Explicit Construction of Multiple Access Channel Resolvability Codes - - PowerPoint PPT Presentation

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Explicit Construction of Multiple Access Channel Resolvability Codes - - PowerPoint PPT Presentation

Dec 20, 2023 6 likes •297 views

Explicit Construction of Multiple Access Channel Resolvability Codes from Source Resolvability Codes Rumia Sultana R emi A. Chou Wichita State University 2020 IEEE International Symposium on Information Theory 1/29 Outline Introduction

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SLIDE 1

1/29

Explicit Construction of Multiple Access Channel Resolvability Codes from Source Resolvability Codes

Rumia Sultana R´ emi A. Chou Wichita State University 2020 IEEE International Symposium on Information Theory

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SLIDE 2

2/29

Outline

  • Introduction and Model
  • Coding Scheme
  • Coding Scheme Analysis
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SLIDE 3

3/29

Channel Resolvability Problem (P2P) [Han-Verd´ u’93]

3

Channel Resolvability Problem (P2P) [Han-Verdú 93]

ENC ( )

N

f S ~ ( ) S Unif

| Y X

q

1: 1

~

N N X i

X q

  • 1:N

Y

  • 1:N

Y

1: 1: N N

Y Y

p q

  • |

Y X

q

?

  • denotes the uniform distribution over
  • P2P Channel .

.

  • 1:N

Y

p

  • 1:N

Y

q

|

( , , )

  • ( )
  • P2P channel (X, qY |X, Y)
  • Unif (S) denotes the uniform distribution over S
slide-4
SLIDE 4

4/29

Channel Resolvability (MAC) [Steinberg’98]

4

Channel Resolvability (MAC) [Steinberg’98]

ENC

1, 1

( )

N

f S

1: 1: 1

( , ) ~

N N N XY i

X Y q

  • 1:N

Z

  • 1:

1: N N

z z

p q

  • and are uniformly distributed sequences.
  • DMMAC .

ENC

| Z XY

q

| Z XY

q

1:N

Z

2, 2

( )

N

f S

1:N

Z

p

  • 1:N

Z

q

|

( , , )

  • 1

1

~ ( ) S Unif ฀

2 2

~ ( ) S Unif ฀

1 2

?

  • DMMAC (X × Y, qZ|XY , Z)
  • S1 and S2 are uniformly distributed sequences
slide-5
SLIDE 5

5/29

Applications

  • Strong secrecy MAC-WT [Pierrot-Bloch’11], [Yassaee-Aref’10]
  • Cooperative jamming [Pierrot-Bloch’11]
  • Semantic security for MAC-WT [Frey-Bjelakovic-Stanczak’17]
  • Strong coordination in networks [Bloch-Kliewer’13]
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SLIDE 6

6/29

Related Works (Non-exhaustive) P2P Resolvability

  • Existence results

⇒ [Han-Verdu’93]

  • Explicit and low-complexity code constructions

⇒ Polar codes for resolvability over BSC and DMC [Bloch-Luzzi-Kliewer’12], [Chou-Bloch-Kliewer’18] ⇒ Injective group homomorphisms for resolvability over BSC [Hayashi-Matsumoto’16]

  • Non-explicit, linear coding

⇒ Resolvability over DMC [Amjad-Kramer’15]

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SLIDE 7

7/29

Related Works MAC Resolvability

  • Existence results

⇒ MAC [Steinberg’98] ⇒ MAC resolvability region [Frey-Bjelakovic-Stanczak’17]

  • Explicit low-complexity coding scheme

⇒ Symmetric MAC [Chou-Bloch-Kliewer’14] − Invertible extractors − Injective group homomorphisms

  • Explicit low-complexity coding scheme

⇒ Polar codes for arbitrary MAC [Sultana-Chou’19]

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SLIDE 8

8/29

This Work

  • Explicit and low-complexity codes for MAC resolvability

⇒ Arbitrary DMMAC ⇒ Input alphabets are binary

  • Coding scheme ideas

⇒ Reduce MAC resolvability coding problem to a simpler source resolvability coding problem ⇒ Randomness recycling with distributed hashing coupled with block-Markov encoding ⇒ Source resolvability codes used in a black box manner

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SLIDE 9

9/29

Outline

  • Introduction and Model
  • Coding Scheme
  • Coding Scheme Analysis
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SLIDE 10

10/29

Reduction of the MAC Resolvability Region to a Simpler Region

To achieve the MAC resolvability region RqZ , it is sufficient to achieve regions of the form RX,Y for some fixed distribution pXpY RX,Y

      

(R1, R2) : I(XY ; Z) < R1 + R2, I(X; Z) < R1, I(Y ; Z) < R2

      

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SLIDE 11

11/29

Reduction of the MAC Resolvability Region to a Simpler Region

Case 1. I(XY ; Z) = I(X; Z) + I(Y ; Z)

R1

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R2

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I(XY ; Z)

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I(X; Z)

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I(Y ; Z)

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C

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RX,Y

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I(XY, Z)

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slide-12
SLIDE 12

12/29

Reduction of the MAC Resolvability Region to a Simpler Region

Case 2. I(XY ; Z) > I(X; Z) + I(Y ; Z)

D

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D

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R1

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R2

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I(XY ; Z)

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I(XY ; Z)

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I(Y ; Z)

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I(Y ; Z|X)

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I(X; Z|Y )

<latexit sha1_base64="jY3wMuF3CEfFN9FJAmiSiwnVW4=">AB8XicbVA9TwJBEJ3DL8Qv1NJmI5hgQ+6wkMSGxEY7TORD4EL2lj3YsLd32d0zISf/wsZCY2z9N3b+Gxe4QsGXTPLy3kxm5nkRZ0rb9reVWVvf2NzKbud2dvf2D/KHR0VxpLQBgl5KNseVpQzQRuaU7bkaQ48DhtePrmd96pFKxUNzrSUTdA8F8xnB2kid4m2pfdV5ejgv9vMFu2zPgVaJk5ICpKj381+9QUjigApNOFaq69iRdhMsNSOcTnO9WNEIkzEe0q6hAgdUucn84ik6M8oA+aE0JTSaq78nEhwoNQk80xlgPVL3kz8z+vG2q+6CRNRrKkgi0V+zJEO0ex9NGCSEs0nhmAimbkVkRGWmGgTUs6E4Cy/vEqalbJzUa7cVQq1ahpHFk7gFErgwCXU4Abq0ACAp7hFd4sZb1Y79bHojVjpTPH8AfW5w+Gvo96</latexit>

RX,Y

<latexit sha1_base64="u+G/yn4iQeq3wbKGVlO9NC5NRos=">AB+nicbVDLSsNAFL3xWesr1aWbwSK4kJUQZdFNy6r2Ie0IUym03boZBJmJkqJ/RQ3LhRx65e482+ctFlo64GBwzn3cs+cIOZMacf5tpaWV1bX1gsbxc2t7Z1du7TXVFEiCW2QiEeyHWBFORO0oZnmtB1LisOA01Ywusr81gOVikXiTo9j6oV4IFifEayN5Nulboj1kGCObv20fXI/Kfp2ak4U6BF4uakDnqv3V7UkCanQhGOlOq4Tay/FUjPC6aTYTRSNMRnhAe0YKnBIlZdOo0/QkVF6qB9J84RGU/X3RopDpcZhYCazoGrey8T/vE6i+xdeykScaCrI7FA/4UhHKOsB9ZikRPOxIZhIZrIiMsQSE23aykpw57+8SJrVintaqd6clWuXeR0FOIBDOAYXzqEG1CHBhB4hGd4hTfryXqx3q2P2eiSle/swx9Ynz8dFJM/</latexit>

