Treating Interference as Noise is Optimal for Covert Communication over Interference Channels Kang-Hee Cho and Si-Hyeon Lee School of Electrical Engineering KAIST 2020 ISIT 1 / 14
Covert Communication ^ Y n W Decoder X n W P × n Encoder Y;Z j X H 0 : Q × n Z n 0 Warden H 1 : ^ Q Z n Reliable with low probability of detection by an adversary (warden) Optimal hypothesis testing by the warden 0 ) or H 1 : active (output dist. is ˆ H 0 : no communication (output dist. is Q × n Q Z n ) � D ( ˆ Q Z n � Q × n π 1 | 0 + π 0 | 1 ≥ 1 − 0 ) Square-root law: Throughput over n channel uses ∝ √ n in AWGN [Bash et al . 2013] and many other discrete cases [Bloch 2016; Wang et al . 2016; Tan-Lee 2019] Low transmit power compared to the noise level (AWGN) 2 / 14
Main Result for Discrete Memoryless Channels (DMCs) with a Warden Literature [Bloch 2016] Y n ^ W Decoder X n W P × n Encoder Y;Z j X H 0 : Q × n Z n 0 Warden H 1 : ^ Q Z n Binary input DMC to the decoder ( X , P Y | X , Y ), to the warden ( X , P Z | X , Z ) Off input symbol 0: Send when no communication occurs Q 1 = P Z | X ( ·| 1) and Q 0 = P Z | X ( ·| 0), P 1 = P Y | X ( ·| 1) and P 0 = P Y | X ( ·| 0) Assume Q 1 ≪ Q 0 , (i.e., supp( Q 1 ) ⊆ supp( Q 0 )). Otherwise, covert comm. is impossible Covertness requirement ⇒ The number of symbol 1 is restricted by c √ n Code rate: log M ≈ c √ n × D ( P 1 � P 0 ) Channel resolvability: Sufficient number of codewords for nearly IID dist. at the warden Message × key rate: log MK ≈ c √ n × D ( Q 1 � Q 0 ) 3 / 14
Extensions MIMO AWGN channels [Abdelaziz and Koksal 2017 ] Square root law on the blocklength still holds Scales exponentially with the number of transmitting antennas in massive MIMO limit Non-coherent Rayleigh-fading channels [Tahmasbi et al . 2020] Amplitude-constrained input distribution with finite number of input points is optimal Multiple-access channels [Arumugam and Bloch 2019 ] No sum-rate bound Broadcast channels [Tan and Lee 2019 ] Time-division is optimal over a broad class of channels Some results are quite different or simpler from the results without the covertness 4 / 14
Our Work - Interference Channels ^ X n Y n W 1 W 1 1 1 Encoder 1 Decoder 1 P × n Y 1 ;Y 2 j X 1 ;X 2 X n Y n ^ W 2 W 2 2 2 Encoder 2 Decoder 2 Consider discrete memoryless interference channels (DM-IC) with a warden The capacity region of interference channels without warden is not known in general, except some special cases e.g., Strong interference channels [Sato 1978] Injective deterministic interference channels [El Gamal and Costa 1982] Complicated coding scheme for the best known inner bound [Han-Kobayashi 1981] Our result: Treating interference as noise is optimal for covert communication over interference channels 5 / 14
