Mobility-Assisted Covert Communication over Wireless Ad Hoc Networks Hyeon-seong Im 1 and Si-Hyeon Lee 2 1 Department of Electrical Engineering, POSTECH 2 School of Electrical Engineering, KAIST ISIT 2020 1 / 20
Overview Introduction 1 Problem Statement 2 Result 3 Proof Idea 4 Extension 5 Conclusion 6 2 / 20
Covert Communication Communication should be not detected from a warden. Hypothesis test: communicating ( H 1 ) or non-communicating ( H 0 ) � D ( Q Z l � Q l P ( H 0 | H 1 ) + P ( H 1 | H 0 ) > 1 − 0 ) (1) Covertness constraint: D ( Q Z l � Q l 0 ) ≤ δ AWGN channel: Sufficiently small transmission power compared with noise level is required. [Bash et al . 2013, Wang et al . 2016]. 3 / 20
Covert Communication over a Wireless Ad Hoc Network Unit area network: n nodes and n w wardens are randomly distributed. n source-destination pairs are randomly determined. n nodes should communicate while satisfying the covertness constraint from each warden. Focus: Proving the capacity scaling law 4 / 20
Capacity Scaling with Fixed Node Location [Cho, Lee, and Tan 2019] Fixed node location : n w = Θ( n s ) ( s > 0): The number of wardens Covertness constraint: Warden’s received power should not be large. Preservation region: Transmission of nodes is not permitted. Throughput scaling SNR s 1 = n (1 / 2 − s / 2)( α − 2) : Short range SNR (length of Θ( n − 1 / 2 )) √ l 5 / 20
Capacity Scaling with Mobile Node Location Our model: Nodes have mobility. Mobility is essential in some cases. ex) military communication No covertness constraint: Capacity is linearly scaled over n regardless of the mobility of the nodes [Tse et al . 2002, Ozgur et al . 2007]. Q) If covertness constraint exists, then does mobility improves throughput scaling? A) Mobility improves throughput scaling! (Why?) 6 / 20
Problem Statement: Network Model Unit disk network n nodes: uniformly and independently distributed in each time t Location of nodes: SSS and ergodic across time t Each node is a source and a destination simultaneously. n source-destination pairs are randomly determined. n w = Θ( n s ) (0 < s < 1) non-colluding wardens: Same distribution with nodes (or fixed location) 7 / 20
Problem Statement: Network Model Received signal at node j : Y j [ t ] = � n k =1 H jk [ t ] X k [ t ] + N j [ t ] X k [ t ]: Transmitted signal by node k N j [ t ] ∼ CN (0 , N 0 ): Gaussian random noise √ G H jk [ t ] = ( d jk [ t ]) α/ 2 exp( j θ jk [ t ]): Large scale path loss α > 2, θ jk [ t ]: uniformly and independently distributed phase Received signal at warden w : Z w [ t ] = � n k =1 H ′ wk [ t ] X k [ t ] + N ′ w [ t ] CSI is available only at the receivers. 8 / 20
Problem Statement: Covertness Constraint Each warden observes l channel outputs. Test hypothesis whether nodes are communicating or not. Covertness constraint for all wardens with threshold δ > 0: w � Q × l D ( Q Z l w ) ≤ δ for w = 1 , 2 , ..., n w N ′ D ( ·�· ): Relative entropy w : Distribution of the received signal at warden w over l channel Q Z l uses (communicating). Q × l w : Distribution of the received signal at warden w over l channel N ′ uses (non-communicating). 9 / 20
Problem Statement: Long-Term Throughput Long-term throughput λ ( n , s ) is feasible if T 1 � R jk ( n , s , t ) ≥ λ ( n , s ) (2) lim T T →∞ t =1 for all source-destination pairs ( j , k ). R jk ( n , s , t ): Throughput of a source-destination pair Goal: Characterize the scaling of the maximally achievable aggregate throughput T ( n , s ) = n λ ( n , s ). 10 / 20