I(X, Z)

<latexit sha1_base64="Gl+ERZo6M7Z0i9/nXqXgLlnD/Q8=">AB7XicbVBNSwMxEJ2tX7V+VT16CRahgpTdKuix6EVvFewHtkvJptk2NpsSVYoS/+DFw+KePX/ePfmLZ70OqDgcd7M8zMC2LOtHdLye3tLyupZfL2xsbm3vFHf3mlomitAGkVyqdoA15UzQhmG03asKI4CTlvB6Grqtx6p0kyKOzOqR/hgWAhI9hYqXlTbp/cH/eKJbfizoD+Ei8jJchQ7xU/u31JkogKQzjWuO5sfFTrAwjnE4K3UTGJMRHtCOpQJHVPvp7NoJOrJKH4VS2RIGzdSfEymOtB5Hge2MsBnqRW8q/ud1EhNe+CkTcWKoIPNFYcKRkWj6OuozRYnhY0swUczeisgQK0yMDahgQ/AWX/5LmtWKd1qp3p6VapdZHk4gEMogwfnUINrqEMDCDzAE7zAqyOdZ+fNeZ+35pxsZh9+wfn4BigDjI=</latexit>
slide-13
SLIDE 13

13/29

Reduction of the MAC Resolvability Region to a Simpler Region

Rate splitting [Grant-Rimoldi-Urbanke-Whiting’01]

  • Choose f : Y × Y → Y, (u, v) → max(u, v)
  • Split (Y, pY ) to form (Y × Y, pUǫpVǫ), ǫ ∈ [0, 1]

Rate Splitting

Y

<latexit sha1_base64="NRADBWhidLrGKbqT5Fr94wkW5hw=">AB6HicbVDLTgJBEOzF+IL9ehlIjHxRHbRI9ELx4hkYeBDZkdemFkdnYzM2tCF/gxYPGePWTvPk3DrAHBSvpFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsG4EthOFNAoEtoLR7cxvPaHSPJb3ZpygH9GB5CFn1Fip/tArltyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1Gu1C9L1ZsjycwCmcgwdXUIU7qEDGCA8wyu8OY/Oi/PufCxac042cwx/4Hz+ALiLjOE=</latexit>

U, V

<latexit sha1_base64="URUGy/poHvon4ofLJjocXV6Kic=">AB6nicbVBNS8NAEJ34WetX1aOXxSJ4kJUQY9FLx4rmrbQhrLZTtqlm03Y3Qil9Cd48aCIV3+RN/+N2zYHbX0w8Hhvhpl5YSq4Nq7aysrq1vbBa2its7u3v7pYPDhk4yxdBniUhUK6QaBZfoG24EtlKFNA4FNsPh7dRvPqHSPJGPZpRiENO+5BFn1FjpwT9vdEtlt+LOQJaJl5My5Kh3S1+dXsKyGKVhgmrd9tzUBGOqDGcCJ8VOpjGlbEj72LZU0h1MJ6dOiGnVumRKFG2pCEz9fEmMZaj+LQdsbUDPSiNxX/89qZia6DMZdpZlCy+aIoE8QkZPo36XGFzIiRJZQpbm8lbEAVZcamU7QheIsvL5NGteJdVKr3l+XaTR5HAY7hBM7AgyuowR3UwQcGfXiGV3hzhPivDsf89YVJ585gj9wPn8AwjKNcw=</latexit>

I(XY ; Z) = R1 + RU + RV with R1 : ǫ → I(X; Z|U), RU : ǫ → I(U; Z), RV : ǫ → I(V ; Z|UX),

slide-14
SLIDE 14

14/29

Randomness used by Transmitter 1

Objective: achieve the rate R1 = I(X; Z|U)

Block1 Block 2 Fresh

H(X) − H(X|UZ)

<latexit sha1_base64="FW80/DsWAzJh584bNO9EXpyW7RY=">AB83icbVBNTwIxEJ3FL8Qv1KOXRmICB8kumuiR6IUjJi4QYUO6pQsN3e6m7ZqQlb/hxYPGePXPePfWGAPCr5kJi/vzaT58ecKW3b31ZubX1jcyu/XdjZ3ds/KB4etVSUSEJdEvFIdnysKGeCupTjuxpDj0OW3749uZ36kUrFI3OtJTL0QDwULGMHaSL1GuVM5N+3Jfaj0iyW7as+BVomTkRJkaPaLX71BRJKQCk04Vqr2LH2Uiw1I5xOC71E0RiTMR7SrqECh1R56fzmKTozygAFkTQlNJqrvzdSHCo1CX0zGWI9UsveTPzP6yY6uPZSJuJEU0EWDwUJRzpCswDQgElKNJ8Ygolk5lZERlhiok1MBROCs/zlVdKqVZ2Lau3uslS/yeLIwmcQhkcuI6NKAJLhCI4Rle4c1KrBfr3fpYjOasbOcY/sD6/AG8tpAw</latexit>

Fresh

H(X)

<latexit sha1_base64="wSv64E/SFNRW+hnjEPAGipo9+Ew=">AB63icbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9NJjBfsB7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMC2LOtHdb6ewsbm1vVPcLe3tHxwelY9POjpKFKFtEvFI9QKsKWeStg0znPZiRbEIO0G0/vM7z5RpVkH80spr7AY8lCRrDJpGa1dzksV9yauwBaJ15OKpCjNSx/DUYRSQSVhnCsd9zY+OnWBlGOJ2XBomMSZTPKZ9SyUWVPvp4tY5urDKCIWRsiUNWqi/J1IstJ6JwHYKbCZ61cvE/7x+YsJbP2UyTgyVZLkoTDgyEcoeRyOmKDF8ZgkmitlbEZlghYmx8ZRsCN7qy+ukU695V7X6w3WlcZfHUYQzOIcqeHADWhC9pAYALP8ApvjnBenHfnY9lacPKZU/gD5/MHDzuNlw=</latexit>

Sum Recycled

H(X|UZ)

<latexit sha1_base64="nML5K8Lwp3xAvfIrRBjsRq7zvKc=">AB7nicbVBNT8JAEJ3iF+IX6tHLRmKCF9KiR6JXjhiYoEIDdkuW9iw3Ta7WxNS+RFePGiMV3+PN/+NW+hBwZdM8vLeTGbm+TFnStv2t1VYW9/Y3Cpul3Z29/YPyodHbRUlklCXRDySXR8rypmgrma024sKQ59Tjv+5DbzO49UKhaJez2NqRfikWABI1gbqdOsdp/ch/NBuWLX7DnQKnFyUoEcrUH5qz+MSBJSoQnHSvUcO9ZeiqVmhNZqZ8oGmMywSPaM1TgkCovnZ87Q2dGaIgkqaERnP190SKQ6WmoW86Q6zHatnLxP+8XqKDay9lIk40FWSxKEg40hHKfkdDJinRfGoIJpKZWxEZY4mJNgmVTAjO8surpF2vORe1+t1lpXGTx1GEziFKjhwBQ1oQgtcIDCBZ3iFNyu2Xqx362PRWrDymWP4A+vzB0gZjuA=</latexit>

H(X)