Channel Model & Assumptions S K ^ Y 1 W 1 ( W 1 ; S 1 ) X 1 Rx 1 Tx 1 W × n Y K j X K ^ W K Y K Rx K S K H 0 : Q × n Z V × n 0 Warden ( W K ; S K ) X K Z j X K H 1 : ^ Q n Tx K K -user-pair binary-input DM-ICs ( X K , W Y K | X K , Y K ) X K := ( X 1 , . . . , X K ) = { 0 , 1 } K , where K := { 1 , 2 , . . . . , K } Symbol 0: Off symbol that is sent when no communication occurs The warden monitors the channel outputs of the DM-MAC ( X K , V Z | X K , Z ) Q U ≪ Q 0 for all U ⊆ K Q U : The output distribution at the warden when only Txs i , i ∈ U send symbol 1 Otherwise, covert communication is restricted to some kinds of symbol combination Q 0 cannot be represented as any convex combination of Q U for some U ⊆ K Otherwise, positive rate is achievable (we do not focus on) W ( k ) ≪ W ( k ) for all k ∈ K and for all U ⊆ K 0 U W ( k ) U : The output distribution at Rx k when only Txs i , i ∈ U send symbol 1 6 / 14
Definitions of a Code & Covert Capacity Region An ( M K , J K , n ) code for the K -user-pair DM-IC with a warden consists of K message sets [1 : M k ] for k ∈ K ; K secret key sets [1 : J k ] for k ∈ K ; K Txs x k ( w k , s k ) : [1 : M k ] × [1 : J k ] → X n for k ∈ K , (uniformly distributed); w k ( y k , s K ) : Y n K Rxs ˆ k × ( × k ∈K [1 : J k ]) → [1 : M k ] for k ∈ K . �� K � k =1 { ˆ covertness measure: D ( ˆ Q n � Q × n Probability of error: P n e := Pr W k � = W k } , 0 ) A tuple ( R K , L K ) ∈ R 2 K + is achievable if there exists a sequence of codes satisfying log M k lim inf ≥ R k , ∀ k ∈ K , � n →∞ nD ( ˆ Q n � Q × n 0 ) log J k lim sup ≤ L k , ∀ k ∈ K , � n →∞ nD ( ˆ Q n � Q × n 0 ) n →∞ P n n →∞ D ( ˆ Q n � Q × n lim e = 0 , lim 0 ) = 0 . Covert capacity region: Closure of { R K ∈ R K + : ( R K , L K ) is achievable for some L K } 7 / 14
Main Results (1) - The Number of Symbol 1 Covertness requirement ⇒ The total number of symbol 1 of the Txs, N is O ( √ n ) Fraction vector α = ( α 1 , . . . , α K ) ∈ [0 , 1] K such that � k ∈K α k = 1: The allocation of the ratio of total symbol 1 at each Tx Total # of symbol 1, N depends on α and channels to the warden, especially the 2 ( k ∈K α k Q k ( z ) − Q 0 ( z ) ) � common factor χ 2 ( α ) := � � χ 2 ( α ) in the way N ∝ 1 / z Q 0 ( z ) This chi-square distance is related to the detectability of the warden 1 N / p Total # of symbol 1 : N χ 2 ( α ) α 1 N Tx 1 α 2 N Tx 2 α K N Tx K ) χ 2 ( α ) Warden's output dist. Q 1 Q K Q 2 8 / 14
Main Results (2) - Covert Capacity Region Theorem 1 The covert capacity region is the set of the rate tuple R K satisfying R k ≤ α k D ( W ( k ) � W ( k ) ) 0 k , ∀ k ∈ K � χ 2 ( α ) / 2 2 ( k ∈K α k Q k ( z ) − Q 0 ( z ) ) for some α ∈ [0 , 1] K such that � � k ∈K α k = 1 , where χ 2 ( α ) := � . z Q 0 ( z ) Sparse interference signals are negligible compared to inherent channel uncertainty (Remark) W ( k ) : output dist. at Rx k when only Tx k send symbol 1 (P2P nature) k Given # of symbol 1, each user can transmit the maximal number of reliable bits 1 p N / Total # of symbol 1 : N χ 2 ( α ) log M 1 ≈ α 1 ND ( W (1) k W (1) α 1 N Tx 1 ) 1 0 log M 2 ≈ α 2 ND ( W (2) k W (2) α 2 N Tx 2 ) 2 0 log M K ≈ α K ND ( W ( K ) k W ( K ) α K N Tx K ) 0 K ) χ 2 ( α ) Warden's output dist. Q 1 Q K Q 2 9 / 14
