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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

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SLIDE 1

Introduction Noiseless Cooperation Cooperation over Noisy Channel

Resolvability of the Multiple Access Channel with Two-Sided Cooperation

  • N. Helal1, M. Bloch2 and A. Nosratinia1

1University of Texas at Dallas 2Georgia Institute of Technology

  • N. Helal, M. Bloch and A. Nosratinia

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Introduction Noiseless Cooperation Cooperation over Noisy Channel

1

Introduction Channel Resolvability Channel Resolvability in Information Theory Multi-user Channel Resolvability

2

Noiseless Cooperation The MAC with a common message The MAC with conferencing

3

Cooperation over Noisy Channel The MAC with Feedback The MAC with Generalized Feedback

  • N. Helal, M. Bloch and A. Nosratinia

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SLIDE 3

Introduction Noiseless Cooperation Cooperation over Noisy Channel

Introduction

  • N. Helal, M. Bloch and A. Nosratinia

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SLIDE 4

Introduction Noiseless Cooperation Cooperation over Noisy Channel

Channel Resolvability

Approximation of output statistics

Given i.i.d. QX generating i.i.d. QZ at the output of a DMC WZ|X Attempt to simulate same output statistics using codebook of rate R Goal: find the minimum R such that limn→∞ (PZn ||Q⊗n

Z ) = 0

| Enc ∈ [1, ] 2 ∼

  • ()
  • |

∼

  • ∼

=

  • ⊗
  • Channel resolvability [Wyner 75, Han-Verdú 93, Cuff 09, Hou-Kramer 13]:

inf

  • R : code of rate R s.t. lim

n→∞ (PZn ||Q⊗n Z ) = 0

  • =

min

PXWZ|X:marginal QZ

I(X; Z)

  • N. Helal, M. Bloch and A. Nosratinia

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SLIDE 5

Introduction Noiseless Cooperation Cooperation over Noisy Channel

Channel Resolvability in Information Theory

1975 1985 1995 2005 2015 Common information [Wyner 75] Source coding [Steinberg 94] Approximating output statistics

  • resolvability [Wyner 75, Han 93]
  • MAC resolvability [Steinberg 98]

Rate distortion [Steinberg 96, Liu 15] Strong secrecy

  • from resolvability [Hayashi 06, Bloch 13]
  • non-cooperating encoders [Yassaee 10,

Goldfeld 17, Wang 18]

  • state information [Han 19]

Stealth and covertness [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

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SLIDE 6

Introduction Noiseless Cooperation Cooperation over Noisy Channel

Multi-user Channel Resolvability ⊲ MAC with non-cooperating encoders [Steinberg 98, Frey et al. 17]

Enc 1 Enc 2 Enc 1 Enc 2

R =

  • PQX1X2Z ∈P

       (R1, R2) :

R1 ≥ I(X1; Z|Q) R2 ≥ I(X2; Z|Q) R1 + R2 ≥ I(X1, X2; Z|Q)

       P = {PQPX1 |QPX2 |QWZ|X1X2 : marginal QZ} ⊲ MAC with cribbing [Helal et al. 18]

Enc 1 Enc 2

delay

delay

R =

  • PX1X2Z ∈P

       (R1, R2) :

R1 ≥ I(X1; Z) R2 ≥ I(X1X2; Z) − H(X1) R1 + R2 ≥ I(X1, X2; Z)

       P = {PX1X2WZ|X1X2 : marginal QZ}

for causal cribbing

⊲ Relay channel [Helal et al. 19]

