Introduction Noiseless Cooperation Cooperation over Noisy Channel Resolvability of the Multiple Access Channel with Two-Sided Cooperation N. Helal 1 , M. Bloch 2 and A. Nosratinia 1 1 University of Texas at Dallas 2 Georgia Institute of Technology N. Helal, M. Bloch and A. Nosratinia 1 / 22
Introduction Noiseless Cooperation Cooperation over Noisy Channel Introduction 1 Channel Resolvability Channel Resolvability in Information Theory Multi-user Channel Resolvability Noiseless Cooperation 2 The MAC with a common message The MAC with conferencing Cooperation over Noisy Channel 3 The MAC with Feedback The MAC with Generalized Feedback N. Helal, M. Bloch and A. Nosratinia 1 / 22
Introduction Noiseless Cooperation Cooperation over Noisy Channel Introduction N. Helal, M. Bloch and A. Nosratinia 2 / 22
Introduction Noiseless Cooperation Cooperation over Noisy Channel Channel Resolvability Approximation of output statistics Given i.i.d. Q X generating i.i.d. Q Z at the output of a DMC W Z | X Attempt to simulate same output statistics using codebook of rate R Goal: find the minimum R such that lim n →∞ � ( P Z n || Q ⊗ n Z ) = 0 � � � � � ⊗ � ∼ � � � � � | � � � � ∼ = � � � ( � ) Enc 2 �� � � � � | � � � � � ∈ [1, ] ∼ Channel resolvability [Wyner 75, Han-Verdú 93, Cuff 09, Hou-Kramer 13]: � � n →∞ � ( P Z n || Q ⊗ n inf R : � code of rate R s.t. lim Z ) = 0 min I ( X ; Z ) = P X W Z | X :marginal Q Z N. Helal, M. Bloch and A. Nosratinia 3 / 22
Introduction Noiseless Cooperation Cooperation over Noisy Channel Channel Resolvability in Information Theory Strong secrecy - from resolvability [Hayashi 06, Bloch 13] Approximating output statistics - non-cooperating encoders [Yassaee 10, Common information - resolvability [Wyner 75, Han 93] Goldfeld 17, Wang 18] [Wyner 75] - MAC resolvability [Steinberg 98] - state information [Han 19] 1975 1995 2005 1985 2015 Stealth and covertness Source coding Rate distortion [Steinberg 94] [Steinberg 96, Liu 15] [Hou-Kramer 13, Wang et al. 16, Bloch'16] Strong coordination [Cuff 10] Strong and semantic secrecy follow naturally Secrecy proofs conceptually clean Strong secrecy in multi-user network with cooperating encoders!! N. Helal, M. Bloch and A. Nosratinia 4 / 22
Enc 1 Introduction Noiseless Cooperation Cooperation over Noisy Channel Enc 2 Multi-user Channel Resolvability ⊲ MAC with non-cooperating encoders [Steinberg 98, Frey et al. 17] R 1 ≥ I ( X 1 ; Z | Q ) Enc 1 � ( R 1 , R 2 ) : R 2 ≥ I ( X 2 ; Z | Q ) R = R 1 + R 2 ≥ I ( X 1 , X 2 ; Z | Q ) P QX 1 X 2 Z ∈P Enc 2 P = { P Q P X 1 | Q P X 2 | Q W Z | X 1 X 2 : marginal Q Z } ⊲ MAC with cribbing [Helal et al. 18] R 1 ≥ I ( X 1 ; Z ) � Enc 1 ( R 1 , R 2 ) : R 2 ≥ I ( X 1 X 2 ; Z ) − H ( X 1 ) R = R 1 + R 2 ≥ I ( X 1 , X 2 ; Z ) P X 1 X 2 Z ∈P delay delay P = { P X 1 X 2 W Z | X 1 X 2 : marginal Q Z } Enc 2 for causal cribbing ⊲ Relay channel [Helal et al. 19] N. Helal, M. Bloch and A. Nosratinia 5 / 22
Introduction Noiseless Cooperation Cooperation over Noisy Channel Noiseless Cooperation N. Helal, M. Bloch and A. Nosratinia 6 / 22
Introduction Noiseless Cooperation Cooperation over Noisy Channel The MAC with a common message Enc 1 Enc 2 f 1 : M 0 × M 1 → X n f 2 : M 0 × M 2 → X n and 1 2 Theorem R 0 ≥ I ( U ; Z ) R 0 + R 1 ≥ I ( U , X 1 ; Z ) R 0 + R 2 ≥ I ( U , X 2 ; Z ) R 0 + R 1 + R 2 ≥ I ( X 1 , X 2 ; Z ) P U P X 1 | U P X 2 | U W Z | X 1 , X 2 N. Helal, M. Bloch and A. Nosratinia 7 / 22
Introduction Noiseless Cooperation Cooperation over Noisy Channel The MAC with conferencing Enc 1 Enc 2 g 1 k : M 1 × V k − 1 g 2 k : M 2 × V k − 1 and → V → V 1 k 2 k 2 1 f 1 : M 1 × V K 2 → X n f 2 : M 2 × V K 1 → X n and 1 2 K K � � log |V 1 k | ≤ nC 12 and log |V 2 k | ≤ nC 21 k = 1 k = 1 Theorem C 12 + C 21 ≥ I ( U ; Z ) R 1 ≥ I ( U , X 1 ; Z ) − C 21 R 2 ≥ I ( U , X 2 ; Z ) − C 12 R 1 + R 2 ≥ I ( X 1 , X 2 ; Z ) P U P X 1 | U P X 2 | U W Y , Z | X 1 , X 2 N. Helal, M. Bloch and A. Nosratinia 8 / 22
Introduction Noiseless Cooperation Cooperation over Noisy Channel Cooperation over Noisy Channel N. Helal, M. Bloch and A. Nosratinia 9 / 22
