Secure Multiterminal Source Coding ... - joffrey-villard.fr · Joffrey Villard and Pablo Piantanida SUPELEC, Dpt. of Telecommunications, Gif-sur-Yvette, France. Email: {joffrey.villard,

Secure Multiterminal Source Codingwith Side Information at the Eavesdropper

Joffrey Villard and Pablo Piantanida

SUPELEC, Dpt. of Telecommunications, Gif-sur-Yvette, France.

Email: {joffrey.villard, pablo.piantanida}@supelec.fr

1st International ICST Workshop on Secure Wireless Networks

Securenets 2011 Secure Multiterminal Source Coding with SI at Eve 1 / 31

Introduction

Context

An Alice Bob E[d(An, An)

J (rate RA)

En Eve 1n H(An|JEn) & ∆

Cn CharlieK (rate RC)

Tradeoff: Min. rates + Min. distortion + Max. equivocation

Our Aim: Find all achievable tuples (RA,RC,D,∆)

Introduction

Context

J (rate RA)

Introduction

Context

An Alice BobE[d(An, An)

J (rate RA)

Introduction

Context

An Alice BobE[d(An, An)

J (rate RA)

Introduction

ReferencesMultiterminal source coding.D. Slepian and J. Wolf. Noiseless coding of correlated information sources. IEEE Trans. IT, 19(4):471–480, 1973.

T. Berger. Multiterminal source coding. in The information theory approach to communications, 1977.

Source coding with side-information.A. Wyner and J. Ziv. The rate-distortion function for source coding with side information at the decoder. IEEE Trans. IT,22(1):1–10, 1976.

Information-theoretic security.C.E. Shannon. Communication theory of secrecy systems. BSTJ, 28:656–715, 1949.

A.D. Wyner. The wire-tap channel. BSTJ, 54(8):1355–1387, 1975.

I. Csiszar and J. Korner. Broadcast channels with confidential messages. , 24(3):339–348, 1978.

Y. Liang, H.V. Poor, and S. Shamai. Information theoretic security. Now Publishers, 2009.

Secure source coding.H. Yamamoto. Rate-distortion theory for the Shannon cipher system. IEEE Trans. IT, 43(3):827–835, 1997.

V. Prabhakaran and K. Ramchandran. On secure distributed source coding. In Proc. ITW, p. 442–447, 2007.

D. Gunduz, E. Erkip, and H.V. Poor. Lossless compression with security constraints. In Proc. ISIT, p. 111–115, 2008.

R. Tandon, S. Ulukus, and K. Ramchandran. Secure source coding with a helper. In Proc. Allerton, p. 1061–1068, 2009.

N. Merhav. Shannon’s secrecy system with informed receivers and its application to systematic coding for wiretappedchannels. IEEE Trans. IT, 54(6):2723–2734, 2008.

Introduction

Outline

1 Definitions and First ResultsDefinitionsInner and Outer BoundsInner Bound–Insight

2 Results of OptimalityUncoded Side InformationLossless Compression of Both SourcesAlternative Characterizations

3 Application Example (Uncoded Side Information)

Definitions and First Results Definitions

Outline

Definitions

A, C and E : three finite sets

(Ai,Ci,Ei)i≥1: i.i.d random variables on A× C × Ewith known joint distribution p(a, b, e)

d : A×A → [0 ; dmax]: a finite distortion measure

An (n,RA,RC)-code for source coding in this setup is defined by

Two encoding functions at Alice and CharliefA : An → {1, . . . , 2nRA} and fC : Cn → {1, . . . , 2nRC}, resp.

A decoding function at Bobg : {1, . . . , 2nRA} × {1, . . . , 2nRC} → An

Definitions (cont.)

J (rate RA)

A tuple (RA,RC,D,∆) ∈ R4+ is achievable if,

for any ε > 0, there exists an (n,RA + ε,RC + ε)-code (fA, fC, g)such that:

E[d(An, g(fA(An), fC(Cn)))

]≤ D + ε

H(An|fA(An),En) ≥ ∆− ε

Definitions and First Results Inner and Outer Bounds

Outline

Inner and Outer Bounds

Theorem (Inner bound)

(RA,RC,D,∆) ∈ R4+ is achievable if there exist

r.v. U, V, W on some finite sets U , V,W, resp., s.t.p(uvwace) = p(u|v)p(v|a)p(w|c)p(ace) ,

a function A : V ×W → A, s.t.