<latexit sha1_base64="wSv64E/SFNRW+hnjEPAGipo9+Ew=">AB63icbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9NJjBfsB7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMC2LOtHdb6ewsbm1vVPcLe3tHxwelY9POjpKFKFtEvFI9QKsKWeStg0znPZiRbEIO0G0/vM7z5RpVkH80spr7AY8lCRrDJpGa1dzksV9yauwBaJ15OKpCjNSx/DUYRSQSVhnCsd9zY+OnWBlGOJ2XBomMSZTPKZ9SyUWVPvp4tY5urDKCIWRsiUNWqi/J1IstJ6JwHYKbCZ61cvE/7x+YsJbP2UyTgyVZLkoTDgyEcoeRyOmKDF8ZgkmitlbEZlghYmx8ZRsCN7qy+ukU695V7X6w3WlcZfHUYQzOIcqeHADWhC9pAYALP8ApvjnBenHfnY9lacPKZU/gD5/MHDzuNlw=</latexit>

Sum

H(X)

<latexit sha1_base64="wSv64E/SFNRW+hnjEPAGipo9+Ew=">AB63icbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9NJjBfsB7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMC2LOtHdb6ewsbm1vVPcLe3tHxwelY9POjpKFKFtEvFI9QKsKWeStg0znPZiRbEIO0G0/vM7z5RpVkH80spr7AY8lCRrDJpGa1dzksV9yauwBaJ15OKpCjNSx/DUYRSQSVhnCsd9zY+OnWBlGOJ2XBomMSZTPKZ9SyUWVPvp4tY5urDKCIWRsiUNWqi/J1IstJ6JwHYKbCZ61cvE/7x+YsJbP2UyTgyVZLkoTDgyEcoeRyOmKDF8ZgkmitlbEZlghYmx8ZRsCN7qy+ukU695V7X6w3WlcZfHUYQzOIcqeHADWhC9pAYALP8ApvjnBenHfnY9lacPKZU/gD5/MHDzuNlw=</latexit>

Block k Fresh

H(X) − H(X|UZ)

<latexit sha1_base64="FW80/DsWAzJh584bNO9EXpyW7RY=">AB83icbVBNTwIxEJ3FL8Qv1KOXRmICB8kumuiR6IUjJi4QYUO6pQsN3e6m7ZqQlb/hxYPGePXPePfWGAPCr5kJi/vzaT58ecKW3b31ZubX1jcyu/XdjZ3ds/KB4etVSUSEJdEvFIdnysKGeCupTjuxpDj0OW3749uZ36kUrFI3OtJTL0QDwULGMHaSL1GuVM5N+3Jfaj0iyW7as+BVomTkRJkaPaLX71BRJKQCk04Vqr2LH2Uiw1I5xOC71E0RiTMR7SrqECh1R56fzmKTozygAFkTQlNJqrvzdSHCo1CX0zGWI9UsveTPzP6yY6uPZSJuJEU0EWDwUJRzpCswDQgElKNJ8Ygolk5lZERlhiok1MBROCs/zlVdKqVZ2Lau3uslS/yeLIwmcQhkcuI6NKAJLhCI4Rle4c1KrBfr3fpYjOasbOcY/sD6/AG8tpAw</latexit>

Sum Recycled

H(X|UZ)

<latexit sha1_base64="nML5K8Lwp3xAvfIrRBjsRq7zvKc=">AB7nicbVBNT8JAEJ3iF+IX6tHLRmKCF9KiR6JXjhiYoEIDdkuW9iw3Ta7WxNS+RFePGiMV3+PN/+NW+hBwZdM8vLeTGbm+TFnStv2t1VYW9/Y3Cpul3Z29/YPyodHbRUlklCXRDySXR8rypmgrma024sKQ59Tjv+5DbzO49UKhaJez2NqRfikWABI1gbqdOsdp/ch/NBuWLX7DnQKnFyUoEcrUH5qz+MSBJSoQnHSvUcO9ZeiqVmhNZqZ8oGmMywSPaM1TgkCovnZ87Q2dGaIgkqaERnP190SKQ6WmoW86Q6zHatnLxP+8XqKDay9lIk40FWSxKEg40hHKfkdDJinRfGoIJpKZWxEZY4mJNgmVTAjO8surpF2vORe1+t1lpXGTx1GEziFKjhwBQ1oQgtcIDCBZ3iFNyu2Xqx362PRWrDymWP4A+vzB0gZjuA=</latexit>

H(X)

<latexit sha1_base64="wSv64E/SFNRW+hnjEPAGipo9+Ew=">AB63icbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9NJjBfsB7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMC2LOtHdb6ewsbm1vVPcLe3tHxwelY9POjpKFKFtEvFI9QKsKWeStg0znPZiRbEIO0G0/vM7z5RpVkH80spr7AY8lCRrDJpGa1dzksV9yauwBaJ15OKpCjNSx/DUYRSQSVhnCsd9zY+OnWBlGOJ2XBomMSZTPKZ9SyUWVPvp4tY5urDKCIWRsiUNWqi/J1IstJ6JwHYKbCZ61cvE/7x+YsJbP2UyTgyVZLkoTDgyEcoeRyOmKDF8ZgkmitlbEZlghYmx8ZRsCN7qy+ukU695V7X6w3WlcZfHUYQzOIcqeHADWhC9pAYALP8ApvjnBenHfnY9lacPKZU/gD5/MHDzuNlw=</latexit>

Hash Function O/P Hash Function O/P

For Block i ∈ 1, k, Source resolvability for (X, qX) using randomness rate H(X).

slide-15
SLIDE 15

15/29

Randomness used by Transmitter 2 (for Input U)

Objective: achieve the rate RU = I(U; Z)

Block1 Block 2 Fresh Fresh

Sum Recycled Sum

Block k Fresh

Sum Recycled

Hash Function O/P Hash Function O/P

H(U)

<latexit sha1_base64="9DIne64j5JS48NOQ4c5IGHs3NA=">AB63icbVBNTwIxEJ3FL8Qv1KOXRmKCF7KLJnokeuGIiQsksCHd0oWGtrtpuyaE8Be8eNAYr/4hb/4bu7AHBV8yct7M5mZFyacaeO6305hY3Nre6e4W9rbPzg8Kh+ftHWcKkJ9EvNYdUOsKWeS+oYZTruJoliEnHbCyX3md56o0iyWj2a0EDgkWQRI9hkUrPqXw7KFbfmLoDWiZeTCuRoDcpf/WFMUkGlIRxr3fPcxAQzrAwjnM5L/VTBJMJHtGepRILqoPZ4tY5urDKEWxsiUNWqi/J2ZYaD0Voe0U2Iz1qpeJ/3m91ES3wYzJDVUkuWiKOXIxCh7HA2ZosTwqSWYKGZvRWSMFSbGxlOyIXirL6+Tdr3mXdXqD9eVxl0eRxHO4Byq4MENKAJLfCBwBie4RXeHOG8O/Ox7K14OQzp/AHzucPCqyNlA=</latexit>

H(U)

