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MIMO Gaussian Broadcast Channels withCommon, Private and Confidential Messages
Ziv Goldfeld
Ben Gurion University
IEEE Information Theory Workshop
September, 2016
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 1
Motivation
Gaussian MIMO channels - model wireless communication.
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 2
Motivation
Gaussian MIMO channels - model wireless communication.
Susceptibility of wireless communication to eavesdropping.
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 2
Motivation
Gaussian MIMO channels - model wireless communication.
Susceptibility of wireless communication to eavesdropping.
Eavesdroppers are not always a malicious entity:
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 2
Motivation
Gaussian MIMO channels - model wireless communication.
Susceptibility of wireless communication to eavesdropping.
Eavesdroppers are not always a malicious entity:
◮ Legitimate recipient of some messages.
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 2
Motivation
Gaussian MIMO channels - model wireless communication.
Susceptibility of wireless communication to eavesdropping.
Eavesdroppers are not always a malicious entity:
◮ Legitimate recipient of some messages.
◮ Eavesdropper of other.
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 2
Motivation
Gaussian MIMO channels - model wireless communication.
Susceptibility of wireless communication to eavesdropping.
Eavesdroppers are not always a malicious entity:
◮ Legitimate recipient of some messages.
◮ Eavesdropper of other.
Modern BC scenario - Common, Private and Confidential
messages.
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 2
Motivation
Gaussian MIMO channels - model wireless communication.
Susceptibility of wireless communication to eavesdropping.
Eavesdroppers are not always a malicious entity:
◮ Legitimate recipient of some messages.
◮ Eavesdropper of other.
Modern BC scenario - Common, Private and Confidential
messages.
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 2
Motivation - Banking Site
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 3
Motivation - Banking Site
Common
Common - Advertisement.
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 3
Motivation - Banking Site
CommonPrivate
Common - Advertisement.
Private - On-demand Public info (programs, reports, forecasts).
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 3
Motivation - Banking Site
CommonPrivate
Confidential
Common - Advertisement.
Private - On-demand Public info (programs, reports, forecasts).
Confidential - Online banking (access account, transactions).Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 3
MIMO Gaussian BC - Problem Setup
X
G1
G2
Z1
Z2
Y1
Y2
(M0,M1,M2)
(
M(1)0
,M1
)
(
M(2)0
,M2
)
M1
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 4
MIMO Gaussian BC - Problem Setup
X
G1
G2
Z1
Z2
Y1
Y2
(M0,M1,M2)
(
M(1)0
,M1
)
(
M(2)0
,M2
)
M1
User j = 1,2 Observes: Yj = GjX + Zj .
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 4
MIMO Gaussian BC - Problem Setup
X
G1
G2
Z1
Z2
Y1
Y2
(M0,M1,M2)
(
M(1)0
,M1
)
(
M(2)0
,M2
)
M1
User j = 1,2 Observes: Yj = GjX + Zj .
Dimensions: X, Y1, Y2, Z1, Z2 ∈ Rt ; G1, G2 ∈ R
t×t.
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 4
MIMO Gaussian BC - Problem Setup
X
G1
G2
Z1
Z2
Y1
Y2
(M0,M1,M2)
(
M(1)0
,M1
)
(
M(2)0
,M2
)
M1
User j = 1,2 Observes: Yj = GjX + Zj .
Dimensions: X, Y1, Y2, Z1, Z2 ∈ Rt ; G1, G2 ∈ R
t×t.
Noise Processes: i.i.d. samples of Zj ∼ N (0, It), j = 1, 2.
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 4
MIMO Gaussian BC - Problem Setup
X
G1
G2
Z1
Z2
Y1
Y2
(M0,M1,M2)
(
M(1)0
,M1
)
(
M(2)0
,M2
)
M1
User j = 1,2 Observes: Yj = GjX + Zj .
Dimensions: X, Y1, Y2, Z1, Z2 ∈ Rt ; G1, G2 ∈ R
t×t.
Noise Processes: i.i.d. samples of Zj ∼ N (0, It), j = 1, 2.
Input Covariance Constraint: 1n
∑ni=1 E
[
X(i)X⊤(i)]
� K.
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 4
MIMO Gaussian BC - Problem Setup
X
G1
G2
Z1
Z2
Y1
Y2
(M0,M1,M2)
(
M(1)0
,M1
)
(
M(2)0
,M2
)
M1
User j = 1,2 Observes: Yj = GjX + Zj .
Dimensions: X, Y1, Y2, Z1, Z2 ∈ Rt ; G1, G2 ∈ R
t×t.
Noise Processes: i.i.d. samples of Zj ∼ N (0, It), j = 1, 2.
