Capacity of a Broadcast Channel with Luo, Gohary, Yanikomeroglu - PowerPoint PPT Presentation

Capacity of a Broadcast Channel with Gaussian Jamming and a Friendly Eavesdropper 1 Capacity of a Broadcast Channel with Luo, Gohary, Yanikomeroglu Gaussian Jamming and a Friendly Channel Eavesdropper Model Relaying Schemes Kevin Luo, Ramy Gohary, Halim Yanikomeroglu Carleton University, Ottawa, ON, Canada Oct 2015

Capacity of a Broadcast A Battlefield Communication Channel with Gaussian Jamming and Scenario a Friendly Eavesdropper 2 • a drone sends common information to two ground Luo, Gohary, Yanikomeroglu troops Channel • a malicious jammer transmits high power Gaussian Model signal to disrupt the communication Relaying Schemes

Capacity of a Broadcast Literature Review (related work) Channel with Gaussian Jamming and a Friendly Eavesdropper 3 • Basar, Gaussian channel with jamming, 1983; Luo, Gohary, Yanikomeroglu • Kashyap, Basar and Srikant, correlated jamming on Channel MIMO Gaussian fading channels, 2004; Model • Tekin and Yener, Gaussian multiple access channel Relaying Schemes with two-way wiretap, 2008; • Shafie and Ulukus, mutual information games in multiuser channels with correlated jamming, 2009; • Lai and H. E. Gamal, relay-eavesdropper channel, 2008. In particular, the relay can be seen as a "friendly jammer" who forwards noise to the malicious eavesdropper to improve secrecy.

Capacity of a Broadcast Eavesdropper Channel with Gaussian Jamming and a Friendly We introduce a “friendly eavesdropper" who Eavesdropper 4 • picks up the jammer’s signal Luo, Gohary, Yanikomeroglu • assists the communication on an orthogonal channel Channel Model Relaying Schemes

Capacity of a Broadcast System Model-1 Channel with Gaussian We conceive the role of the eavesdropper as a relay Jamming and a Friendly Eavesdropper • drone → source 5 • troops → receivers Luo, Gohary, Yanikomeroglu • jammer → noise or interference Channel • eavesdropper → relay Model a 1 Y 1 Y s , 1 Relaying Schemes Receiver 1 b 1 Z 1 ∼ N ( 0 , N 1 ) X e J Y e X Friendly Jammer Transmitter Eavesdropper R e 1 b 2 Z 2 ∼ N ( 0 , N 2 ) a 2 Y s , 2 Y 2 Receiver 2

Capacity of a Broadcast System Model-2 Channel with Gaussian Jamming and a Friendly Eavesdropper 6 Model details of the eavesdropper as a relay: Luo, Gohary, Yanikomeroglu • The eavesdropper has average power constraint • The eavesdropper-to-receiver links are noisy links Channel Model • The eavesdropper’s transmission is rate-limited by the Relaying Schemes higher capacity of the two eavesdropper-to-receiver links • The jammer does not cooperate with the eavesdropper, i.e., the jammer’s codebook is not exposed to the eavesdropper What is the eavesdropper’s optimal strategy to enable the maximum rate to be reliably communicated between the source and the receivers?

Capacity of a Broadcast Relaying Schemes Channel with Gaussian Jamming and a Friendly Eavesdropper 7 Luo, Gohary, Yanikomeroglu Two types of relaying schemes: Channel • Relaying schemes that require the eavesdropper (relay) Model to know the the jammer’s codebook: Relaying Schemes • decode-and-forward (DF) and hash-and-forward (HF) • DF and HF are not suitable for this channel • Relaying schemes that do not require the eavesdropper (relay) to know the the jammer’s codebook: • amplify-and-forward (AF) • compress-and-forward (CF) and variants

Capacity of a Broadcast Cut-set Bound Channel with Gaussian Jamming and • We first show a capacity upper bound by deriving the a Friendly Eavesdropper cut-set bound: 8 C ≤ min Luo, Gohary, � C ( γ 1 ) + C ( γ e,1 ) , C ( γ 2 )+ C ( γ e,2 ) � . Yanikomeroglu where γ i is the signal-to-jamming ratio of the Channel source-to-receiver i link and γ e , i is the signal-to-noise Model ratio of the eavesdropper-to-receiver i link. Relaying Schemes Y 1 Y s , 1 Receiver 1 γ 1 γ e , 1 Z 1 Y e X e J X Friendly Jammer Transmitter Eavesdropper R e Z 2 γ 2 γ e , 2 Receiver 2 Y 2 Y s , 2

Capacity of a Broadcast AF: Suboptimal Channel with Gaussian Jamming and a Friendly Eavesdropper 9 Luo, Gohary, Yanikomeroglu • The achievable rate expression of the AF scheme can Channel Model be obtained by Relaying Schemes R AF = min � � i = 1 , 2 {C γ i ( 1 + γ e , i ) } . which is below the cut-set bound in general. • AF does not achieve the cut-set bound in general.