<latexit sha1_base64="9DIne64j5JS48NOQ4c5IGHs3NA=">AB63icbVBNTwIxEJ3FL8Qv1KOXRmKCF7KLJnokeuGIiQsksCHd0oWGtrtpuyaE8Be8eNAYr/4hb/4bu7AHBV8yct7M5mZFyacaeO6305hY3Nre6e4W9rbPzg8Kh+ftHWcKkJ9EvNYdUOsKWeS+oYZTruJoliEnHbCyX3md56o0iyWj2a0EDgkWQRI9hkUrPqXw7KFbfmLoDWiZeTCuRoDcpf/WFMUkGlIRxr3fPcxAQzrAwjnM5L/VTBJMJHtGepRILqoPZ4tY5urDKEWxsiUNWqi/J2ZYaD0Voe0U2Iz1qpeJ/3m91ES3wYzJDVUkuWiKOXIxCh7HA2ZosTwqSWYKGZvRWSMFSbGxlOyIXirL6+Tdr3mXdXqD9eVxl0eRxHO4Byq4MENKAJLfCBwBie4RXeHOG8O/Ox7K14OQzp/AHzucPCqyNlA=</latexit>

H(U)

<latexit sha1_base64="9DIne64j5JS48NOQ4c5IGHs3NA=">AB63icbVBNTwIxEJ3FL8Qv1KOXRmKCF7KLJnokeuGIiQsksCHd0oWGtrtpuyaE8Be8eNAYr/4hb/4bu7AHBV8yct7M5mZFyacaeO6305hY3Nre6e4W9rbPzg8Kh+ftHWcKkJ9EvNYdUOsKWeS+oYZTruJoliEnHbCyX3md56o0iyWj2a0EDgkWQRI9hkUrPqXw7KFbfmLoDWiZeTCuRoDcpf/WFMUkGlIRxr3fPcxAQzrAwjnM5L/VTBJMJHtGepRILqoPZ4tY5urDKEWxsiUNWqi/J2ZYaD0Voe0U2Iz1qpeJ/3m91ES3wYzJDVUkuWiKOXIxCh7HA2ZosTwqSWYKGZvRWSMFSbGxlOyIXirL6+Tdr3mXdXqD9eVxl0eRxHO4Byq4MENKAJLfCBwBie4RXeHOG8O/Ox7K14OQzp/AHzucPCqyNlA=</latexit>

H(U)

<latexit sha1_base64="9DIne64j5JS48NOQ4c5IGHs3NA=">AB63icbVBNTwIxEJ3FL8Qv1KOXRmKCF7KLJnokeuGIiQsksCHd0oWGtrtpuyaE8Be8eNAYr/4hb/4bu7AHBV8yct7M5mZFyacaeO6305hY3Nre6e4W9rbPzg8Kh+ftHWcKkJ9EvNYdUOsKWeS+oYZTruJoliEnHbCyX3md56o0iyWj2a0EDgkWQRI9hkUrPqXw7KFbfmLoDWiZeTCuRoDcpf/WFMUkGlIRxr3fPcxAQzrAwjnM5L/VTBJMJHtGepRILqoPZ4tY5urDKEWxsiUNWqi/J2ZYaD0Voe0U2Iz1qpeJ/3m91ES3wYzJDVUkuWiKOXIxCh7HA2ZosTwqSWYKGZvRWSMFSbGxlOyIXirL6+Tdr3mXdXqD9eVxl0eRxHO4Byq4MENKAJLfCBwBie4RXeHOG8O/Ox7K14OQzp/AHzucPCqyNlA=</latexit>

H(U) − H(U|Z)

<latexit sha1_base64="UanXE7JkCNKRoxNqSiampJ65VKE=">AB8nicbVBNTwIxEO3iF+IX6tFLIzGBg2QXTfRI9MIRExeIy4Z0S4G7nbTzpqQlZ/hxYPGePXePfWGAPCr6kzct7M5mZF8SCa7Dtbyu3tr6xuZXfLuzs7u0fFA+PWlomijKXSiFVJyCaCR4xFzgI1okVI2EgWDsY38789iNTmsvoHiYx80MyjPiAUwJG8hplt3JuvqeHSq9Ysqv2HiVOBkpoQzNXvGr25c0CVkEVBCtPceOwU+JAk4Fmxa6iWYxoWMyZJ6hEQmZ9tP5ylN8ZpQ+HkhlXgR4rv7uSEmo9SQMTGVIYKSXvZn4n+clMLj2Ux7FCbCILgYNEoFB4tn9uM8VoyAmhCquNkV0xFRhIJqWBCcJZPXiWtWtW5qNbuLkv1myOPDpBp6iMHSF6qiBmshFEn0jF7RmwXWi/VufSxKc1bWc4z+wPr8AQqlj8s=</latexit>

H(U) − H(U|Z)

<latexit sha1_base64="UanXE7JkCNKRoxNqSiampJ65VKE=">AB8nicbVBNTwIxEO3iF+IX6tFLIzGBg2QXTfRI9MIRExeIy4Z0S4G7nbTzpqQlZ/hxYPGePXePfWGAPCr6kzct7M5mZF8SCa7Dtbyu3tr6xuZXfLuzs7u0fFA+PWlomijKXSiFVJyCaCR4xFzgI1okVI2EgWDsY38789iNTmsvoHiYx80MyjPiAUwJG8hplt3JuvqeHSq9Ysqv2HiVOBkpoQzNXvGr25c0CVkEVBCtPceOwU+JAk4Fmxa6iWYxoWMyZJ6hEQmZ9tP5ylN8ZpQ+HkhlXgR4rv7uSEmo9SQMTGVIYKSXvZn4n+clMLj2Ux7FCbCILgYNEoFB4tn9uM8VoyAmhCquNkV0xFRhIJqWBCcJZPXiWtWtW5qNbuLkv1myOPDpBp6iMHSF6qiBmshFEn0jF7RmwXWi/VufSxKc1bWc4z+wPr8AQqlj8s=</latexit>

H(U|Z)

<latexit sha1_base64="JHrgSUE0iHymwy0tca7mRslj2ck=">AB7XicbVBNTwIxEJ3FL8Qv1KOXRmKCF7KrJnokeuGIiQtE2JBuKVDptpu2a0JW/oMXDxrj1f/jzX9jgT0o+JXt6bycy8MOZMG9f9dnIrq2vrG/nNwtb2zu5ecf+goWiCPWJ5FK1QqwpZ4L6hlOW7GiOAo5bYajm6nfKRKMynuzDimQYQHgvUZwcZKjVrZf7o/7RZLbsWdAS0TLyMlyFDvFr86PUmSiApDONa67bmxCVKsDCOcTgqdRNMYkxEe0LalAkdUB+ns2gk6sUoP9aWyJQyaqb8nUhxpPY5C2xlhM9SL3lT8z2snpn8VpEzEiaGCzBf1E46MRNPXUY8pSgwfW4KJYvZWRIZYWJsQAUbgrf48jJpnFW8rZ7UWpep3FkYcjOIYyeHAJVahBHXwg8ADP8ApvjnRenHfnY96ac7KZQ/gD5/MHm8WOfg=</latexit>

H(U|Z)

<latexit sha1_base64="JHrgSUE0iHymwy0tca7mRslj2ck=">AB7XicbVBNTwIxEJ3FL8Qv1KOXRmKCF7KrJnokeuGIiQtE2JBuKVDptpu2a0JW/oMXDxrj1f/jzX9jgT0o+JXt6bycy8MOZMG9f9dnIrq2vrG/nNwtb2zu5ecf+goWiCPWJ5FK1QqwpZ4L6hlOW7GiOAo5bYajm6nfKRKMynuzDimQYQHgvUZwcZKjVrZf7o/7RZLbsWdAS0TLyMlyFDvFr86PUmSiApDONa67bmxCVKsDCOcTgqdRNMYkxEe0LalAkdUB+ns2gk6sUoP9aWyJQyaqb8nUhxpPY5C2xlhM9SL3lT8z2snpn8VpEzEiaGCzBf1E46MRNPXUY8pSgwfW4KJYvZWRIZYWJsQAUbgrf48jJpnFW8rZ7UWpep3FkYcjOIYyeHAJVahBHXwg8ADP8ApvjnRenHfnY96ac7KZQ/gD5/MHm8WOfg=</latexit>

