Joint Source-Channel Secrecy Using Hybrid Coding Eva Song, Paul - - PowerPoint PPT Presentation

Joint Source-Channel Secrecy Using Hybrid Coding

Eva Song, Paul Cuff, and H. Vincent Poor

Department of Electrical Engineering Princeton University

June 19, 2015

A source-channel coding setting

Encoder fn PYZ|X Decoder gn Eve t = 1, . . . , n Sn X n Y n Z n ˆ St ˇ St St−1

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

A source-channel coding setting

Encoder fn PYZ|X Decoder gn Eve t = 1, . . . , n Sn X n Y n Z n ˆ St ˇ St St−1 Quality of reconstruction: d(Sn, ˆ Sn), d(Sn, ˇ Sn)

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

A source-channel coding setting

Encoder fn PYZ|X Decoder gn Eve t = 1, . . . , n Sn X n Y n Z n ˆ St ˇ St St−1 Quality of reconstruction: d(Sn, ˆ Sn), d(Sn, ˇ Sn) Why causal disclosure?

◮ Stronger formulation: to the favor of eavesdropper ◮ Can generalize equivocation Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 2 / 22

In this talk...

Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 3 / 22

In this talk...

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

◮ E

• d(Sn, ˆ

Sn)

• ≤n Db

◮ min{Pˇ

St |ZnSt−1}n t=1 E[d(Sn, ˇ

Sn)] ≥n De

Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 3 / 22

In this talk...

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

◮ E

• d(Sn, ˆ

Sn)

• ≤n Db

◮ min{Pˇ

St |ZnSt−1}n t=1 E[d(Sn, ˇ

Sn)] ≥n De

Analysis uses The Likelihood Encoder

◮ Total variation distance ◮ Soft-covering lemma Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 3 / 22

What is a likelihood encoder?

a stochastic source encoder: fn : X n → M Encoder fn Decoder gn X n M Y n

Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 4 / 22

What is a likelihood encoder?

a stochastic source encoder: fn : X n → M Encoder fn Decoder gn X 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).

Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 4 / 22

Warm up – soft-covering lemma

Lemma

Given 1) PUXZ 2) random C(n) of sequences Un(m) ∼ n

t=1 PU(ut), m ∈ [1 : 2nR]

Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 5 / 22

Warm up – soft-covering lemma

Lemma

Given 1) 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)

Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 5 / 22

Warm up – soft-covering lemma

Lemma

Given 1) 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.

Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 5 / 22

Problem setup

Encoder fn PYZ|X Decoder gn Eve t = 1, . . . , n Sn X n Y n Z n ˆ St ˇ St St−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 → X n (possibly stochastic) Legitimate receiver decoder gn : Yn → ˆ Sn (possibly stochastic) Eavesdropper decoders {Pˇ

St|Z nSt−1}n t=1

Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 6 / 22

Definition

Definition

A distortion pair (Db, De) is achievable if there exists a sequence of source-channel encoders and decoders (fn, gn) such that E[d(Sn, ˆ Sn)] ≤n Db and min

{Pˇ

St |ZnSt−1}n t=1

E[d(Sn, ˇ Sn)] ≥n De.

Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 7 / 22

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

Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 8 / 22

Scheme O – operational separate

Theorem

A distortion pair (Db, De) is achievable if I(S; U1) < I(U2; Y ) I(S; ˆ S|U1) < I(V2; Y |U2) − I(V2; Z|U2) Db ≥ E

• d(S, ˆ

S)

• De ≤ η min

a∈ ˆ S

E[d(S, a)] + (1 − η) min

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

for some distribution PSP ˆ

S|SPU1|ˆ SPU2PV2|U2PX|V2PYZ|X, where

η = [I(U2; Y ) − I(U2; Z)]+ I(S; U1) .

Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 9 / 22

Hybrid coding

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

at least optimal for P2P communication [Minero et al.] achieves best known bounds in multiuser settings Secrecy: need stochastic symbol-by-symbol mapping

Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 10 / 22

Scheme I – basic hybrid coding

Theorem

A distortion pair (Db, De) is achievable if I(U; S) < I(U; Y ) Db ≥ E[d(S, φ(U, Y ))] De ≤ β min

ψ0(z) E[d(S, ψ0(Z))]

+(1 − β) min

ψ1(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(·, ·).

Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 11 / 22

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]

Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 12 / 22

Scheme I – achievability scheme – continued

Encoder

◮ likelihood encoder PLE(m|sn) with

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

◮ produces channel input through a random transformation:

n

t=1 PX|SU(xt|st, Ut(m))

✶

Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 13 / 22

Scheme I – achievability scheme – continued

Encoder

◮ likelihood encoder PLE(m|sn) with

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

◮ produces channel input through a random transformation:

n

t=1 PX|SU(xt|st, Ut(m))

Decoder

◮ good channel decoder PD1( ˆ

m|y n) w.r.t. codebook {un(a)}a and memoryless channel PY |U

◮ deterministic mapping φn(un, y n) is the concatenation of

{φ(ut, yt)}n

t=1:

PD2(ˆ sn| ˆ m, y n) ✶{ˆ sn = φn(un( ˆ m), y n)}

Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 13 / 22

Analysis outline – at legitimate receiver

System induced distribution P Idealized distribution Q QMUnSnX nY nZ n(m, un, sn, xn, yn, zn)

• 1

2nR ✶{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, ˆ

Sn)

• ≤ EP[d(S, φ(U, Y ))] + δn

Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 14 / 22

Analysis outline – at eavesdropper

auxiliary distribution ˇ Q(i)

SiZ n(si, zn) n

• t=1

PZ(zt)

i

• j=1

PS|Z(sj|zj) soft-covering: R > I(Z; U) ⇒ ˇ Q(i)

Z nSi ≈ QZ nSi

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

I(S;U|Z)

phase transition in distortion

◮ before βn: ◮ min{ψ0i(si−1,zn)}i EP

• 1

k

k

i=1 d(Si, ψ0i(Si−1, Z n))

• ≥

minψ0(z) EP [d(S, ψ0(Z))] − ǫn

◮ after βn: ◮ min{ψ1i(si−1,zn)} EP

• 1

k

n

i=j d(Si, ψ1i(Si−1, Z n))

• ≥

minψ1(u,z) EP [d(S, ψ1(U, Z))] − ǫn

Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 15 / 22

Scheme II – superposition hybrid coding

Theorem

A distortion pair (Db, De) is achievable if I(V ; S) < I(UV ; Y ) Db ≥ E [d(S, φ(V , Y ))] De ≤ min{β, α} min

ψ0(z) E [d(S, ψ0(Z))]

+ (α − min{β, α}) min

ψ1(u,z) E [d(S, ψ1(U, Z))]

+(1 − α) min

ψ2(v,z) E [d(S, ψ2(V , Z))]

for some distribution PSPV |SPU|V PX|SUV PYZ|X and function φ(·, ·).

Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 16 / 22

Scheme II – achievability proof

Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 17 / 22

Scheme II – achievability proof

Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 17 / 22

Relations among schemes

Scheme II generalizes Scheme I Scheme II generalizes Scheme O

Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 18 / 22

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 ≤ min

a∈ ˆ S

E[d(S, a)] for some distribution PSPU|SPX|SUPYZ|X and function φ(·, ·).

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

Numerical example

Source: i.i.d. Bern(p) Channels: BSC with crossover probabilities p1, p2 Legitimate receiver: lossless decoding Eavesdropper: Hamming distortion

Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 20 / 22

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 O Scheme I No Encoding Perfect Secrecy Outer Bound

Figure: Distortion at the eavesdropper as a function of source distribution p with p1 = 0, p2 = 0.3

Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 21 / 22

Summary

have done:

◮ achieved better performance in joint source-channel secrecy with hybrid

coding

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

and operationally separate SC coding (O)

have not done:

◮ Can I outperform O? ◮ Is II strictly better than I? ◮ non-trivial outer bound? Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 22 / 22