Multiple Antenna Secret Broadcast over Multiple Antenna Secret Broadcast over Wireless Networks Wireless Networks Ruoheng Liu and H. Vincent Poor Ruoheng Liu and H. Vincent Poor Princeton University Princeton University IAB IAB – – May, 2007 May, 2007 1
ACKNOWLEDGEMENTS ACKNOWLEDGEMENTS Ivana Marvic Marvic, , Zang Zang Li, Li, Predrag Predrag Spasojevic Spasojevic, Roy Yates, and Wade Trappe , Roy Yates, and Wade Trappe Ivana
Outline of the Talk Outline of the Talk • motivation • motivation • physical- -layer secret system layer secret system • physical • secret broadcast over wireless networks • secret broadcast over wireless networks
Reliable and Secret Communication over Wireless Network Reliable and Secret Communication over Wireless Network reliability confidentiality confidential message W • can be successfully decoded by the desired receiver (user 1) • cannot be understood by anyone else (e.g. user 2)
Secret Communication Secret Communication information-theoretic security crypto-system (physical-layer) (secret key) Bob Alice Eve • Alice sends a message to sends a message to Bob Bob • Alice • Eve accesses the channel and eavesdrops the information accesses the channel and eavesdrops the information • Eve
Traditional Crypto- -System System Traditional Crypto Eve encryptor encoder decoder decrypter Bob Alice disadvantage: • it is very difficult to distribute initial key in a large wireless network • Eve becomes smarter with time and lifetime of secret key is shorter
Physical- -Layer Secret System Layer Secret System Physical Eve secure encoder secure decoder Bob Alice advantage: • secret communication can be achieved without a key • perfect communication secrecy can be ensured
How Physical Physical- -Layer Secret System Works Layer Secret System Works How Bob Alice Eve Bob has a “better” channel than Eve
How Physical Physical- -Layer Secret System Works Layer Secret System Works How 0 0 1 1 y =[0000000000] Bob Alice w =[0000000000] z =[0110011100] Eve 0.5 0 0 0.5 1 1 0.5 • Bob can get the message ( y = w ) • Eve is kept ignorant with respect to the message ( w and z are independent)
How to Measure Secrecy Level How to Measure Secrecy Level ^ Bob: Y � W Alice Y X W H(W | Z ) Eve Z physical-layer secret system can achieve: both reliability and confidentiality • reliability is evaluated in terms of error probability • secrecy level is measured by the equivocation rate: H(W | Z )/n • perfect secrecy: H(W | Z )/n � H(W)/n
Wire- -Tap Channel Tap Channel Wire W [Wyner Wyner, 1975] , 1975] [ • user 1 is a desired desired receiver receiver • user 1 is a • • user 2 is an eavesdropper user 2 is an eavesdropper • confidential onfidential m message essage W W to User 1 to User 1 • c • degraded b broadcast roadcast c channel hannel • degraded
Broadcast Channel with Single CM Set Broadcast Channel with Single CM Set [Csisz Csiszá ár r & & K Kö örner rner, 1978] , 1978] [ • W 1 is a confidential message to user 1 • W 1 is a confidential message to user 1 • W 0 is a common message to both users to both users • W 0 is a common message • • non non- -degraded degraded b broadcast roadcast c channel hannel
Communication of Confidential Messages Communication of Confidential Messages [Wyner Wyner, 1975] , 1975] [ [Csisz [ Csiszá ár r & & K Kö örner rner, 1978] , 1978] Can we send confidential messages to both users instantaneously? ?
Single Antenna Gaussian Broadcast Channel (GBC) Single Antenna Gaussian Broadcast Channel (GBC) Channel 1 Channel 2 • Channel 1 is “ “better better” ” than Channel 2 (degraded BC) than Channel 2 (degraded BC) • Channel 1 is • this case reduces to Gaussian wire- -tap channel tap channel • this case reduces to Gaussian wire ≤ − max [ ( ; ) ( ; )] R I X Y I X Y 1 2 1 ( ) p x = 0 R 2
Multi ulti- -Antenna Antenna G Gaussian aussian B Broadcast roadcast C Channel ( hannel (MGBC MGBC) ) M Transmitter 1, 2 non-degraded BC A B 1 2 1 2 User 2 User 1 • (W 1 , W 2 ): independent, confidential confidential messages messages • (W 1 , W 2 ): independent, • how to achieve reliable, secret secret communication over the communication over the MGBC MGBC- -CM CM? ? • how to achieve reliable, • what is the capacity region for the MGBC MGBC- -CM CM? ? • what is the capacity region for the