Main Results (2) - Covert Capacity Region Theorem 1 The covert capacity region is the set of the rate tuple R K satisfying R k ≤ α k D ( W ( k ) � W ( k ) ) 0 k , ∀ k ∈ K � χ 2 ( α ) / 2 2 ( k ∈K α k Q k ( z ) − Q 0 ( z ) ) for some α ∈ [0 , 1] K such that � � k ∈K α k = 1 , where χ 2 ( α ) := � . z Q 0 ( z ) If Q k ( z ) = Q ( z ) , ∀ k , z (symmetric to the warden), χ 2 ( α ) and total # of 1 is fixed Time-division scheme is optimal ( α plays a role of time fraction) 1 p N / Total # of symbol 1 : N χ 2 ( α ) log M 1 ≈ α 1 ND ( W (1) k W (1) α 1 N Tx 1 ) 1 0 log M 2 ≈ α 2 ND ( W (2) k W (2) α 2 N Tx 2 ) 2 0 log M K ≈ α K ND ( W ( K ) k W ( K ) α K N Tx K ) 0 K ) χ 2 ( α ) Warden's output dist. Q 1 Q K Q 2 9 / 14
Main Results (3) - Secret Key Length Theorem 2 α k D ( W ( k ) � W ( k ) ) √ Given α , for R k = k 0 , ∀ k ∈ K , a tuple ( R K , L K ) is achievable if and only if χ 2 ( α ) / 2 L k ≥ α k [ D ( Q k � Q 0 ) − D ( W ( k ) � W ( k ) )] + k 0 , ∀ k ∈ K . � χ 2 ( α ) / 2 Using channel resolvability approach for covertness analysis requiring that R k + L k ≥ α k D ( Q k � Q 0 ) , ∀ k ∈ K � χ 2 ( α ) / 2 Sufficient number of codewords ⇒ Output distribution at the warden is nearly IID Treating interference as noise ⇒ Key sharing between only each Tx-Rx pair If D ( Q k � Q 0 ) ≤ D ( W ( k ) � W ( k ) ) (i.e. channel from Tx k to the warden is worse than k 0 the channel from Tx k to Rx k ), a secret key between user pair k is unnecessary. 10 / 14
Achievability Sketch - Reliability Fix α . Random coding and joint typicality decoding with treating interference as noise At transmitter k , the probability of symbol 1 is α k γ n Upper bound on the probability of error at receiver 1 over codebook ensemble 1 Equivalent point-to-point channel Tx 1 Tx 2 Rx 1 Rx 1 W (1) W (1) ¯ = Tx 1 Y 1 j X K Y 1 j X 1 Tx K 2 Approximate marginal channel without interference Tx 1 Tx 2 Rx 1 Rx 1 W (1) ≈ 0 W (1) ¯ Tx 1 Y 1 j X 1 Y 1 j X K Tx K 0 log M k = (1 − ǫ ) n α k γ n D ( W ( k ) � W ( k ) ) is achievable with E [ P n e ] ≤ e − cn γ n , ∀ k k 0 for arbitrarily small ǫ > 0 and a constant c > 0. Code rate ∝ the number of symbol 1 × P2P channel quality We can choose a proper γ n 11 / 14
Achievability Sketch - Covertness Channel resolvability approach for covertness analysis log M k J k = (1 + ǫ ) n α k γ n D ( Q k � Q 0 ) is achievable ∀ k and arbitrarily small ǫ > 0 with Required codebook size ∝ the number of symbol 1 × channel resolution at the warden � � D ( ˆ Q n � Q × n ≤ e − cn γ n : Output distribution at the warden is nearly IID α ,γ n ) E � nD ( ˆ Q n � Q × n � Thus, ) ≈ n γ n χ 2 ( α ) / 2 0 Achievable rate = (1 − ǫ ) α k D ( W ( k ) � W ( k ) log M k ) 0 k lim , ∀ k , � � n →∞ χ 2 ( α ) / 2 nD ( ˆ Q n � Q × n 0 ) log M k J k = (1 + ǫ ) α k D ( Q k � Q 0 ) lim , ∀ k . � � χ 2 ( α ) / 2 n →∞ nD ( ˆ Q n � Q × n 0 ) 12 / 14
Recommend
More recommend