Input Covariance Constraint: 1n
∑ni=1 E
[
X(i)X⊤(i)]
� K.
Security Criterion: 1n
I(M1; Yn2 ) −−−→
n→∞0.
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 4
MIMO Gaussian BC - Goals
X
G1
G2
Z1
Z2
Y1
Y2
(M0,M1,M2)
(
M(1)0
,M1
)
(
M(2)0
,M2
)
M1
Known inner and outer bounds on secrecy-capacity region.
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 5
MIMO Gaussian BC - Goals
X
G1
G2
Z1
Z2
Y1
Y2
(M0,M1,M2)
(
M(1)0
,M1
)
(
M(2)0
,M2
)
M1
Known inner and outer bounds on secrecy-capacity region.
Q: Do they match for the MIMO Gaussian case?
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 5
MIMO Gaussian BC - Goals
X
G1
G2
Z1
Z2
Y1
Y2
(M0,M1,M2)
(
M(1)0
,M1
)
(
M(2)0
,M2
)
M1
Known inner and outer bounds on secrecy-capacity region.
Q: Do they match for the MIMO Gaussian case?
Q: Do Gaussian inputs achieve boundary points?
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 5
MIMO Gaussian BC - Some Literature
MIMO Gaussian BCs with Eavesdropping Receivers:
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 6
MIMO Gaussian BC - Some Literature
MIMO Gaussian BCs with Eavesdropping Receivers:
M0 M1 M2 Solution
— Private Private Weingarten-Steinberg-Shamai 2006
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 6
MIMO Gaussian BC - Some Literature
MIMO Gaussian BCs with Eavesdropping Receivers:
M0 M1 M2 Solution
— Private Private Weingarten-Steinberg-Shamai 2006
Public Private Private Geng-Nair 2014
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 6
MIMO Gaussian BC - Some Literature
MIMO Gaussian BCs with Eavesdropping Receivers:
M0 M1 M2 Solution
— Private Private Weingarten-Steinberg-Shamai 2006
Public Private Private Geng-Nair 2014
Public Secret — Ly-Liu-Liang 2010
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 6
MIMO Gaussian BC - Some Literature
MIMO Gaussian BCs with Eavesdropping Receivers:
M0 M1 M2 Solution
— Private Private Weingarten-Steinberg-Shamai 2006
Public Private Private Geng-Nair 2014
Public Secret — Ly-Liu-Liang 2010
— Secret Secret Liu-Liu-Poor-Shamai 2010
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 6
MIMO Gaussian BC - Some Literature
MIMO Gaussian BCs with Eavesdropping Receivers:
M0 M1 M2 Solution
— Private Private Weingarten-Steinberg-Shamai 2006
Public Private Private Geng-Nair 2014
Public Secret — Ly-Liu-Liang 2010
— Secret Secret Liu-Liu-Poor-Shamai 2010
Public Secret Secret Ekrem-Ulukus 2012
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 6
MIMO Gaussian BC - Some Literature
MIMO Gaussian BCs with Eavesdropping Receivers:
M0 M1 M2 Solution
— Private Private Weingarten-Steinberg-Shamai 2006
Public Private Private Geng-Nair 2014
Public Secret — Ly-Liu-Liang 2010
— Secret Secret Liu-Liu-Poor-Shamai 2010
Public Secret Secret Ekrem-Ulukus 2012
— Secret Private
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 6
MIMO Gaussian BC - Some Literature
MIMO Gaussian BCs with Eavesdropping Receivers:
M0 M1 M2 Solution
— Private Private Weingarten-Steinberg-Shamai 2006
Public Private Private Geng-Nair 2014
Public Secret — Ly-Liu-Liang 2010
— Secret Secret Liu-Liu-Poor-Shamai 2010
Public Secret Secret Ekrem-Ulukus 2012
— Secret Private
Public Secret Private
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 6
MIMO Gaussian BC - Some Literature
MIMO Gaussian BCs with Eavesdropping Receivers:
M0 M1 M2 Solution
— Private Private Weingarten-Steinberg-Shamai 2006
Public Private Private Geng-Nair 2014
Public Secret — Ly-Liu-Liang 2010
— Secret Secret Liu-Liu-Poor-Shamai 2010
Public Secret Secret Ekrem-Ulukus 2012
— Secret Private This work
Public Secret Private This work
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 6
MIMO Gaussian BC - Some Literature
MIMO Gaussian BCs with Eavesdropping Receivers:
M0 M1 M2 Solution
— Private Private Weingarten-Steinberg-Shamai 2006
Public Private Private Geng-Nair 2014
Public Secret — Ly-Liu-Liang 2010
— Secret Secret Liu-Liu-Poor-Shamai 2010
Public Secret Secret Ekrem-Ulukus 2012
— Secret Private This work
Public Secret Private This work
⋆ Solution for two last unsolved cases via Upper Concave Envelopes ⋆
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 6
MIMO Gaussian BC - Secrecy-Capacity ResultsWithout a Common Message: M1 - Confidential ; M2 - Private
Theorem (ZG 2016)
The secrecy-capacity region for a covariance constraint K � 0 is
CK =⋃
0�K⋆�K
(R1, R2) ∈ R2+
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
R1 ≤ 12 log
∣
∣
∣
∣
∣
I + G1K⋆G⊤1
I + G2K⋆G⊤2
∣
∣
∣
∣
∣
R2 ≤ 12 log
∣
∣
∣
∣
∣
I + G2KG⊤2
I + G2K⋆G⊤2
∣
∣
∣
∣
∣
.