Capacity of a Broadcast CF with Standard Decoding: Channel with Gaussian Jamming and Suboptimal a Friendly Eavesdropper 10 Luo, Gohary, • Conventional CF Yanikomeroglu • The relay bin index is recovered by finding a unique Channel codeword representing the bin index in the joint Model typicality set with the received signal at the receiver. Relaying Schemes • The achievable rate expression of the conventional CF can be obtained by R CF ≤ min i = 1 , 2 C ( γ i ) + min i = 1 , 2 C ( γ e , i ) , which is also below the cut-set bound in general. • CF with standard decoding does not achieve the cut-set bound in general either.

Capacity of a Broadcast CF with List Decoding: Capacity Channel with Gaussian Jamming and Achieving a Friendly Eavesdropper 11 Luo, Gohary, Yanikomeroglu Channel Model Relaying Schemes • Modified CF • Use the same codebook structure and encoding as conventional CF.

Capacity of a Broadcast CF with List Decoding: Capacity Channel with Gaussian Jamming and Achieving a Friendly Eavesdropper 12 Luo, Gohary, • Modified CF (continued) Yanikomeroglu • In decoding block j , receiver i finds unique Channel y n e , j − 1 , x n j − 1 , x n (ˆ e , j ) such that Model y n e , j − 1 , x n j − 1 ) } jointly typical with y n Relaying • { (ˆ i ( j − 1 ) (not necessarily Schemes unique); and • x n e , j jointly typical with y n s , ij , i = 1 , 2. ( x n e , j is the y n eavesdropper codeword corresponding to ˆ e , j − 1 ) X n X n Y n ˆ e e y n s , ij y n i ( j − 1 )

Capacity of a Broadcast CF with List Decoding: Capacity Channel with Gaussian Jamming and Achieving a Friendly Eavesdropper 13 Luo, Gohary, Yanikomeroglu Channel • The eavesdropper uses Model • Standard CF encoding, with Relaying Schemes • Gaussian codebook. This signaling strategy is able to achieve the cut-set bound and hence C = min � C ( γ 1 ) + C ( γ e,1 ) , C ( γ 2 )+ C ( γ e,2 ) � .

Capacity of a Broadcast Benefit of Eavesdropper Channel with Gaussian Jamming and a Friendly Eavesdropper 14 • Capacity without eavesdropper Luo, Gohary, Yanikomeroglu C No Eavesdropper = min i = 1 , 2 C ( γ i ) . Channel Model Relaying Schemes • The rate gain provided by the friendly eavesdropper is given by � � min ∆ + C ( γ e,1 ) , C ( γ e , 2 ) γ 1 ≥ γ 2 , , � � min C ( γ e,1 ) , ∆ + C ( γ e , 2 ) , γ 1 < γ 2 , where ∆ � |C ( γ 1 ) − C ( γ 2 ) | .

Capacity of a Broadcast Numerical Result Channel with Gaussian Jamming and • Consider the instance γ 2 = 4, γ e , 1 = 3, γ e , 2 = 2 and a Friendly Eavesdropper γ 1 ∈ [ 0 , 4 . 5 ] . 15 Luo, Gohary, Yanikomeroglu 2 Channel Model 1.8 rate (bits per channel use) Relaying 1.6 Schemes 1.4 1.2 1 0.8 0.6 Capacity with Eavesdropper 0.4 Capacity without Eavesdropper CF with standard decoding 0.2 Amplify-and-Forward 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 γ 1

Capacity of a Broadcast Summary and Conclusions Channel with Gaussian Jamming and a Friendly Eavesdropper 16 Luo, Gohary, Yanikomeroglu Channel • Channel with multiple receivers, a jammer and a Model friendly eavesdropper. Relaying Schemes • AF and conventional CF with Gaussian codebooks do not achieve cut-set bound. • CF with list decoding and Gaussian codebooks achieves cut-set bound.

Capacity of a Broadcast Comments on NNC and SNNC Channel with Gaussian Jamming and a Friendly Eavesdropper 17 Luo, Gohary, Yanikomeroglu • NNC and SNNC uses a 1-to-1 mapping between the codewords in ˆ Y e and in X e . Channel Model • This is in contrast with conventional CF and modified Relaying CF, both of which use the N -to-1 mapping from Schemes Wyner-Ziv binning. • The 1-to-1 mapping induces an additional constraint on the estimation noise at the eavesdropper since X e is rate limited in the considered channel. • This constraint results in a rate loss in general.

Capacity of a Broadcast Comments on HF Channel with Gaussian Jamming and a Friendly Eavesdropper 18 Luo, Gohary, Yanikomeroglu Channel Model • In HF, the relay constructs a mapping from J to the Relaying Schemes bins ( X e ). • Mapping available at receivers. Hence, not applicable in jamming case.