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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
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
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
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
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.
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
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.
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
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 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).
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 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).
Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 4 / 22
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.
Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 5 / 22
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.
Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 5 / 22
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.
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 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
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 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 .
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 ; 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).
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(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
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 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(·, ·).
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
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)}
Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 13 / 22
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)}
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 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
Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 14 / 22
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
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} 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(·, ·).
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 ≤ mina∈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 OScheme INo EncodingPerfect Secrecy Outer Bound
Figure: Distortion at the eavesdropper as a function of source distribution p withp1 = 0, p2 = 0.3
Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 21 / 22
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?
Song, Cuff, Poor (Princeton University) Joint Source-Channel Secrecy June 19, 2015 22 / 22