  • N. Helal, M. Bloch and A. Nosratinia

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SLIDE 7

Introduction Noiseless Cooperation Cooperation over Noisy Channel

Noiseless Cooperation

  • N. Helal, M. Bloch and A. Nosratinia

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SLIDE 8

Introduction Noiseless Cooperation Cooperation over Noisy Channel

The MAC with a common message

Enc 1 Enc 2 f1 : M0 × M1 → Xn

1

and f2 : M0 × M2 → Xn

2

Theorem R0 ≥ I(U; Z) R0 + R1 ≥ I(U, X1; Z) R0 + R2 ≥ I(U, X2; Z) R0 + R1 + R2 ≥ I(X1, X2; Z) PUPX1 |UPX2 |UWZ|X1,X2

  • N. Helal, M. Bloch and A. Nosratinia

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SLIDE 9

Introduction Noiseless Cooperation Cooperation over Noisy Channel

The MAC with conferencing

Enc 1 Enc 2 g1k : M1 × Vk−1

2

→ V

1k

and g2k : M2 × Vk−1

1

→ V

2k

f1 : M1 × VK

2 → Xn 1

and f2 : M2 × VK

1 → Xn 2 K

  • k=1

log |V

1k | ≤ nC12

and

K

  • k=1

log |V

2k | ≤ nC21

Theorem C12 + C21 ≥ I(U; Z) R1 ≥ I(U, X1; Z) − C21 R2 ≥ I(U, X2; Z) − C12 R1 + R2 ≥ I(X1, X2; Z) PUPX1 |UPX2 |UWY,Z|X1,X2

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SLIDE 10

Introduction Noiseless Cooperation Cooperation over Noisy Channel

Cooperation over Noisy Channel

  • N. Helal, M. Bloch and A. Nosratinia

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SLIDE 11

Introduction Noiseless Cooperation Cooperation over Noisy Channel

The MAC with Feedback

Enc 1 Enc 2

delay delay

f1i : M1 × Zi−1 → X1i and f2i : M2 × Zi−1 → X2i Theorem R1 ≥ I(X1; Z|U) R2 ≥ I(X2; Z|U) R1 + R2 ≥ I(X1, X2; Z|U) PUPX1 |UPX2 |UWZ|X1,X2

Feedback does not improve resolvability of MAC!

  • N. Helal, M. Bloch and A. Nosratinia

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SLIDE 12

Introduction Noiseless Cooperation Cooperation over Noisy Channel

The MAC with Generalized Feedback

Enc 1 Enc 2

delay delay

f1i : M1 × Zi−1

1

→ X1i

and f2i : M2 × Zi−1

2

→ X2i

  • N. Helal, M. Bloch and A. Nosratinia

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SLIDE 13

Introduction Noiseless Cooperation Cooperation over Noisy Channel

The MAC with Generalized Feedback (Decode-and-Forward) Proposition (achievability) R1 ≥ I(X1, X2; Z) − I(X2; Z1|X1, U) R2 ≥ I(X1, X2; Z) − I(X1; Z2|X2, U) R1 + R2 ≥ I(X1, X2; Z) I(X1; Z2|X2, U)+I(X2; Z1|X1, U) > I(X1, X2; Z) PUPX1 |UPX2 |UWZ1,Z2,Z|X1,X2 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

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SLIDE 14

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

2

( ) () ( , , )

  • 1 ()

′()

1

″()

1

( , , )

  • 2 ()

′()

2

″()

2

′

1

″

1

′

2

″

2

  • N. Helal, M. Bloch and A. Nosratinia

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SLIDE 15

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.

′

1

″

1

′

2

″

2

User 1 User 2

Block

′

1

″

1

′

2

″

2 Block + 1

( + − ) ″

1

″

2

(1 − )( + − ) ″

1

″

2

= + − ( + − ) 1 ′

1

″

1

″

1

″

2

= + − (1 − )( + − ) 2 ′

2

″

2

″

1

″

2

  • N. Helal, M. Bloch and A. Nosratinia

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SLIDE 16

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.

(PZn||Q⊗n

Z ) ≤ 2 B

  • b=1

(PM′′(b)

1

M′′(b)

2

Zr

b||PM′′(b) 1

PM′′(b)

2

Q⊗r

Z )

+ B

  • 2H(P(b)

e ) + P(b) e r(ρ′′ 1 + ρ′′ 2 )

  • P(b)

e is the average

error probability of decoding

  • ver the wiretap channel
  • N. Helal, M. Bloch and A. Nosratinia

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SLIDE 17

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.

