Joint Source-Channel Secrecy Using Hybrid Codingstorage.googleapis.com/wzukusers/user-20000200/documents/... · 2016-02-15 · Joint Source-Channel Secrecy Using Hybrid Coding Eva

Joint Source-Channel Secrecy Using Hybrid Coding

Eva Song, Paul Cuff, and H. Vincent Poor

Department of Electrical EngineeringPrinceton University

June 19, 2015

A source-channel coding setting

Encoder fn PYZ |X

Decoder gn

Eve

t = 1, . . . , n

Sn X nY n

Zn

St

St

S t−1

Quality of reconstruction: d(Sn, Sn), d(Sn, Sn)

Why causal disclosure?I Stronger formulation: to the favor of eavesdropperI Can generalize equivocation

Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 2 / 22


Encoder fn PYZ |X

Decoder gn

Eve

t = 1, . . . , n

Sn X nY n

Zn

St

St

S t−1





Encoder fn PYZ |X

Decoder gn

Eve

t = 1, . . . , n

Sn X nY n

Zn

St

St

S t−1




In this talk...

Design source-channel coding schemes for (Db,De) s.t.

I E[d(Sn, Sn)

]≤n Db

I min{PSt |ZnSt−1}nt=1E[d(Sn, Sn)] ≥n De

Analysis uses The Likelihood EncoderI Total variation distanceI Soft-covering lemma


In this talk...


I E[d(Sn, Sn)

]≤n Db




In this talk...


I E[d(Sn, Sn)

]≤n Db




What is a likelihood encoder?

a stochastic source encoder: fn : X n 7→ M

Encoder fn Decoder gnX n M Y n

Given

a codebook {yn(m)}m, m ∈ [1 : 2nR ]

a joint distribution PXY

the likelihood function for each codeword:

L(m|xn) , PX n|Y n(xn|yn(m)) =∏

PX |Y (xn|yn(m))

the likelihood encoder determines the message index according to:

PM|X n(m|xn) =L(m|xn)∑

m′∈[1:2nR ] L(m′|xn)∝ L(m|xn).


What is a likelihood encoder?

a stochastic source encoder: fn : X n 7→ M

Encoder fn Decoder gnX n M Y n

Given

a codebook {yn(m)}m, m ∈ [1 : 2nR ]

a joint distribution PXY

the likelihood function for each codeword:

L(m|xn) , PX n|Y n(xn|yn(m)) =∏

PX |Y (xn|yn(m))

the likelihood encoder determines the message index according to:

PM|X n(m|xn) =L(m|xn)∑

m′∈[1:2nR ] L(m′|xn)∝ L(m|xn).


Warm up – soft-covering lemma

Lemma

Given1) PUXZ

2) random C(n) of sequences Un(m) ∼∏n

t=1 PU(ut), m ∈ [1 : 2nR ]

Let

PMX nZ k (m, xn, zk) ,1

2nR

n∏t=1

PX |U(xt |Ut(m))k∏

t=1

PZ |XU(zt |xt ,Ut(m))

PX nZ k ,n∏

t=1

PX (xt)k∏

t=1

PZ |X (zt |xt)

If R > I (X ;U), then

ECn[∥∥PX nZ k − PX nZ k

∥∥TV

]≤ exp(−γn)→n 0,

for any β < R−I (X ;U)I (Z ;U|X ) , k ≤ βn, γ > 0 depending on this gap.



Lemma

Given1) PUXZ


t=1 PU(ut), m ∈ [1 : 2nR ]

Let


2nR

n∏t=1


t=1


PX nZ k ,n∏

t=1

PX (xt)k∏

t=1

PZ |X (zt |xt)



∥∥TV

]≤ exp(−γn)→n 0,




Lemma

Given1) PUXZ


t=1 PU(ut), m ∈ [1 : 2nR ]

Let


2nR

n∏t=1


t=1


PX nZ k ,n∏

t=1

PX (xt)k∏

t=1

PZ |X (zt |xt)



∥∥TV

]≤ exp(−γn)→n 0,



Problem setup

Encoder fn PYZ |X

Decoder gn

Eve

t = 1, . . . , n

Sn X nY n

Zn

St

St

S t−1

i.i.d. source Sn ∼∏n

t=1 PS(st)

memoryless broadcast channel∏n

t=1 PYZ |X (yt , zt |xt)Encoder fn : Sn 7→ X n (possibly stochastic)

Legitimate receiver decoder gn : Yn 7→ Sn (possibly stochastic)

Eavesdropper decoders {PSt |ZnS t−1}nt=1


Definition

Definition

A distortion pair (Db,De) is achievable if there exists a sequence ofsource-channel encoders and decoders (fn, gn) such that

E[d(Sn, Sn)] ≤n Db

andmin

{PSt |ZnSt−1}nt=1

E[d(Sn, Sn)] ≥n De .


We consider

Scheme O – Operationally separate SC coding [Schieler et al.Allerton 2012]

Scheme I – Joint SC coding using Hybrid Coding

Scheme II – Joint SC coding using superposition Hybrid Coding


Scheme O – operational separate

Theorem

A distortion pair (Db,De) is achievable if

I (S ;U1) < I (U2;Y )

I (S ; S |U1) < I (V2;Y |U2)− I (V2;Z |U2)

Db ≥ E[d(S , S)

]De ≤ ηmin

a∈SE[d(S , a)] + (1− η) min

t(u1)E[d(S , t(U1))]

for some distribution PSP S |SPU1|SPU2PV2|U2PX |V2

PYZ |X , where

η =[I (U2;Y )− I (U2;Z )]+

I (S ;U1).