For Block i ∈ 1, k, Source resolvability for (Y, qU) using randomness rate H(U).

slide-16
SLIDE 16

16/29

Randomness used by Transmitter 2 (for Input V )

Objective: achieve the rate RV = I(V ; Z|UX)

Block1 Block 2 Fresh Fresh

Sum Recycled Sum

Block k Fresh

Sum Recycled

Hash Function O/P Hash Function O/P H(V )

<latexit sha1_base64="nJnJwrVGh2s+VnuG7JVLl2Z6t8M=">AB63icbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9NJjBfsB7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMC2LOtHdb6ewsbm1vVPcLe3tHxwelY9POjpKFKFtEvFI9QKsKWeStg0znPZiRbEIO0G0/vM7z5RpVkH80spr7AY8lCRrDJpGa1czksV9yauwBaJ15OKpCjNSx/DUYRSQSVhnCsd9zY+OnWBlGOJ2XBomMSZTPKZ9SyUWVPvp4tY5urDKCIWRsiUNWqi/J1IstJ6JwHYKbCZ61cvE/7x+YsJbP2UyTgyVZLkoTDgyEcoeRyOmKDF8ZgkmitlbEZlghYmx8ZRsCN7qy+ukU695V7X6w3WlcZfHUYQzOIcqeHADWhC9pAYALP8ApvjnBenHfnY9lacPKZU/gD5/MHDGNlQ=</latexit>

H(V )

<latexit sha1_base64="nJnJwrVGh2s+VnuG7JVLl2Z6t8M=">AB63icbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9NJjBfsB7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMC2LOtHdb6ewsbm1vVPcLe3tHxwelY9POjpKFKFtEvFI9QKsKWeStg0znPZiRbEIO0G0/vM7z5RpVkH80spr7AY8lCRrDJpGa1czksV9yauwBaJ15OKpCjNSx/DUYRSQSVhnCsd9zY+OnWBlGOJ2XBomMSZTPKZ9SyUWVPvp4tY5urDKCIWRsiUNWqi/J1IstJ6JwHYKbCZ61cvE/7x+YsJbP2UyTgyVZLkoTDgyEcoeRyOmKDF8ZgkmitlbEZlghYmx8ZRsCN7qy+ukU695V7X6w3WlcZfHUYQzOIcqeHADWhC9pAYALP8ApvjnBenHfnY9lacPKZU/gD5/MHDGNlQ=</latexit>

H(V )

<latexit sha1_base64="nJnJwrVGh2s+VnuG7JVLl2Z6t8M=">AB63icbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9NJjBfsB7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMC2LOtHdb6ewsbm1vVPcLe3tHxwelY9POjpKFKFtEvFI9QKsKWeStg0znPZiRbEIO0G0/vM7z5RpVkH80spr7AY8lCRrDJpGa1czksV9yauwBaJ15OKpCjNSx/DUYRSQSVhnCsd9zY+OnWBlGOJ2XBomMSZTPKZ9SyUWVPvp4tY5urDKCIWRsiUNWqi/J1IstJ6JwHYKbCZ61cvE/7x+YsJbP2UyTgyVZLkoTDgyEcoeRyOmKDF8ZgkmitlbEZlghYmx8ZRsCN7qy+ukU695V7X6w3WlcZfHUYQzOIcqeHADWhC9pAYALP8ApvjnBenHfnY9lacPKZU/gD5/MHDGNlQ=</latexit>

H(V )

<latexit sha1_base64="nJnJwrVGh2s+VnuG7JVLl2Z6t8M=">AB63icbVBNSwMxEJ2tX7V+VT16CRahXspuFfRY9NJjBfsB7VKyabYNTbJLkhXK0r/gxYMiXv1D3vw3Zts9aOuDgcd7M8zMC2LOtHdb6ewsbm1vVPcLe3tHxwelY9POjpKFKFtEvFI9QKsKWeStg0znPZiRbEIO0G0/vM7z5RpVkH80spr7AY8lCRrDJpGa1czksV9yauwBaJ15OKpCjNSx/DUYRSQSVhnCsd9zY+OnWBlGOJ2XBomMSZTPKZ9SyUWVPvp4tY5urDKCIWRsiUNWqi/J1IstJ6JwHYKbCZ61cvE/7x+YsJbP2UyTgyVZLkoTDgyEcoeRyOmKDF8ZgkmitlbEZlghYmx8ZRsCN7qy+ukU695V7X6w3WlcZfHUYQzOIcqeHADWhC9pAYALP8ApvjnBenHfnY9lacPKZU/gD5/MHDGNlQ=</latexit>

H(V ) − H(V |UZX)

<latexit sha1_base64="7QUokM9k39WlvdWXsbO9aPwNcU=">AB9HicbVBNTwIxEJ3FL8Qv1KOXRmICB8kumuiR6IUjJi4QYUO6pUBDt7u2XRKy8ju8eNAYr/4Yb/4bC+xBwZfM5OW9mXT6/IgzpW3728qsrW9sbmW3czu7e/sH+cOjhgpjSahLQh7Klo8V5UxQVzPNaSuSFAc+p01/dDvzm2MqFQvFvZ5E1AvwQLA+I1gbyasVG6Vz057ch1apmy/YZXsOtEqclBQgRb2b/+r0QhIHVGjCsVJtx460l2CpGeF0muvEikaYjPCAtg0VOKDKS+ZHT9GZUXqoH0pTQqO5+nsjwYFSk8A3kwHWQ7XszcT/vHas+9dewkQUayrI4qF+zJEO0SwB1GOSEs0nhmAimbkVkSGWmGiTU86E4Cx/eZU0KmXnoly5uyxUb9I4snACp1AEB6gCjWogwsEHuEZXuHNGlsv1rv1sRjNWOnOMfyB9fkDZFqQjg=</latexit>

H(V ) − H(V |UZX)

<latexit sha1_base64="7QUokM9k39WlvdWXsbO9aPwNcU=">AB9HicbVBNTwIxEJ3FL8Qv1KOXRmICB8kumuiR6IUjJi4QYUO6pUBDt7u2XRKy8ju8eNAYr/4Yb/4bC+xBwZfM5OW9mXT6/IgzpW3728qsrW9sbmW3czu7e/sH+cOjhgpjSahLQh7Klo8V5UxQVzPNaSuSFAc+p01/dDvzm2MqFQvFvZ5E1AvwQLA+I1gbyasVG6Vz057ch1apmy/YZXsOtEqclBQgRb2b/+r0QhIHVGjCsVJtx460l2CpGeF0muvEikaYjPCAtg0VOKDKS+ZHT9GZUXqoH0pTQqO5+nsjwYFSk8A3kwHWQ7XszcT/vHas+9dewkQUayrI4qF+zJEO0SwB1GOSEs0nhmAimbkVkSGWmGiTU86E4Cx/eZU0KmXnoly5uyxUb9I4snACp1AEB6gCjWogwsEHuEZXuHNGlsv1rv1sRjNWOnOMfyB9fkDZFqQjg=</latexit>

H(V |UZX)

