Secret Communication via Secret Communication via Multi- -antenna - PowerPoint PPT Presentation

Secret Communication via Secret Communication via Multi- -antenna Systems antenna Systems Multi Zang Li Wade Trappe Roy Yates WINLAB, Rutgers University 1

Outline � Information theory background on information security � Problem formulation for multiple antenna system � Solution for a multiple-input-single-output (MISO) system � Numerical evaluation � Conclusion 2

Introduction � Information theoretic secret communication over wireless medium in presence of a passive eavesdropper – Eavesdropper is no better than random guessing the secret message � The noisy nature of the wireless medium can be exploited to achieve information theoretic communication � Multi-antenna system: the extra degrees of freedom facilitates secret communication 3

Scenario Bob Alice Eve � Wireless broadcast channel � Passive eavesdropper � Can Alice talk to Bob secretly? If yes, what is the secrecy rate? 4

Information Secure Secret Communication Bob receives Y n Ŝ Error Probability P(S ≠ Ŝ ) ≤ ε Alice Message: P(Y|X) X n S Eve overhears Z n P(Z|X) Normalized Equivocation H(S|Z n )/H(S)>1- ε � Reliable transmission requirement � Perfect secrecy requirement � Secrecy capacity: maximum reliable rate with perfect secrecy – This rate might be very small, but we only need it to setup the key for subsequent communication 5

Wiretap Channel � Wiretap channel (Wyner75) Alice Bob X Y P(Y|X) P(Z|Y) Z Eve has a degraded channel Eve X → Y → Z = − max ( ; ) ( ; ) C I X Y I X Z sec ( ) P x 6

Broadcast Channel � Broadcast channel (Csiszar & Korner 78) Bob Alice X Y V P(Y|X) P(X|V) P(Z|X) Z Eve V is designed to confuse Eve more than Bob! = = − − max max ( ( ; ; ) ) ( ( ; ; ) ) C I V Y I V Z C I X Y I X Z sec sec → → → V X YZ X YZ 7

When does V = X? More capable condition (Csiszar & Korner 78) : � – I(X; Y) – I(X; Z) ≥ 0 for all input x Bob’s channel is more capable ⇒ V = X � – Wiretap channel satisfies the more capable condition – Gaussian broadcast channel (when Bob’s SNR > Eve’s SNR) • Leung-Yan-Cheong &Hellman 1978 Still a mystery in many other scenarios � 8

Recent Work on Wireless PHY Secrecy � Mitrpant, Vinck, Luo [ISIT 06] Wiretap with noncausal CSI � Barros & Rodrigues [ISIT 06] Outage in Rayleigh Fading � Liang & Poor [Allerton 06] Ergodic Secrecy Capacity in Flat Fading Li, Yates, Trappe [Allerton 06] Parallel Channels � � Gopala, Lai, H. El Gamal [ITA 07] Slow Fading � Khisti, Tchamkerten and Wornell [IT] Secure broadcasting � Relay channel, multiple access channel, interference channel… � Multi-antenna system: Hero 03, Negi et. al. 03, Xiaohua Li et. al. 03, Parada & Blahut 05 … 9

Problem Formulation Alice Y X H Bob Z G = + Y HX W Eve 1 = + Z GX W 2 Multi-antenna system provides gains in both communication rate and error performance. Can multiple antennas facilitate secret communication? What is the secrecy capacity for this system? 10

Why is the Problem Hard? Bob Alice ? Y=HX+W 1 V P(X|V) X =[X 1 � X n ] T Z=GX+W 2 Eve = − Capacity max ( ; ) ( ; ) C I V Y I V Z sec → → V X YZ Issues: More capable condition Preprocessing V → X ? is not satisfied! Optimal Input V ? 11

Simplification: Achievable Secrecy Rate � Take V=X to obtain a secrecy rate lower bound – Achievable Rate: R = max X I(X;Y) - I(X;Z) � Assume H & G are known to all parties � How to maximize the rate over the distribution of X? – Gaussian input characterized by covariance matrix Q Difference of concave functions � 12

Gaussian MISO: M TX antennas, 1 RX antenna/user Alice Bob Y = h T X+W 1 h = [h 1 ••• h M ] T Eve Z = g T X+W 2 X =[X 1 ••• X M ] T g = [g 1 ••• g M ] T Now the outputs are scalars! Coordinate rotation can simplify the expressions without changing the system properties 13