  • ,

,| 1 2 12

Enc1 Enc2 1 2 , ′

1 ″ 1

  • ,

′

2 ″ 2

delay delay

  • ,

,| 1 2 12

Enc1 Enc2 1 2 , ′

1 ″ 1

  • ,

′

2 ″ 2

delay delay

  • ̂ ″

1

  • ̂ ″

2

, ″

1 ″ 2

  • N. Helal, M. Bloch and A. Nosratinia

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SLIDE 18

Introduction Noiseless Cooperation Cooperation over Noisy Channel

The MAC with Generalized Feedback (Randomness Extraction)

Proposition (achievability) R1 ≥ I(X1; Z) − H(Z1|X1, Z), R2 ≥ I(X2; Z) − H(Z2|X2, Z), R1 + R2 ≥ I(X1, X2; Z) − H(Z1, Z2|X1, X2, Z). PX1PX2WZ1,Z2,Z|X1,X2 Proof Outline:

Divide transmission over B blocks. Random binning to carefully extract randomness that stems from channel noise. Re-inject the randomness in the following block and break dependence created by randomness recycling.

  • N. Helal, M. Bloch and A. Nosratinia

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SLIDE 19

Introduction Noiseless Cooperation Cooperation over Noisy Channel

The MAC with Generalized Feedback (Randomness Extraction)

Proof Outline:

Divide transmission over B blocks. Random binning to carefully extract randomness that stems from channel noise. Re-inject the randomness in the following block and break dependence created by randomness recycling.

  • 1

( )

  • 1 ()

1

1

  • 2

( )

  • 2 ()

2

2

  • N. Helal, M. Bloch and A. Nosratinia

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SLIDE 20

Introduction Noiseless Cooperation Cooperation over Noisy Channel

The MAC with Generalized Feedback (Randomness Extraction)

Proof Outline:

Divide transmission over B blocks. Random binning to carefully extract randomness that stems from channel noise. Re-inject the randomness in the following block and break dependence created by randomness recycling.

= 1 ()

1

= 2 ()

1

= 3 ()

1

= ()

1

21 = 1 ()

2

= 2 ()

2

= 3 ()

2

= ()

2

22 ( ( ), )

  • 1 ()

1

  • 1

( ( ), )

  • 2 ()

2

  • 2

k(b)

1

= φ1b(xr

1(m(b) 1 ), zr 1b)

k(b)

2

= φ2b(xr

2(m(b) 1 ), zr 2b)

  • N. Helal, M. Bloch and A. Nosratinia

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SLIDE 21

Introduction Noiseless Cooperation Cooperation over Noisy Channel

The MAC with Generalized Feedback (Randomness Extraction)

Proof Outline:

Divide transmission over B blocks. Random binning to carefully extract randomness that stems from channel noise. Re-inject the randomness in the following block and break dependence created by randomness recycling.

(PZn||Q⊗n

Z ) ≤ 2 B

  • b=1

(PK(b)

1

K(b)

2 Zr b||QK(b) 1 QK(b) 2 Q⊗r

Z )

R1 = ρ1 − ρk1 R2 = ρ2 − ρk2

  • N. Helal, M. Bloch and A. Nosratinia

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SLIDE 22

Introduction Noiseless Cooperation Cooperation over Noisy Channel

The MAC with Generalized Feedback Special cases of the MAC with generalized feedback:

MAC with feedback Z1 = Z2 = Z The relay channel Z1 = const., R2 = 0 MAC with strictly-causal cribbing Z1 = X2, Z2 = X1 MAC Z1 = const., Z2 = const. Enc 1 Enc 2

delay delay

  • N. Helal, M. Bloch and A. Nosratinia

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SLIDE 23

Introduction Noiseless Cooperation Cooperation over Noisy Channel

THANK YOU

  • N. Helal, M. Bloch and A. Nosratinia

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