<latexit sha1_base64="jB95uTDW+1mXtsyS61lrCzLeo=">AB73icbVBNT8JAEJ36ifiFevSykZjghbRokeiF46YWGiEhmyXLWzYbuvu1oRU/oQXDxrj1b/jzX/jAj0o+JXt6bycy8IOFMadv+tlZW19Y3Ngtbxe2d3b390sFhS8WpJNQlMY+lF2BFORPU1Uxz6iWS4ijgtB2MbqZ+5FKxWJxp8cJ9SM8ECxkBGsjeY1K68m9856pbJdtWdAy8TJSRlyNHulr24/JmlEhSYcK9Vx7ET7GZaEU4nxW6qaILJCA9ox1CBI6r8bHbvBJ0apY/CWJoSGs3U3xMZjpQaR4HpjLAeqkVvKv7ndVIdXvkZE0mqSDzRWHKkY7R9HnUZ5ISzceGYCKZuRWRIZaYaBNR0YTgL68TFq1qnNerd1elOvXeRwFOIYTqIADl1CHBjTBQIcnuEV3qwH68V6tz7mrStWPnMEf2B9/gDxo9A</latexit>

H(V |UZX)

<latexit sha1_base64="jB95uTDW+1mXtsyS61lrCzLeo=">AB73icbVBNT8JAEJ36ifiFevSykZjghbRokeiF46YWGiEhmyXLWzYbuvu1oRU/oQXDxrj1b/jzX/jAj0o+JXt6bycy8IOFMadv+tlZW19Y3Ngtbxe2d3b390sFhS8WpJNQlMY+lF2BFORPU1Uxz6iWS4ijgtB2MbqZ+5FKxWJxp8cJ9SM8ECxkBGsjeY1K68m9856pbJdtWdAy8TJSRlyNHulr24/JmlEhSYcK9Vx7ET7GZaEU4nxW6qaILJCA9ox1CBI6r8bHbvBJ0apY/CWJoSGs3U3xMZjpQaR4HpjLAeqkVvKv7ndVIdXvkZE0mqSDzRWHKkY7R9HnUZ5ISzceGYCKZuRWRIZaYaBNR0YTgL68TFq1qnNerd1elOvXeRwFOIYTqIADl1CHBjTBQIcnuEV3qwH68V6tz7mrStWPnMEf2B9/gDxo9A</latexit>

For Block i ∈ 1, k, Source resolvability for (Y, qV ) using randomness rate H(V ).

slide-17
SLIDE 17

17/29

Randomness Recycling via Distributed Hashing

For Block i ∈ 2, k

e X1:N

i−1

<latexit sha1_base64="9pVbq+4XUzMH5hE1n9BaX7oxkrc=">ACAXicbVDLSsNAFJ34rPUVdSO4GSyCG0tSBcV0Y0rqWAf0MYwmdy2QycPZiZKCXHjr7hxoYhb/8Kdf+O0zUJbD1w4nHMv97jxZxJZVnfxtz8wuLScmGluLq2vrFpbm03ZJQICnUa8Ui0PCKBsxDqikOrVgACTwOTW9wOfKb9yAki8JbNYzBCUgvZF1GidKSa+52HpgPinEfcMtN2ZGd3aX2+XmiWrbI2BZ4mdkxLKUXPNr4f0SAUFOpGzbVqyclAjFKIes2EkxIQOSA/amoYkAOmk4w8yfKAVH3cjoStUeKz+nkhJIOUw8HRnQFRfTnsj8T+vnajumZOyME4UhHSyqJtwrCI8igP7TABVfKgJoYLpWzHtE0Go0qEVdQj29MuzpFEp28flys1JqXqRx1FAe2gfHSIbnaIqukI1VEcUPaJn9IrejCfjxXg3Piatc0Y+s4P+wPj8AaOdls=</latexit>

e U 1:N

i−1

<latexit sha1_base64="Su9AWX8jR4TqhDA3uFG35dgNZzY=">ACAXicbVBNS8NAEN34WetX1YvgZbEIXixJFRPRS+epIJpC20Nm82kXbrZhN2NUkK9+Fe8eFDEq/Cm/G7cdBWx8MPN6bYWaen3CmtG1/W3PzC4tLy7mV/Ora+sZmYWu7puJUnBpzGPZ8IkCzgS4mkOjUQCiXwOdb93OfTr9yAVi8Wt7ifQjkhHsJBRo3kFXZbDywAzXgA2PUyduQM7jLn/HrgFYp2yR4BzxJnQopogqpX+GoFMU0jEJpyolTsRPdzojUjHIY5FupgoTQHulA01BIlDtbPTBAB8YJcBhLE0JjUfq74mMREr1I90RkR31bQ3FP/zmqkOz9oZE0mqQdDxojDlWMd4GAcOmASqed8QiUzt2LaJZJQbULmxCc6ZdnSa1co5L5ZuTYuViEkcO7aF9dIgcdIoq6ApVkYsoekTP6BW9WU/Wi/VufYxb56zJzA76A+vzB57tlg=</latexit>

e V 1:N

i−1

<latexit sha1_base64="1N1u/iyDCxbZbTFlHxAvmghiCxY=">ACAXicbVDLSsNAFJ34rPUVdSO4GSyCG0tSBcV0Y0rqWAf0MYwmdy2QycPZiZKCXHjr7hxoYhb/8Kdf+O0zUJbD1w4nHMv97jxZxJZVnfxtz8wuLScmGluLq2vrFpbm03ZJQICnUa8Ui0PCKBsxDqikOrVgACTwOTW9wOfKb9yAki8JbNYzBCUgvZF1GidKSa+52HpgPinEfcMN2ZGd3aX2+XmiWrbI2BZ4mdkxLKUXPNr4f0SAUFOpGzbVqyclAjFKIes2EkxIQOSA/amoYkAOmk4w8yfKAVH3cjoStUeKz+nkhJIOUw8HRnQFRfTnsj8T+vnajumZOyME4UhHSyqJtwrCI8igP7TABVfKgJoYLpWzHtE0Go0qEVdQj29MuzpFEp28flys1JqXqRx1FAe2gfHSIbnaIqukI1VEcUPaJn9IrejCfjxXg3Piatc0Y+s4P+wPj8AaB9lk=</latexit>

e Ei

<latexit sha1_base64="D7P8Xv/VF0pv09Z/RihGzR83XOg=">AB+HicbVBNS8NAEN34WetHox69LBbBU0mqoMeiCB4r2A9oQ9hsJu3SzSbsbpQa+ku8eFDEqz/Fm/GbZuDtj4YeLw3w8y8IOVMacf5tlZW19Y3Nktb5e2d3b2KvX/QVkmKbRowhPZDYgCzgS0NMcuqkEgcOsHoeup3HkAqloh7PU7Bi8lAsIhRo3k25X+IwtBMx4CvEZ9u2qU3NmwMvELUgVFWj69lc/TGgWg9CUE6V6rpNqLydSM8phUu5nClJCR2QAPUMFiUF5+ezwCT4xSoijRJoSGs/U3xM5iZUax4HpjIkeqkVvKv7n9TIdXo5E2mQdD5oijWCd4mgIOmQSq+dgQiUzt2I6JQbIqmxDcxZeXSbtec89q9bvzauOqiKOEjtAxOkUukANdIuaqIUoytAzekVv1pP1Yr1bH/PWFauYOUR/YH3+AQTkqk=</latexit>