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 7
MIMO Gaussian BC - Secrecy-Capacity ResultsWithout a Common Message: M1 - Confidential ; M2 - Private
Theorem (ZG 2016)
The secrecy-capacity region for a covariance constraint K � 0 is
CK =⋃
0�K⋆�K
(R1, R2) ∈ R2+
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
R1 ≤ 12
log
∣
∣
∣
∣
∣
I + G1K⋆G⊤1
I + G2K⋆G⊤2
∣
∣
∣
∣
∣
R2 ≤ 12 log
∣
∣
∣
∣
∣
I + G2KG⊤2
I + G2K⋆G⊤2
∣
∣
∣
∣
∣
.
R1 Bound - MIMO Gaussian WTC Secrecy-capacity:User 1 - Legitimate with input covariance K⋆ ; User 2- Eavesdropper.
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 7
MIMO Gaussian BC - Secrecy-Capacity ResultsWithout a Common Message: M1 - Confidential ; M2 - Private
Theorem (ZG 2016)
The secrecy-capacity region for a covariance constraint K � 0 is
CK =⋃
0�K⋆�K
(R1, R2) ∈ R2+
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
R1 ≤ 12 log
∣
∣
∣
∣
∣
I + G1K⋆G⊤1
I + G2K⋆G⊤2
∣
∣
∣
∣
∣
R2 ≤ 12
log
∣
∣
∣
∣
∣
I + G2KG⊤2
I + G2K⋆G⊤2
∣
∣
∣
∣
∣
.
R1 Bound - MIMO Gaussian WTC Secrecy-capacity:User 1 - Legitimate with input covariance K⋆ ; User 2- Eavesdropper.
R2 Bound - Capacity of MIMO Gaussian PTP to User 2:Input covariance K − K⋆ ; Noise covariance I + G2K⋆G⊤
2 .
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 7
MIMO Gaussian BC - Secrecy-Capacity ResultsM0 - Common ; M1 - Confidential ; M2 - Private
Theorem (ZG 2016)
The secrecy-capacity region for a covariance constraint K � 0 is
CK =⋃
0�K1,K2:
K1+K2�K
R0 ≤ minj=1,2
{
12 log
∣
∣
∣
∣
∣
I + GjKG⊤j
I + Gj(K1 + K2)G⊤j
∣
∣
∣
∣
∣
}
R1 ≤ 12 log
∣
∣
∣
∣
∣
I + G1K1G⊤1
I + G2K1G⊤2
∣
∣
∣
∣
∣
R2 ≤ 12 log
∣
∣
∣
∣
∣
I + G2(K1 + K2)G⊤2
I + G2K1G⊤2
∣
∣
∣
∣
∣
.
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 8
MIMO Gaussian BC - Secrecy-Capacity ResultsM0 - Common ; M1 - Confidential ; M2 - Private
Theorem (ZG 2016)
The secrecy-capacity region for a covariance constraint K � 0 is
CK =⋃
0�K1,K2:
K1+K2�K
R0 ≤ minj=1,2
{
12 log
∣
∣
∣
∣
∣
I + GjKG⊤j
I + Gj(K1 + K2)G⊤j
∣
∣
∣
∣
∣
}
R1 ≤ 12
log
∣
∣
∣
∣
∣
I + G1K1G⊤1
I + G2K1G⊤2
∣
∣
∣
∣
∣
R2 ≤ 12 log
∣
∣
∣
∣
∣
I + G2(K1 + K2)G⊤2
I + G2K1G⊤2
∣
∣
∣
∣
∣
.
