Stealth Communication with Vanishing Power over Binary Symmetric Channels Diego Lentner, Gerhard Kramer Technical University of Munich Institute for Communications Engineering ISIT 2020
Covert Communication Covert Communication vs. Secrecy: • In secrecy, we want to hide the content of a message, i.e., we minimize the mutual information between Alice’s message and Eve’s Alice Bob channel output. I s A l i c e c o m • In covert communication/stealth, we want to hide m u n i c a w t i i t n h g B o b ? ( y e s the presence of communication! / n o ) Other names: • Low probability of detection (LPD), • Hiding information in noise. Warren Diego Lentner, Gerhard Kramer (TUM) 2
Hypothesis testing • Binary hypothesis test: ◮ Null hypothesis H 0 : z n ∼ P 0 , ◮ Alternative hypothesis H 1 : z n ∼ P 1 . • Optimal test (e.g., Neyman-Pearson ) minimizes PFA + PMD where PFA and PMD are the probabilities of false alarm and missed detection, respectively. • Bounds on the performance of an optimal test: ◮ Variational distance: PFA + PMD = 1 − V T ( P 0 , P 1 ) . ◮ KL divergence (Pinsker’s inequality): � 1 V T ( P 0 , P 1 ) ≤ 2 D ( P 0 � P 1 ) . Diego Lentner, Gerhard Kramer (TUM) 3
Measuring Covertness X n Y n • Alice sends ”0” when she is not communicating P Y | X Alice Bob • Warren observes his channel output z n and performs a binary hypothesis test : ◮ H 0 : Alice is not communicating ⇔ z n ∼ P n Z | X ( ·| 0 ) , ◮ H 1 : Alice is communicating ⇔ z n ∼ P Z n . Z n P Z | X Warren • Alice can bound Warren’s detection performance when she is communicating to Bob by ensuring D ( P Z n � P n Z | X ( ·| 0 )) ≤ δ for a small δ > 0. Diego Lentner, Gerhard Kramer (TUM) 4
Covert Communication: Main Results Square root law (Bash et al. ’13, Wang et al. ’16, Bloch ’16) • Let n be the total number of channel uses. • Warren observes Z n and compares its statistics to the null hypothesis. • Alice can reliably transmit O ( √ n ) bits to Bob without being detected by Warren. This means that the rate is zero. ⇒ lim n →∞ O ( √ = n ) / n = 0 ! Secret key lengths (Bash et al. ’13, Bloch ’16) • Alice and Bob pre-share a secret key of length K . • In general, O ( √ n log n ) pre-shared key bits are necessary. If Alice knows the statistics of her channel to Warren, O ( √ n ) key bits suffice. • If Warren’s channel is noisier than Bob’s, then covert communication can be achieved without key. • If Warren has uncertainty, e.g., about the transmission time or his noise model, then these results can be improved. Diego Lentner, Gerhard Kramer (TUM) 5
Stealth Communication Hou and Kramer ’14 • Alice does not have to remain silent when not communicating information to Bob. • She is free to send any other symbols than the ”zero-symbol”. Main idea: Alice confuses Warren by sending obfuscation symbols i.i.d. ∼ P X o , n • Warren tests his observations against P Z o , n = P Z | X · P X o , n . • Alice must ensure that D ( P Z n � P n Z o ) ≤ δ for a small constant δ > 0. Then: Positive-rate stealth communication possible! Diego Lentner, Gerhard Kramer (TUM) 6
Vanishing Power Communication Model We are interested in stealth communication • with vanishing power (VP) as without obfuscation • with energy that scales as n α , 0 ≤ α < 1, with blocklenght n . = ⇒ Average block power: � � n ≤ an α 1 � X 2 n , 0 ≤ α < 1 , a > 0 . n E i i = 1 P X , n P Y , n Assume X Y � ¯ 1 − p � 1 − an α 1 − an α p + an α • ¯ p = 1 − p , 0 ≤ α ≤ 1, and 0 < a < 1. n p 0 0 n n p • n is fixed (one-shot analysis!). How much information can we transmit p � � an α 1 − an α p + an α n ¯ with VP over n channel uses? p 1 1 n n 1 − p Diego Lentner, Gerhard Kramer (TUM) 7
VP Gallager Exponents • Rate R : • Scaling constant R α for a fixed α : M = e n α R α . M = e nR . • For M ρ = e nR ρ , DMS P X , and DMC P Y | X : • For M ρ = e n α R α ρ , DMS P X , n , and DMC P Y | X : P ( E ) ≤ e − n α ˆ P ( E ) ≤ e − nE G ( R , P X ) E α G ( R α , P X , n ) with Gallager exponent (Gallager ’68) with VP Gallager exponent � � ˆ ˆ E G ( R , P X ) = max 0 ≤ ρ ≤ 1 [ E 0 ( ρ, P X ) − ρ R ] E α E α G ( R α , P X , n ) = max 0 ( ρ, P X , n ) − ρ R α 0 ≤ ρ ≤ 1 and and �� � 1 + ρ n ˆ E α � 0 ( ρ, P X , n ) = lim n α E 0 ( ρ, P X , n ) 1 E 0 ( ρ, P X ) = − log P ( x ) P ( y | x ) 1 + ρ n →∞ y x Diego Lentner, Gerhard Kramer (TUM) 8