e Di

<latexit sha1_base64="SDoF7x1BTN/Nw257B3nYtrwYu+w=">AB+HicbVBNS8NAEN34WetHox69LBbBU0mqoMeiHjxWsB/QhrDZTNqlm03Y3Sg19Jd48aCIV3+KN/+N2zYHbX0w8Hhvhpl5QcqZ0o7zba2srq1vbJa2yts7u3sVe/+grZJMUmjRhCeyGxAFnAloaY5dFMJA4dILR9dTvPIBULBH3epyCF5OBYBGjRBvJtyv9RxaCZjwEfOMz7NtVp+bMgJeJW5AqKtD07a9+mNAsBqEpJ0r1XCfVXk6kZpTDpNzPFKSEjsgAeoYKEoPy8tnhE3xilBHiTQlNJ6pvydyEis1jgPTGRM9VIveVPzP62U6uvRyJtJMg6DzRVHGsU7wNAUcMglU87EhEpmbsV0SCSh2mRVNiG4iy8vk3a95p7V6nfn1cZVEUcJHaFjdIpcdIEa6BY1UQtRlKFn9IrerCfrxXq3PuatK1Yxc4j+wPr8AQKMkqg=</latexit>

e Fi

<latexit sha1_base64="ZD4DjOegSVsi/Q+eXEWpPMzi3ug=">AB+HicbVBNS8NAEN34WetHox69LBbBU0mqoMeiIB4r2A9oQ9hsJu3SzSbsbpQa+ku8eFDEqz/Fm/GbZuDtj4YeLw3w8y8IOVMacf5tlZW19Y3Nktb5e2d3b2KvX/QVkmKbRowhPZDYgCzgS0NMcuqkEgcOsHoeup3HkAqloh7PU7Bi8lAsIhRo3k25X+IwtBMx4CvEZ9u2qU3NmwMvELUgVFWj69lc/TGgWg9CUE6V6rpNqLydSM8phUu5nClJCR2QAPUMFiUF5+ezwCT4xSoijRJoSGs/U3xM5iZUax4HpjIkeqkVvKv7n9TIdXo5E2mQdD5oijWCd4mgIOmQSq+dgQiUzt2I6JQbIqmxDcxZeXSbtec89q9bvzauOqiKOEjtAxOkUukANdIuaqIUoytAzekVv1pP1Yr1bH/PWFauYOUR/YH3+AWakqo=</latexit>

GU : {0, 1}N − ! {0, 1}rU

<latexit sha1_base64="X24R7z4GQWAjrVhR6rnZMau2QWY=">ACFnicbVDLSgMxFM34rPU16tJNsAgutMxUQXFVdKErqeC0hU4dMmnahmaSIckoZivcOvuHGhiFtx59+YPhBtPXDhcM693HtPGDOqtON8WTOzc/MLi7ml/PLK6tq6vbFZVSKRmHhYMCHrIVKEU48TUj9VgSFIWM1MLe+cCv3RGpqOA3uh+TZoQ6nLYpRtpIgX1wEXinfursu352m15l0GeCdyTtdDWSUtzDH08GXhbYBafoDAGniTsmBTBGJbA/ZbASUS4xgwp1XCdWDdTJDXFjGR5P1EkRriHOqRhKEcRUc10+FYGd43Sgm0hTXENh+rviRFSvWj0HRGSHfVpDcQ/MaiW6fNFPK40QTjkeL2gmDWsBRrBFJcGa9Q1BWFJzK8RdJBHWJsm8CcGdfHmaVEtF97BYuj4qlM/GceTANtgBe8AFx6AMLkEFeACDB/AEXsCr9Wg9W2/W+6h1xhrPbIE/sD6+AUI7ntU=</latexit>

GX : {0, 1}N − ! {0, 1}rX

<latexit sha1_base64="72a5EqIgy4zlvlIjaTezdvUv2NA=">ACFnicbVDLSgMxFM34rPU16tJNsAgutMxUQXFVdKErqWAf0BmHTJq2oZlkSDJKGeYr3Pgrblwo4lbc+TemD0RbD1w4nHMv94Txowq7Thf1szs3PzCYm4pv7yurZub2zWlEgkJlUsmJCNECnCKCdVTUjVgSFIWM1MPe+cCv3xGpqOA3uh8TP0IdTtsUI2kwD64CBqnXursu152m15l0GOCdyTtdDWSUtzDH08GjSywC07RGQJOE3dMCmCMSmB/ei2Bk4hwjRlSquk6sfZTJDXFjGR5L1EkRriHOqRpKEcRUX46fCuDu0ZpwbaQpriGQ/X3RIoipfpRaDojpLtq0huI/3nNRLdP/JTyONGE49GidsKgFnCQEWxRSbBmfUMQltTcCnEXSYS1STJvQnAnX54mtVLRPSyWro8K5bNxHDmwDXbAHnDBMSiDS1ABVYDBA3gCL+DVerSerTfrfdQ6Y41ntsAfWB/fS9Se2w=</latexit>

GV : {0, 1}N − ! {0, 1}rV

<latexit sha1_base64="cQ3982d4J81/5/JEHCWGX6ZcbC8=">ACFnicbVDLSgMxFM34rPVdekmWAQXWmaqoLgqutCVLAP6NQhk2amoZlkSDJKGeYr3Pgrblwo4lbc+TemD0RbD1w4nHMv97jx4wqbdtf1szs3PzCYm4pv7yurZe2NisK5FITGpYMCGbPlKEU5qmpGmrEkKPIZafi984HfuCNSUcFvdD8m7QiFnAYUI20kr3Bw4dVP3dTed9zsNr3KoMsEDyUNuxpJKe7hjye9euYVinbJHgJOE2dMimCMqlf4dDsCJxHhGjOkVMuxY91OkdQUM5Ll3USRGOEeCknLUI4iotrp8K0M7hqlAwMhTXENh+rviRFSvUj3RGSHfVpDcQ/NaiQ5O2inlcaIJx6NFQcKgFnCQEexQSbBmfUMQltTcCnEXSYS1STJvQnAmX54m9XLJOSyVr4+KlbNxHDmwDXbAHnDAMaiAS1AFNYDBA3gCL+DVerSerTfrfdQ6Y41ntsAfWB/fRW6e1w=</latexit>

Inputs of Hash Functions Outputs of Hash Functions Hash Functions

rX ≈ NH(X|UZ), rU ≈ NH(U|Z), rV ≈ NH(V |UZX)

slide-18
SLIDE 18

18/29

Dependencies in two consecutive blocks

Fi+1

<latexit sha1_base64="BnBeDldJdMmfyQE3wLJtLTS3cE4=">AB7nicbVBNS8NAEJ34WetX1aOXxSIQkmqoCcpCOKxgv2ANpTNdtIu3WzC7kYoT/CiwdFvPp7vPlv3LY5aOuDgcd7M8zMCxLBtXHdb2dldW19Y7OwVdze2d3bLx0cNnWcKoYNFotYtQOqUXCJDcONwHaikEaBwFYwup36rSdUmsfy0YwT9CM6kDzkjBorte56GT/3Jr1S2a24M5Bl4uWkDnqvdJXtx+zNEJpmKBadzw3MX5GleFM4KTYTUmlI3oADuWShqh9rPZuRNyapU+CWNlSxoyU39PZDTSehwFtjOiZqgXvan4n9dJTXjtZ1wmqUHJ5ovCVBATk+nvpM8VMiPGlCmuL2VsCFVlBmbUNG4C2+vEya1Yp3Uak+XJZrN3kcBTiGEzgD6gBvdQhwYwGMEzvMKbkzgvzrvzMW9dcfKZI/gD5/MHsdqPIg=</latexit>