Result: Achievable Aggregate Throughput Throughput scaling SNR s 1 := n (1 / 2 − s / 2)( α − 2) : Short range ( n − 1 / 2 ) SNR for 0 < s < 1 √ l Non-covertness constraint: SNR s 1 = 1 Red box: p ( R pair ( n , s ) = Θ(1)) R pair ( n , s ) : Throughput of a sender-receiver pair Throughput is linearly scaled in n if l is sufficiently small. 11 / 20
Result: Mobility Improves Throughput Scaling Throughput scaling for 2 < α ≤ 3 HC scheme: Long range communication and several hops Our scheme: Short range communications and two hops Covertness constraint: Transmission power is limited. Long range communication has throughput loss! For 2 < α ≤ 3, our scheme has throughput gain. 12 / 20
Result: Mobility Improves Throughput Scaling Throughput scaling for α > 3 MH scheme: Short range communications and multi (Θ( n 1 / 2 )) hops Our scheme: Short range communications and two hops Fewer hops have throughput gain. For α > 3, our scheme has throughput gain. Mobility improves throughput scaling! 13 / 20
Result: Upper bound on Aggregate Throughput Proving a non-trivial upper bound is not easy. Distances between senders and wardens → upper bound on the transmit power Distances between senders and receivers → transmission rate These two things are independently vary over time. Assumption: Nodes distant from every warden to a certain extent use the same power. Throughput scaling Tight under the assumption! 14 / 20
Mobility-Assisted Two-Hop Scheme Inspired by the Two-Hop scheme [Tse et al . 2002] Scheme for mobile nodes without any covertness constraint Nodes are partitioned by senders and receivers in each time t . Each sender communicates with the nearest receiver (sender-receiver pair). Preservation region [S.-W.Jeon et al . 2011]: Transmission of nodes is not permitted. Overall communication is divided into two phases: 15 / 20
Mobility-Assisted Two-Hop Scheme Phase 1: Active in odd time ( t = 1 , 3 , 5 , ... ) Sender → source, receiver → relay Each sender transmits its own source data to its paired receiver. Senders in a preservation region: No transmission 16 / 20
Mobility-Assisted Two-Hop Scheme Phase 2: Active in even time ( t = 2 , 4 , 6 , ... ) Sender → relay, receiver → destination Each sender selects and transmits destined data to its paired receiver. Senders in a preservation region: No transmission Sender might not have destined data → steady state is assumed Data transmission in each period: source → relay → destination Only using two hops Mobile nodes: Short range communications are sufficient. 17 / 20
Mobility-Assisted Two-Hop Scheme Mobility-assisted two-hop scheme ensures us T ( n , s ) = θ n (1 − ǫ ( n )) · λ pair ( n , s ) = Θ( n ) · λ pair ( n , s ) . 2 T ( n , s ): Achievable aggregate throughput θ : Proportion of senders ǫ ( n ): Region of total preservation region. λ pair ( n , s ): Feasible throughput of a sender-receiver pair Proof of λ pair ( n , s ): Different proof technique is required. (Why?) Allowable transmit power is precisely evaluated by covertness constraint. Distance between sender-receiver pair affects the order of λ pair ( n , s ). 18 / 20
Extension to s ≥ 1 Extension to s ≥ 1 SNR s 2 := n α (1 / 2 − s / 2) : Short range ( n − 1 / 2 ) SNR for s ≥ 1 √ l Not tight (Why?) Distance between a warden and the nearest sender without a preservation region is different between the pessimistic and optimistic derivations. 19 / 20
Conclusion Consider covert communications over a wireless ad hoc network with mobility. Propose mobility-assisted two-hop scheme. Capacity is linearly scaled in n for 0 < s < 1 if testing channel length is sufficiently small. Mobility improves throughput scaling. Tight for 0 < s < 1 under some mild assumption. Detailed proofs can be found on arXiv [2004.08852]. 20 / 20
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