Coordinate Transform Y = h T X+W 1 X Z = g T X+W 2 Useless to put power in the space orthogonal to h & g = ||h|| e 1 h e 1 = X 1 e 1 + X 2 e 2 X’ g T h α = ⋅ || || || || g h e 2 = ||g|| ( α e 1 +(1- α 2 ) 1/2 e 2 ) g 14

Jamming View of the MISO Problem Bob Alice = + || || Y h X W 1 1 = α + − α + 2 X’ = [x 1 , x 2 ] T || || || || 1 Z g X g X W 1 2 2 Eve � X 1 is signal for Bob, with power P 1 � X 2 is jamming signal to annoy Eve, with power P 2 � Similar to correlated jamming [Medard 97], [Shafiee+Ulukus 05] – except X 1 and X 2 are designed and transmitted by TX, – P 1 +P 2 ≤ P � Questions: – How to signal? – How to allocate power between X 1 and X 2 ? 15

Gaussian MISO with Gaussian Input = + || || Y h X W 1 1 = α + − α + 2 || || || || 1 Z g X g X W 1 2 2 X 2 should be linear to X 1 for cancellation at Eve � � When P 1 is small, we should zero force Eve − α – Choose = X X 2 1 − α 2 1 – Eve receives Z = W 2 ⇒ pure noise – R ZF = I(X; Y) = log (1+||h|| 2 P 1 ) For zero-forcing to be possible, P 1 · P * = (1- α 2 )P � 16

Gaussian MISO with Gaussian Input = + || || Y h X W 1 1 = α + − α + 2 || || || || 1 Z g X g X W 1 2 2 Largest rate obtained by zero-forcing: � – R ZF * = log (1+||h|| 2 P * ) � But this is not optimal! – Very conservative, same rate regardless of Eve’s channel gain For P 1 > P * , choose X 2 = -c α X 1 and P 2 = P-P 1 to cancel X 1 as � much as possible ⎛ ⎞ + ⎜ 2 ⎟ || || 1 h P = ( ) 1 log – Rs(P 1 ) = I(X; Y) – I(X; Z) ⎜ ⎟ ⎜ 2 ⎟ α − − α − + 2 2 2 || || ( 1 )( ) 1 ⎝ g P P P ⎠ 1 1 17

Secrecy Rate R S (P 1 ) α = 0.7, P = 10, || h| | = 1. * R s ⎛ ⎞ + ⎜ 2 ⎟ || || 1 h P = ( ) 1 ( ) log R s P ⎜ ⎟ 1 ⎜ ⎟ 2 Largest ZF rate α − − α − + 2 2 2 || || ( 1 )( ) 1 ⎝ g P P P ⎠ 1 1 obtained at P 1 =P* ZF Cancellation is not enough R s (P 1 ) = log (1+||h|| 2 P 1 ) and Eve has a better channel ⎛ ⎞ + ⎜ 2 ⎟ || || 1 h P = * ( ) 1 max log R ⎜ ⎟ s ⎜ ⎟ 2 ≤ ≤ * α − − α − + P P P 18 2 2 2 || || ( 1 )( ) 1 1 ⎝ g P P P ⎠ 1 1

Optimal Secrecy Rate R s * P =10, ||h|| = 1 (bits per channel use) Converge to ZF rate R ZF =log (1+||h|| 2 P * ) P* = (1- α 2 )P Eavesdropper Link Gain ||g|| 19

Conclusions � The extra dimensions provided by multi-antenna system can enhance the secrecy rate � Derived the secrecy rate for MISO Gaussian broadcast channel – Coordinate transform – Partial cancellation at Eve – This rate was shown to be the capacity recently (Khisti et al ISIT2007) 20

Thanks! Any Questions? 21

Coding Procedure � Stochastic encoding, joint typical decoding (Csiszar&Korner 78) To ensure correct decoding at Bob w 2 nR S = 1 (Bob finds only one typical 1 sequence in the whole table.) V n R + R’ < I(V;Y) To ensure full equivocation at Eve (Eve finds at least one typical 2 nR’ sequence in every column.) X n = f(V n ) R’ > I(V;Z) R < I(V;Y) - I(V;Z) 22