Fi

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Di

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Di+1

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Ei+1

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Ei

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Ni

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Ni+1

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Block i

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Block i+1

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e Fi

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e Di

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e Ei

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e Ei+1

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e Di+1

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e Fi+1

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e Vi+1

<latexit sha1_base64="FJqxsc/MRmJBo3QigrjwjNm+UZ8=">AB/XicbVDLSsNAFJ3UV62v+Ni5GWwFQShJXeiy6MZlBfuANoTJ5LYdOpmEmYlSQ/FX3LhQxK3/4c6/cdpmodUDFw7n3Mu9wQJZ0o7zpdVWFpeWV0rpc2Nre2d+zdvZaKU0mhSWMey05AFHAmoKmZ5tBJAo4NAORldTv30HUrFY3OpxAl5EBoL1GSXaSL59UOndsxA04yHglp+xU3dS8e2yU3VmwH+Jm5MytHw7c9eGNM0AqEpJ0p1XSfRXkakZpTDpNRLFSEjsgAuoYKEoHystn1E3xslBD3Y2lKaDxTf05kJFJqHAWmMyJ6qBa9qfif101/8LmEhSDYLOF/VTjnWMp1HgkEmgmo8NIVQycyumQyIJ1SawkgnBXz5L2nVqu5ZtXZTK9cv8ziK6BAdoRPkonNUR9eogZqIogf0hF7Qq/VoPVtv1vu8tWDlM/voF6yPbzKUlGY=</latexit>

e Ui+1

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e Xi+1

<latexit sha1_base64="iDBtVwmGXrFkHAg/bYuzeru0Hk=">AB/XicbVDLSsNAFJ3UV62v+Ni5GWwFQShJXeiy6MZlBfuANoTJ5LYdOpmEmYlSQ/FX3LhQxK3/4c6/cdpmodUDFw7n3Mu9wQJZ0o7zpdVWFpeWV0rpc2Nre2d+zdvZaKU0mhSWMey05AFHAmoKmZ5tBJAo4NAORldTv30HUrFY3OpxAl5EBoL1GSXaSL59UOndsxA04yHgjp+xU3dS8e2yU3VmwH+Jm5MytHw7c9eGNM0AqEpJ0p1XSfRXkakZpTDpNRLFSEjsgAuoYKEoHystn1E3xslBD3Y2lKaDxTf05kJFJqHAWmMyJ6qBa9qfif101/8LmEhSDYLOF/VTjnWMp1HgkEmgmo8NIVQycyumQyIJ1SawkgnBXz5L2nVqu5ZtXZTK9cv8ziK6BAdoRPkonNUR9eogZqIogf0hF7Qq/VoPVtv1vu8tWDlM/voF6yPbzWqlGg=</latexit>

e Yi+1

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e Zi+1

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e Zi

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e Yi

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e Ui

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e Xi

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For Block i ∈ 2, k,

  • Fi, Di, Ei denote fresh randomness
  • Fi,

Di, Ei denote recycled randomness

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Main Results

The proposed coding scheme achieves the region RX,Y for the DMMAC (X × Y, qZ|XY , Z) with binary input alphabets X and Y

The coding scheme uses a combination of source resolvability codes in a black box manner This yields an explicit coding scheme for the MAC resolvability region since explicit codes for source resolvability are known

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Outline

  • Introduction and Model
  • Coding Scheme
  • Coding Scheme Analysis
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Coding Scheme Analysis

The output of the hash functions are almost jointly uniformly distributed Lemma 1 Let punif

¯ E

, punif

¯ D

, punif

¯ F

denote the uniform distributions over {0, 1}rX , {0, 1}rU, {0, 1}rV , respectively. We have V

  • qGX (X 1:N)GU(U1:N)GV (V 1:N)Z 1:N, punif

¯ E

punif

¯ D

punif

¯ F

qZ 1:N

  • ≤ δT(N),

where δT(N) is such that limN→∞ δT(N) = 0.

  • Evaluation of min-entropies
  • Distributed leftover hash lemma
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Coding Scheme Analysis (Continued...)

In each encoding block, the random variables induced by the coding scheme approximate well the target distribution Lemma 2 For Block i ∈ 1, k, V( pU1:N

i

V 1:N

i

X 1:N

i

Y 1:N

i

Z 1:N

i

, qU1:NV 1:NX 1:NY 1:NZ 1:N) ≤ δi(N), where δi(N) is such that lim

N→+∞ δi(N) = 0.

  • Source resolvability in each block
  • Lemma 1
  • Proof by induction
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Coding Scheme Analysis (Continued...)

The recycled randomness in Block i ∈ 2, k is almost independent of the channel output in Block i − 1 Lemma 3 For i ∈ 2, k, V( pZ 1:N

i−1EiDiFi,

pZ 1:N

i−1

pEiDiFi) ≤ δ(1)

i

(N), where δ(1)

i

(N) is such that lim

N→+∞ δ(1) i

(N) = 0.

  • Triangle inequality
  • Lemmas 1, 2
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Coding Scheme Analysis (Continued...)

The recycled randomness in Block i ∈ 2, k is almost independent of the channel outputs in Blocks 1 to i − 1 considered jointly Lemma 4 For i ∈ 2, k, we have V

  • pZ 1:N

1:i−1DiEiFi,

pZ 1:N

1:i−1

pDiEiFi

  • ≤ δ(C)

i

(N), where δ(C)

i

(N) such that lim

N→∞ δ(C) i

(N) = 0.

  • Proof by induction
  • Triangle inequality
  • Lemma 3
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Coding Scheme Analysis (Continued...)

The channel outputs of all the blocks are asymptotically independent Lemma 5 We have V

  • pZ 1:N

1:k ,

k

  • i=1
  • pZ 1:N

i

  • ≤ (k − 1) max

j∈2,k δ(C) j

(N), where (δ(C)

j

(N))j∈2,k is defined in Lemma 4.

  • Triangle inequality
  • Markov Chain

Z 1:N

i

− ( Di, Ei, Fi) − Z 1:N

1:i−1

  • Lemma 4
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Coding Scheme Analysis (Continued...)

The target output distribution is well approximated jointly over all blocks Lemma 6 For block i ∈ 1, k, we have V

  • pZ 1:N

1:k , qZ 1:kN

  • ≤ k
  • max

j∈2,k δ(C) j

(N) + max

j∈1,k δj(N)

  • ,

where (δ(C)

j

(N))j∈2,k is defined in Lemma 5 and (δj(N))j∈1,k is defined in Lemma 2.

  • Triangle inequality
  • Proof by induction
  • Lemmas 2, 5
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Coding Scheme Analysis (Continued...)

Finally, the coding scheme achieves the desired rate pair Lemma 7 Let ǫ0 > 0. For k large enough, the rate pair (R1, RU + RV ) is achievable and lim

N→+∞ R1 = I(X; ZU) + ǫ0,

lim

N→+∞ RU = I(U; Z) + ǫ0,

lim

N→+∞ RV = I(V ; ZUX) + ǫ0.

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Concluding remarks

  • Reduction of the MAC resolvability code construction to

source resolvability code construction

  • Source resolvability codes used in a black box manner
  • Randomness recycling through distributed hashing coupled with

block-Markov coding

  • Explicit codes that achieve the MAC resolvability region for

arbitrary channels with binary input alphabets

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Thank You