brute force search over a dscretzed space. Fg. and show that the varaton of expected secrecy capacty (sold lnes) wth power s smlar to that of capacty (dashed lnes). Thus, secrecy capacty behaves lke capacty n both the scenaros and the dfference between the two represents the loss because of the secrecy requrement. Further, n the case of multple antenna scenaro, Fg. shows that both capacty and secrecy capacty ncrease wth the number of transmt antennas. The varaton of expected secrecy capacty wth power s governed by two factors. quatons (7) and () show that the expected secrecy capacty s lower bounded by the dfference of two terms, I(Z; U ) and I(Y ; U ), where U s found by performng the optmzaton n (). Snce, p k s chosen as p k = h k / h k, I(Z; U ) s equal to the capacty on the transmtter-recever lnk. The only dfference s that the power avalable on ths lnk for nformaton transmsson s σ u, whch s n general less than P. The loss n capacty because of secrecy requrement occurs because of two factors;
the frst one arses because only part of the total power s avalable for nformaton transfer and the other factor s the mutual nformaton I(Y ; U ), whch represents the amount of nformaton that the eavesdropper has about the nformaton bearng sgnal. Smulaton results (Fg. ) show that the combned effect of these two factors remans roughly constant as power P s vared. Fg. and 4 show that as the number of transmt antennas ( ) ncreases, the expected secrecy capacty ncreases whle the outage probablty decreases. Intutvely, as ncreases, the dmensons of the null space of h H k (= 1) also ncreases. However, the range space of h k s stll onedmensonal. Thus, the probablty of g k havng a large component along h k reduces rapdly as ncreases. On the other hand, ts component n the null space of h H k tends to be much larger. Therefore, as ncreases, wth a hgh probablty, gk Hp k s small and so s I(Y ; U ). quaton (7) shows that as I(Y ; U ) decreases, secrecy capacty ncreases. Further, for a fxed σu the capacty on that transmtter-recever lnk ncreases as ncreases because t s a MISO (Multple Input Sngle Output) lnk [6]. Thus, as ncreases, the probablty of secrecy capacty havng a small value reduces rapdly. Ths phenomenon ncreases the expected secrecy capacty and also reduces the outage probablty of secrecy capacty, as shown by Fg. 4. The fgure also shows that farly low outage probabltes are achevable as number of transmt antennas are ncreased. The multple helper scenaro explores the case when the transmtter does not have multple antennas. Instead multple helper nodes are avalable whch help the transmtter n generatng artfcal nose. We study the problem under the constrant on total power transmtted by all nodes. Fg. shows that the expected secrecy capacty reduces as the number of helper nodes ncreases. The prmary reason for ths behavor s that as the number of helper nodes ncreases, the power avalable per node reduces. The power of the nformaton bearng sgnal, σz s lmted and consequently the secrecy capacty reduces. As opposed to the multple antenna scenaro, the helper nodes are not under the drect control of the transmtter. Any cooperaton between the transmtter and the helper nodes (or among the helper nodes themselves) requres multple transmssons. Thus, for every transmsson of the secret nformaton sgnal, several transmssons must occur for the artfcal nose to be produced, whch makes ths scheme neffcent n comparson wth the multple antenna scheme. Fg. 5 shows that the outage probablty ncreases as the number of helper nodes ncreases, prmarly because the power avalable per node becomes more lmted. However, we note n multple helper scenaro, that f there s more than one colludng eavesdropper, we wll need to use more than one helper node to ensure secrecy. The varaton of secrecy capacty and outage probablty wth the number of helper nodes mples that the mnmum requred number of helpers should be used (whch s one more than the number of colludng eavesdroppers). V. CONCLUSION We consdered the problem of secret communcaton n the wreless envronment, n the presence of a passve eavesdropper. We showed how artfcally generated nose can be added to the nformaton bearng sgnal to acheve secrecy. In the multple antenna scenaro, the transmtter can use the multple antennas to generate the artfcal nose ntellgently such that t degrades only the eavesdropper s channel. Further, we showed that even f the transmtter does not have multple transmt antennas, helper nodes can smulate the effect of multple antennas and the transmtter can stll produce artfcal nose. We showed that non-neglgble rates are achevable for secret communcaton. Further, farly low outage probabltes of secrecy capacty can be acheved. Future work wll nvestgate the behavor of secrecy capacty n presence of multple colludng eavesdroppers, and attempt to desgn practcal schemes to acheve the rates of secret communcaton ndcated by ths paper. RFRNCS [1] I. Csszar, J. Korner, roadcast Channels wth Confdental Messages, I Trans. Info. Theory, pp. 9-48, May 1978. [] U. M. Maurer, S. Wolf, Uncondtonally Secure Key Agreement and the Intrnsc Condtonal Informaton, I Trans. Info. Theory, vol. 45, no., pp. 499-514, March 1999. [] U. M. Maurer, Secret Key Agreement by Publc Dscusson from Common Informaton, I Trans. Info. Theory, vol. 9, no., pp. 7-74, May 199. [4] A.. Hero, Secure Space-Tme Communcaton, I Trans. Info. Theory, pp. 5-49, December. [5] H. Koorapaty, A. A. Hassan, S. Chennakeshu Secure Informaton Transmsson for Moble Rado, I Trans. Wreless Communcatons, pp. 5-55, July. [6] G. J. Foschn, M. J. Gans On lmts of wreless communcatons n a fadng envronment when usng multple antennas, Wreless Personal Commun.: Kluwer Academc Press, no 6, pp. 11-5, 1998. [7] G. J. Foschn, D. Chzhk, M. J. Gans, C. Papadas, R. A. Valenzuela, Analyss and Performance of Some asc Space-Tme Archtectures, I J. Select. Areas Comm., vol. 1, no., pp. -, Aprl.