RA ≥ I(V; A|W)

RC ≥ I(W; C|V)

RA + RC ≥ I(VW; AC)

D ≥ E[d(A, A(V,W))

]∆ ≤ H(A|UE)− I(V; A|UW)

∆− RC ≤ H(A|V)− I(A; E|U)− I(W; C|V)

RA ≥ I(V; A|W)

RC ≥ I(W; C|V)

D ≥ E[d(A, A(V,W))

]∆ ≤ H(A|UE)− I(V; A|UW)

∆− RC ≤ H(A|V)− I(A; E|U)− I(W; C|V)

RA ≥ I(V; A|W)

RC ≥ I(W; C|V)

D ≥ E[d(A, A(V,W))

]∆ ≤ H(A|UE)− I(V; A|UW)

∆− RC ≤ H(A|V)− I(A; E|U)− I(W; C|V)

RA ≥ I(V; A|W)

RC ≥ I(W; C|V)

D ≥ E[d(A, A(V,W))

]∆ ≤ H(A|UE)− I(V; A|UW)

∆− RC ≤ H(A|V)− I(A; E|U)− I(W; C|V)

RA ≥ I(V; A|W)

RC ≥ I(W; C|V)

D ≥ E[d(A, A(V,W))

]∆ ≤ H(A|UE)− I(V; A|UW)

∆− RC ≤ H(A|V)− I(A; E|U)− I(W; C|V)

Theorem (Outer bound)

If (RA,RC,D,∆) ∈ R4+ is achievable, then there exist

r.v. U, V, W on some finite sets U , V,W, resp., s.t.p(wace) = p(w|c)p(ace), p(uvace) = p(u|v)p(v|a)p(ace) ,

RA ≥ I(V; A|W)

RC ≥ I(W; C|V)

D ≥ E[d(A, A(V,W))

]∆ ≤ H(A|UE)− I(V; A|UW)

∆− RC ≤ H(A|V)− I(A; E|U)− I(W; C|V)

Auxiliary Variables

Inner Bound

U V A E

Outer Bound

U V A E

Definitions and First Results Inner Bound–Insight

Outline

Three Corner Points

3 three-step schemes to deliver (U,V) and W, usingsuperposition coding (U − V − A− C −W)previously received information used as side informationrandom binningtime-sharing

Corner point (I)

(II) (III)

Comm. order W, U, V

U, W, V U, V, W

RA I(V; A|W)

I(U; A) + I(V; A|UW) I(V; A)

RC I(W; C)

I(W; C|U) I(W; C|V)

D E[d(A, A(V,W))

— —

∆ H(A|UE)− I(V; A|UW)

H(A|UE)− I(V; A|UW) H(A|UE)− I(V; A|U)

Three Corner Points

Corner point (I) (II)

Comm. order W, U, V U, W, V

U, V, W

RA I(V; A|W) I(U; A) + I(V; A|UW)

I(V; A)

RC I(W; C) I(W; C|U)

I(W; C|V)

D E[d(A, A(V,W))

∆ H(A|UE)− I(V; A|UW) H(A|UE)− I(V; A|UW)

H(A|UE)− I(V; A|U)

Three Corner Points

Corner point (I) (II) (III)

Comm. order W, U, V U, W, V U, V, W

RA I(V; A|W) I(U; A) + I(V; A|UW) I(V; A)

RC I(W; C) I(W; C|U) I(W; C|V)

D E[d(A, A(V,W))

]— —

Time-Sharing

Segment (I)–(II)

D = E[d(A, A(V,W))

]∆ = H(A|UE)− I(V; A|UW)

RA + RC = I(VW; AC)

(III)RA

Segment (II)–(III)

D = E[d(A, A(V,W))

]∆− RC = H(A|UE)− I(V; A|U)− I(W; C|V)

RA + RC = I(VW; AC)

Time-Sharing

Segment (I)–(II)

D = E[d(A, A(V,W))

]∆ = H(A|UE)− I(V; A|UW)

RA + RC = I(VW; AC)

(III)RA

Segment (II)–(III)

D = E[d(A, A(V,W))

]∆− RC = H(A|UE)− I(V; A|U)− I(W; C|V)

RA + RC = I(VW; AC)

Achievable Region for Some Fixed D

Results of Optimality Uncoded Side Information

Outline

Context

J (rate RA)

Auxiliary Variables

Inner Bound

U V A E

Outer Bound

U V A E

Uncoded Side Information

Theorem (Rate-Distortion-Equivocation Region)

(RA, D,∆) ∈ R3+ is achievable i.f.f. there exist

r.v. U, V

on some finite sets U , V

, resp., s.t.p(uvace) = p(u|v)p(v|a)p(ace) ,

a function A : V × C → A, s.t.