VP Gallager Exponent Example: a = 0 . 2, α = 0 . 5, p = 0 . 1. 0 . 16 ˆ E α 0 ( ρ, P X , n ) 0 . 14 � ��� � ¯ 0 . 12 � 1 + ρ 1 1 1 − an α 1 + ρ + an α G ( R α , P X , n ) n = lim − log p n p 1 + ρ n α n 0 . 1 n →∞ � 1 + ρ �� �� 0 . 08 � 1 1 R α, max ( P X , n ) 1 − an α 1 + ρ + an α n ¯ + p p 1 + ρ n 0 . 06 E α � � � � � ˆ ρ ρ 1 1 0 . 04 1 + ρ − p 1 + ρ − p ¯ ¯ ( 1 + ρ ) a p p , α < 1 1 + ρ 1 + ρ = 0 . 02 E 0 ( ρ, P X , n ) , α = 1 . 0 0 . 05 0 . 1 0 . 15 0 . 2 0 . 25 0 . 3 0 . 35 0 . 4 0 R α Maximum VP scaling constant for BSCs � � a ( 1 − 2 p ) log ¯ R α, max ( P X , n ) = ∂ ˆ p E α � p , α < 1 0 ( ρ, P X , n ) � = � ∂ρ I ( P X , n ; P Y | X ) , α = 1 . � ρ = 0 Diego Lentner, Gerhard Kramer (TUM) 9
Stealth Communication with VP • P Y | X = BSC ( p ) and P Z | X = BSC ( q ) . • Alice can transmit obfuscation symbols with VP � � n = bn β X n Y n 1 � X 2 n , 0 ≤ β < 1 , 0 < b . P Y | X n E Alice Bob i i = 1 • When transmitting information, she must ensure that ◮ Bob can decode her message (with high probability) Z n ◮ Warren cannot distinguish the message from obfuscation: P Z | X Warren ! D ( P Z n � P n Z o ) ≤ δ. How large can Alice choose ( α, a ) for the given vanishing obfuscation power? Diego Lentner, Gerhard Kramer (TUM) 10
Uncoded Stealth Communication • Assume all Z i are i.i.d.: ! D ( P Z n � P n Z o ) = n D ( P Z , n � P Z o , n ) ≤ δ. • For α, β < 1 and sufficiently large n , we find via Taylor approximation � 2 � an α (¯ q − q ) 2 − bn β D ( P Z , n � P Z o , n ) ≈ 1 q ¯ 2 q n n Uncoded Stealth Communication For sufficiently large n , uncoded stealth communication with α, β < 1 is possible if √ 2 q ¯ √ √ q � an α − bn β � � � ≤ k k = δ. n with ¯ q − q Diego Lentner, Gerhard Kramer (TUM) 11
Uncoded Stealth Communication 1 D ( P Z , n � P Z o , n ) = 0 possible � an α − bn β � � For α, β < 1: 0 . 9 � = 0 ⇔ n D ( P Z , n � P Z o , n ) ≤ δ possible � ≤ k √ 0 . 8 � � an α − bn β � √ ⇔ n � � an α − bn β � � ≤ k n 0 . 7 • Case bn β 0 . 6 n = 0 (covert communication): 0 . 5 α max = 0 . 5 , a ≤ k . α • Case β = 0 . 5: 0 . 4 0 . 3 α max = 0 . 5 , a ≤ k + b . 0 . 2 • Case β = 1 (Hou and Kramer ’14): 0 . 1 α max = 1 . 0 = ⇒ Positive-rate communication! 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 0 1 β Diego Lentner, Gerhard Kramer (TUM) 12
Coded Stealth Communication Coded Stealth Communication We can bound � a ( 1 − 2 p ) log ¯ p log M p , α < 1 < n α I ( P X , n ; P Y | X ) , α = 1 � � + a ( 1 − 2 q ) log ¯ q − a ( 1 − 2 p ) log ¯ q p log K , α < 1 > p � + , n α � I ( P X , n ; P Z | X ) − I ( P X , n ; P Y | X ) α = 1 where • [ x ] + = max( x , 0 ) , � ≤ k √ � � an α − bn β � • ( α, a ) satisfy n for specified ( β, b ) . Note: • In the covert communication case bn β n = 0, we recover the bounds from Bloch ’16 evaluated for BSCs. • For log K = 0, we obtain the bounds for key-less stealth / covert communication. Diego Lentner, Gerhard Kramer (TUM) 13
Proof Sketch • Random coding with binning • Pre-shared key indexes a subcodebook � � � � • Warren has to test against the entire codebook C � P n C � P n + D ( P n Z � P n D ( P Z n | ˜ Z o ) = E D ( P Z n | ˜ Z ) Z o ) E • Reliability: proof via VP Gallager exponents as � �� � � �� � ( b ) ( a ) for noisy channel coding theorem ��� � � log P n Z ( z n ) • Stealth: bound (a),(b),(c) separately. C ( z n | ˜ C ) − P n Z ( z n ) + E . P Z n | ˜ P n Z o ( z n ) (a) Proof with VP resolvability exponents (following z n Hayashi ’06, Hou and Kramer ’13). � �� � ( c ) (b) Uncoded stealth scenario, reuse result! (c) Bound with Pinsker’s and Jensen’s inequalities. Diego Lentner, Gerhard Kramer (TUM) 14
Conclusion Main results: • Unified stealth communication framework that includes covert communication and positive-rate stealth communication as extreme cases. • Simple achievability proof based on suitably modified Gallager exponents. Outlook • Generalization to arbitrary DMCs (done!) and AWGN channels • Practical coding scheme! • Stealth with feedback • ... Diego Lentner, Gerhard Kramer (TUM) 15
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