A Computable Outer Bound for MGBC- -CM CM A Computable Outer Bound for MGBC ~ ~ ⎧ ⎫ ≤ ( ; | ) R I X Y Y C ⊆ ∩ x ∪ 1 1 2 ⎨ ⎬ ~ ~ BCC ≤ ~ ~ ⎩ ( ; | ) ⎭ R I X Y Y ( , | ) ( ) p y y p x 1 2 2 2 1 • basic ideas: decode the message in a cooperative manner evaluate the secrecy level in a individual manner • for GBC, Gaussian input can optimize this outer bound • MGBC can be transferred to an equivalent 2-dimension “Z-broadcast” channel model Ỹ 1 h 1 X 1 h 2 =0 X g 1 X 2 g 2 Ỹ 2
A Computable Outer Bound for MGBC- -CM CM A Computable Outer Bound for MGBC ~ ~ ⎧ ⎫ ≤ ( ; | ) R I X Y Y C ⊆ ∩ x ∪ 1 1 2 ⎨ ⎬ ~ ~ BCC ≤ ~ ~ ⎩ ( ; | ) ⎭ R I X Y Y ( , | ) ( ) p y y p x 1 2 2 2 1 P=10, h =[1, 0] T , g =0.3·[0.20, 0.98] T P=10, h =[1, 0] T , g =2·[0.90, 0.43] T 0.7 2 time-sharing outer bound 0.6 DPC outer bound 1.5 0.5 outer bound 0.4 R 2 1 R 2 0.3 0.2 0.5 time-sharing 0.1 DPC outer bound 0 0 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 R 1 R 1
Transmission Strategy 1: Time Sharing Transmission Strategy 1: Time Sharing basic ideas: • transmission period is divided into two slots of durations t 1 and t 2 • transmitter sends W 1 during time t 1 with power P 1 , • transmitter sends W 2 during time t 2 with power P 2 , • in each slot, the channel reduces to a Gaussian MISO wiretap channel
Transmission Strategy 1: Time Sharing Transmission Strategy 1: Time Sharing achievable region: achievable region: (R1, R2): { t 1 C s,MISO (P 1 ), t 2 C s,MISO (P 2 ) } for all t 1 P 1 +t 2 P 2 =P t 1 +t 2 =1 (R1, R2): { t 1 C s,MISO (P 1 ), t 2 C s,MISO (P 2 ) } for all t 1 P 1 +t 2 P 2 =P t 1 +t 2 =1 C s,MISO is the secrecy capacity of Gaussian MISO wiretap channel Gaussian MISO wiretap channel [Li, [Li, et. al. , CISS 07] C s,MISO is the secrecy capacity of et. al. , CISS 07] P=10, h =[1, 0] T , g =0.3·[0.20, 0.98] T P=10, h =[1, 0] T , g =2·[0.90, 0.43] T 0.7 2 time-sharing outer bound 0.6 DPC outer bound 1.5 0.5 outer bound 0.4 R 2 R 2 1 0.3 time-sharing 0.2 time-sharing 0.5 time-sharing 0.1 DPC outer bound 0 0 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 R 1 R 1 MISO wiretap MISO wiretap
Transmission Strategy 2: Secret D Transmission Strategy 2: Secret Dirty irty P Paper aper C Coding (DPC) oding (DPC) ⎧ ⎫ + + + + H H 1 ( ) 1 ( ) h K K h g K K g ≤ − U V U V ⎪ log log ⎪ R ⎪ + + ⎪ 1 2 2 H H 1 1 g K g h K h R [DPC] = ∪ ⎨ ⎬ V V + H 1 g K g + ≤ ⎪ ⎪ tr ( ) K K P ≤ U V log V R ⎪ ⎪ + 2 2 H ⎩ 1 ⎭ h K h V P=10, h =[1, 0] T , g =0.3·[0.20, 0.98] T P=10, h =[1, 0] T , g =2·[0.90, 0.43] T 0.7 2 time-sharing outer bound 0.6 DPC outer bound 1.5 0.5 DPC outer bound 0.4 R 2 R 2 1 0.3 DPC time-sharing 0.2 time-sharing 0.5 time-sharing 0.1 DPC outer bound 0 0 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 R 1 R 1
Communication of Confidential Messages Communication of Confidential Messages downlink scenario scenario downlink [Liu et al, 2006 & 2007] [Liu et al, 2006 & 2007] [Wyner [ Wyner, 1975] , 1975] [Csisz Csiszá ár r & & K Kö örner rner, 1978] , 1978] [ uplink scenario scenario uplink
Communication of Confidential Messages Communication of Confidential Messages downlink scenario scenario downlink [Liu et al Liu et al, 2006 & 2007] , 2006 & 2007] [ [Wyner [ Wyner, 1975] , 1975] [Csisz Csiszá ár r & & K Kö örner rner, 1978] , 1978] [ uplink scenario scenario uplink W 1 User 1 X 1 Y W 2 Destination ^ ^ W 1 W 2 (Y 2, X 2 ) User 2 W 1 [Liu et al Liu et al, 2006] , 2006] [
Challenge and Opportunity Challenge and Opportunity (a) IC -- CM (b) MAC -- CM W 2 W 1 W 1 W 2 W 1 W 1 W 2 W 2 W 2 W 2 W 1 W 1 (c) BC -- CM W 1 W 2 W 2 W 1 W 1 W 2 nterference C hannel with C onfidential M essages (IC IC- -CM) CM) I nterference C hannel with C onfidential M essages ( (a) I (a) ultiple A ccess C hannel with C onfidential M essages (MAC MAC- -CM) CM) (b) M ultiple A ccess C hannel with C onfidential M essages ( (b) M roadcast C hannel with C onfidential M essages (BC BC- -CM) CM) (c) B roadcast C hannel with C onfidential M essages ( (c) B
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