Hybrid coding

Likelihood Encoder PX |SU PYZ |X Channel Decoder φ(u, y)Sn Un(M) X n Y n Un(M) Sn

at least optimal for P2P communication [Minero et al.]

achieves best known bounds in multiuser settings

Secrecy: need stochastic symbol-by-symbol mapping


Scheme I – basic hybrid coding

Theorem


I (U; S) < I (U;Y )

Db ≥ E[d(S , φ(U,Y ))]

De ≤ β minψ0(z)

E[d(S , ψ0(Z ))]

+(1− β) minψ1(u,z)

E[d(S , ψ1(U,Z ))]

where

β = min

{[I (U;Y )− I (U;Z )]+

I (S ;U|Z ), 1

}for some distribution PSPU|SPX |SUPYZ |X and function φ(·, ·).


Scheme I – achievability scheme

Fix distribution PSPU|SPX |SUPYZ |X

Codebook generation: Independently generate 2nR sequences in Un

according to∏n

t=1 PU(ut) and index by m ∈ [1 : 2nR ]


Scheme I – achievability scheme – continued

EncoderI likelihood encoder PLE (m|sn) with

L(m|sn) = PSn|Un(sn|un(m))

I produces channel input through a random transformation:∏nt=1 PX |SU(xt |st ,Ut(m))

DecoderI good channel decoder PD1(m|yn) w.r.t.

codebook {un(a)}a and memoryless channel PY |UI deterministic mapping φn(un, yn) is the concatenation of{φ(ut , yt)}nt=1:

PD2(sn|m, yn) , 1{sn = φn(un(m), yn)}


Scheme I – achievability scheme – continued

EncoderI likelihood encoder PLE (m|sn) with

L(m|sn) = PSn|Un(sn|un(m))

I produces channel input through a random transformation:∏nt=1 PX |SU(xt |st ,Ut(m))

DecoderI good channel decoder PD1(m|yn) w.r.t.

codebook {un(a)}a and memoryless channel PY |UI deterministic mapping φn(un, yn) is the concatenation of{φ(ut , yt)}nt=1:

PD2(sn|m, yn) , 1{sn = φn(un(m), yn)}


Analysis outline – at legitimate receiver

System induced distribution P

Idealized distribution Q

QMUnSnX nY nZn(m, un, sn, xn, yn, zn)

,1

2nR1{un = Un(m)}

n∏t=1

PS|U(st |ut)

n∏t=1

PX |SU(xt |st , ut)n∏

t=1

PYZ |X (yt , zt |xt).

soft-covering: R > I (U;S) ⇒ P ≈ Q

channel coding: R ≤ I (U;Y )⇒

EC(n)

[EP

[d(Sn, Sn)

]]≤ EP [d(S , φ(U,Y ))] + δn


Analysis outline – at eavesdropper

auxiliary distribution

Q(i)

S iZn(s i , zn) ,n∏

t=1

PZ (zt)i∏

j=1

PS |Z (sj |zj)

soft-covering: R > I (Z ;U) ⇒ Q(i)

ZnS i ≈ QZnS i

i can go up to βn, for any β < R−I (U;Z)I (S ;U|Z)

phase transition in distortionI before βn:

I min{ψ0 i (si−1,zn)}i EP

[1k

∑ki=1 d(Si , ψ0i (S

i−1,Z n))]≥

minψ0(z) EP [d(S , ψ0(Z ))]− εnI after βn:

I min{ψ1 i (si−1,zn)} EP

[1k

∑ni=j d(Si , ψ1i (S

i−1,Z n))]≥

minψ1(u,z) EP [d(S , ψ1(U,Z ))]− εn


Scheme II – superposition hybrid coding

Theorem


I (V ;S) < I (UV ;Y )

Db ≥ E [d(S , φ(V ,Y ))]

De ≤ min{β, α} minψ0(z)

E [d(S , ψ0(Z ))]

+ (α−min{β, α}) minψ1(u,z)

E [d(S , ψ1(U,Z ))]

+(1− α) minψ2(v ,z)

E [d(S , ψ2(V ,Z ))]

for some distribution PSPV |SPU|VPX |SUVPYZ |X and function φ(·, ·).


Scheme II – achievability proof


Scheme II – achievability proof


Relations among schemes

Scheme II generalizes Scheme I

Scheme II generalizes Scheme O


Perfect secrecy outer bound

Theorem

If (Db,De) is achievable, then

I (S ;U) ≤ I (U;Y )

Db ≥ E[d(S , φ(U,Y ))]

De ≤ mina∈S

E[d(S , a)]

for some distribution PSPU|SPX |SUPYZ |X and function φ(·, ·).


Numerical example

Source: i.i.d. Bern(p)

Channels: BSC with crossover probabilities p1, p2

Legitimate receiver: lossless decoding

Eavesdropper: Hamming distortion


Numerical example

0.0 0.1 0.2 0.3 0.4 0.5

p

0.0

0.1

0.2

0.3

0.4

0.5

De

Scheme OScheme INo EncodingPerfect Secrecy Outer Bound

Figure: Distortion at the eavesdropper as a function of source distribution p withp1 = 0, p2 = 0.3


Summary

have done:I achieved better performance in joint source-channel secrecy with hybrid

codingI superposition hybrid coding (II) fully generalizes basic hybrid coding (I)

and operationally separate SC coding (O)

have not done:I Can I outperform O?I Is II strictly better than I?I non-trivial outer bound?


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Joint Source-Channel Secrecy Using Hybrid Codingstorage.googleapis.com/wzukusers/user-20000200/documents/... · 2016-02-15 · Joint Source-Channel Secrecy Using Hybrid Coding Eva