Introduction Noiseless Cooperation Cooperation over Noisy Channel The MAC with Feedback delay Enc 1 Enc 2 delay f 1 i : M 1 × Z i − 1 → X 1 i f 2 i : M 2 × Z i − 1 → X 2 i and Theorem R 1 ≥ I ( X 1 ; Z | U ) R 2 ≥ I ( X 2 ; Z | U ) R 1 + R 2 ≥ I ( X 1 , X 2 ; Z | U ) P U P X 1 | U P X 2 | U W Z | X 1 , X 2 Feedback does not improve resolvability of MAC! N. Helal, M. Bloch and A. Nosratinia 10 / 22
Introduction Noiseless Cooperation Cooperation over Noisy Channel The MAC with Generalized Feedback delay Enc 1 Enc 2 delay f 1 i : M 1 × Z i − 1 f 2 i : M 2 × Z i − 1 and → X 1 i → X 2 i 1 2 N. Helal, M. Bloch and A. Nosratinia 11 / 22
Introduction Noiseless Cooperation Cooperation over Noisy Channel The MAC with Generalized Feedback (Decode-and-Forward) Proposition (achievability) R 1 ≥ I ( X 1 , X 2 ; Z ) − I ( X 2 ; Z 1 | X 1 , U ) R 2 ≥ I ( X 1 , X 2 ; Z ) − I ( X 1 ; Z 2 | X 2 , U ) R 1 + R 2 ≥ I ( X 1 , X 2 ; Z ) I ( X 1 ; Z 2 | X 2 , U ) + I ( X 2 ; Z 1 | X 1 , U ) > I ( X 1 , X 2 ; Z ) P U P X 1 | U P X 2 | U W Z 1 , Z 2 , Z | X 1 , X 2 Proof Outline: Block-Markov encoding to handle strict causality constraint. Break the dependence between blocks created by the block-Markov encoding. Careful recycling of randomness via wiretap coding between Encoder 1 and Encoder 2. N. Helal, M. Bloch and A. Nosratinia 12 / 22
Introduction Noiseless Cooperation Cooperation over Noisy Channel The MAC with Generalized Feedback (Decode-and-Forward) Proof Outline: Block-Markov encoding to handle strict causality constraint. Break the dependence between blocks created by the block-Markov encoding. Careful recycling of randomness via wiretap coding between Encoder 1 and Encoder 2. Transmit over B blocks. � v 1 � 0 � � � ( � ) ( ) 0 � ′ � ″ 1 1 1 � ( � ) � ′ ( � ) � ″ ( � ) � � ( , , ) 0 1 1 � � � ′ � ″ 2 2 2 � ( � ) � ′ ( � ) � ″ ( � ) � � ( , , ) 0 2 2 � 2 N. Helal, M. Bloch and A. Nosratinia 13 / 22
Introduction Noiseless Cooperation Cooperation over Noisy Channel The MAC with Generalized Feedback (Decode-and-Forward) Proof Outline: Block-Markov encoding to handle strict causality constraint. Break the dependence between blocks created by the block-Markov encoding. Careful recycling of randomness via wiretap coding between Encoder 1 and Encoder 2. User 1 User 2 � ′ � ″ � ′ � ″ � 0 � 0 1 1 2 2 Block � � ″ � ″ � ( + − � 0 ) � 0 1 2 � ″ � ″ (1 − � )( + − � 0 ) 1 2 Block � + 1 � ′ � ″ � ′ � ″ � 0 1 � 0 1 2 2 � ′ � ″ � ″ � ″ � ′ � ″ � ″ � ″ � 1 = + − � ( + − � 0 ) � 2 = + − (1 − � )( + − � 0 ) 1 1 1 2 2 2 1 2 N. Helal, M. Bloch and A. Nosratinia 14 / 22
Introduction Noiseless Cooperation Cooperation over Noisy Channel The MAC with Generalized Feedback (Decode-and-Forward) Proof Outline: Block-Markov encoding to handle strict causality constraint. Break the dependence between blocks created by the block-Markov encoding. Careful recycling of randomness via wiretap coding between Encoder 1 and Encoder 2. B � � ( P Z n || Q ⊗ n Q ⊗ r Z ) ≤ 2 � ( P M ′′( b ) b || P M ′′( b ) P M ′′( b ) Z ) M ′′( b ) Z r 1 2 1 2 b = 1 � � 2 H ( P ( b ) e ) + P ( b ) + B e r ( ρ ′′ 1 + ρ ′′ 2 ) � ����������������������������� �� ����������������������������� � P ( b ) e is the average error probability of decoding over the wiretap channel N. Helal, M. Bloch and A. Nosratinia 15 / 22
Introduction Noiseless Cooperation Cooperation over Noisy Channel The MAC with Generalized Feedback (Decode-and-Forward) Proof Outline: Block-Markov encoding to handle strict causality constraint. Break the dependence between blocks created by the block-Markov encoding. Careful recycling of randomness via wiretap coding between Encoder 1 and Encoder 2. delay delay � ′ 1 � ″ , � 1 � � ′ 1 � ″ Enc1 1 Enc1 � 1 � , 1 ̂ ″ � 2 � � � 1 � 2 , , � | � 1 � 2 � � 1 � 2 , , � | � 1 � 2 � ̂ ″ � 2 � Enc2 � Enc2 � 2 � 1 � ′ 2 � ″ , � ″ 1 � ″ 2 � ′ 2 � ″ , , 2 2 delay delay N. Helal, M. Bloch and A. Nosratinia 16 / 22
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