RA ≥ I(V; A|C)

D ≥ E[d(A, A(V,C))

]∆ ≤ H(A|UE)− I(V; A|UC)

Uncoded Side Information (cont.)

Achievability: point (I) with W = C

(I)(II)

Converse: new proof

Wyner-Ziv coding achieves the optimal performanceif one side information is less noisy than the other(optimal choice: U∗ = ∅ or U∗ = V)

Results of Optimality Lossless Compression of Both Sources

Outline

Context

An Alice Bob (An, Cn) ≈ (An,Cn)J (rate RA)

Auxiliary Variables

Inner Bound

U V A E

Outer Bound

U V A E

Lossless Compression of Both Sources

Theorem (Compression-Equivocation Rates Region)

(RA,RC, ∆) ∈ R3+ is achievable i.f.f. there exists

r.v. U

, V, W

on some finite set

, V,W, resp.,

s.t.p(uace) = p(u|a)p(ace),

RA ≥ H(A|C)

RC ≥ H(C|U)

RA + RC ≥ H(AC)

∆ ≤ H(A|UE)− H(A|UC)

Lossless Compression of Both Sources (cont.)

Achievability: points (I) and (II) with V = A and W = C

(III)RA

Converse: new proof

Slepian-Wolf coding is sufficientif E is less noisy than C (U∗ = A, and ∆ = 0)

Results of Optimality Alternative Characterizations

Outline

Results of Optimality Alternative Characterizations

Giving U to Eve is also optimal

Alice can enable Eve to decode the common message U:

RA ≥ (·) + [I(U; C)− I(U; E)]+ ,

with no loss on secrecy

Achievability: OKConverse: new proof

cf. broadcast channel with confidential messages[Csiszàr & Körner–1978]

optimal choice U∗:part of V which conveys “more information” about E than C

Application Example (Uncoded Side Information)

Outline

Binary Source with BEC and BSC Side Informations

1− p

1− ε

1− p

1− ε

Neither Bob nor Eve is a lessnoisy decoder for all values of (p, ε):

0 2p 4p(1− p) h2(p) 1 ε

E stoch. degr.A − C − E

C less noisyU − A− (C,E)⇒I(U; C) ≥ I(U; E)

C more capableI(A; C) ≥ I(A; E)

1− p

1− ε

Neither Bob nor Eve is a lessnoisy decoder for all values of (p, ε):

0 2p 4p(1− p) h2(p) 1 ε

E stoch. degr.

A − C − E

C less noisy

U − A− (C,E)⇒I(U; C) ≥ I(U; E)

C more capable

I(A; C) ≥ I(A; E)

Wyner-Ziv is optimal Wyner-Ziv is insufficient

1− p

1− ε

distortion d: Hamming distance

source A: uniformly distributed

Illustration (p = 0.1, ε = h2(p) ≈ 0.469)

10−4 10−3 10−2 10−1

10−2

10−1

Optimal U, VWyner-Ziv

Equivocation rate at Eve as a function of the distortion level at Bob

Conclusion

Summary and Discussion

Single-letter inner and outer bounds on the generalrates-distortion-equivocation region

Results of optimalityuncoded side informationdistributed lossless compression

Ongoing work:

Source-channel coding with security constraints

This is the end...

with P. PiantanidaSecure Multiterminal Source Coding with Side Information at the Eavesdroppersubmitted to IEEE Trans. on Inf. Theory, available on arXiv:1105.1658.

with P. Piantanida and S. ShamaiSecure Lossy Source-Channel Wiretapping with Side Information at theReceiving Terminalsto be presented at ISIT 2011.

Thank you for your attention.

This is the end...

with P. PiantanidaSecure Multiterminal Source Coding with Side Information at the Eavesdroppersubmitted to IEEE Trans. on Inf. Theory, available on arXiv:1105.1658.

with P. Piantanida and S. ShamaiSecure Lossy Source-Channel Wiretapping with Side Information at theReceiving Terminalsto be presented at ISIT 2011.

Thank you for your attention.

Secure Multiterminal Source Coding ... - joffrey-villard.fr · Joffrey Villard and Pablo Piantanida SUPELEC, Dpt. of Telecommunications, Gif-sur-Yvette, France. Email: {joffrey.villard,

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