R1 Bound: MIMO Gaussian WTC with input K1
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 8
MIMO Gaussian BC - Secrecy-Capacity ResultsM0 - Common ; M1 - Confidential ; M2 - Private
Theorem (ZG 2016)
The secrecy-capacity region for a covariance constraint K � 0 is
CK =⋃
0�K1,K2:
K1+K2�K
R0 ≤ minj=1,2
{
12 log
∣
∣
∣
∣
∣
I + GjKG⊤j
I + Gj(K1 + K2)G⊤j
∣
∣
∣
∣
∣
}
R1 ≤ 12 log
∣
∣
∣
∣
∣
I + G1K1G⊤1
I + G2K1G⊤2
∣
∣
∣
∣
∣
R2 ≤ 12
log
∣
∣
∣
∣
∣
I + G2(K1 + K2)G⊤2
I + G2K1G⊤2
∣
∣
∣
∣
∣
.
R1 Bound: MIMO Gaussian WTC with input K1
R2 Bound: MIMO Gaussian PTP with input K2 (K1 is noise).
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 8
MIMO Gaussian BC - Secrecy-Capacity ResultsM0 - Common ; M1 - Confidential ; M2 - Private
Theorem (ZG 2016)
The secrecy-capacity region for a covariance constraint K � 0 is
CK =⋃
0�K1,K2:
K1+K2�K
R0 ≤ minj=1,2
{
12
log
∣
∣
∣
∣
∣
I + GjKG⊤j
I + Gj(K1 + K2)G⊤j
∣
∣
∣
∣
∣
}
R1 ≤ 12 log
∣
∣
∣
∣
∣
I + G1K1G⊤1
I + G2K1G⊤2
∣
∣
∣
∣
∣
R2 ≤ 12 log
∣
∣
∣
∣
∣
I + G2(K1 + K2)G⊤2
I + G2K1G⊤2
∣
∣
∣
∣
∣
.
R1 Bound: MIMO Gaussian WTC with input K1
R2 Bound: MIMO Gaussian PTP with input K2 (K1 is noise).R0 Bound: MIMO Gaussian PTP with remaining covarianceK − (K1 + K2) (K1,K2 are noises).
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 8
Secrecy-Capacity without M0 - Proof Outline
Outer Bound:
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 9
Secrecy-Capacity without M0 - Proof Outline
Outer Bound: Fix a covariance constraint K� 0.
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 9
Secrecy-Capacity without M0 - Proof Outline
Outer Bound: Fix a covariance constraint K� 0.
1 [ZG-Kramer-Permuter 2016]
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 9
Secrecy-Capacity without M0 - Proof Outline
Outer Bound: Fix a covariance constraint K� 0.
1 [ZG-Kramer-Permuter 2016] =⇒ IK ⊆ CK ⊆ OK
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 9
Secrecy-Capacity without M0 - Proof Outline
Outer Bound: Fix a covariance constraint K� 0.
1 [ZG-Kramer-Permuter 2016] =⇒ IK ⊆ CK ⊆ OK
2 OK bounded & convex
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 9
Secrecy-Capacity without M0 - Proof Outline
Outer Bound: Fix a covariance constraint K� 0.
1 [ZG-Kramer-Permuter 2016] =⇒ IK ⊆ CK ⊆ OK
2 OK bounded & convex =⇒ Supporting hyperplanesmax
(R1,R2)∈OK
λ1R1 + λ2R2
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 9
Secrecy-Capacity without M0 - Proof Outline
Outer Bound: Fix a covariance constraint K� 0.
1 [ZG-Kramer-Permuter 2016] =⇒ IK ⊆ CK ⊆ OK
2 OK bounded & convex =⇒ Supporting hyperplanesmax
(R1,R2)∈OK
λ1R1 + λ2R2
3 max(R1,R2)∈OK
λ1R1 + λ2R2
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 9
Secrecy-Capacity without M0 - Proof Outline
Outer Bound: Fix a covariance constraint K� 0.
1 [ZG-Kramer-Permuter 2016] =⇒ IK ⊆ CK ⊆ OK
2 OK bounded & convex =⇒ Supporting hyperplanesmax
(R1,R2)∈OK
λ1R1 + λ2R2
3 max(R1,R2)∈OK
λ1R1 + λ2R2 ≤ Upper Concave Envelope
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 9
Secrecy-Capacity without M0 - Proof Outline
Outer Bound: Fix a covariance constraint K� 0.
1 [ZG-Kramer-Permuter 2016] =⇒ IK ⊆ CK ⊆ OK
2 OK bounded & convex =⇒ Supporting hyperplanesmax
(R1,R2)∈OK
λ1R1 + λ2R2
3 max(R1,R2)∈OK
λ1R1 + λ2R2 ≤ Upper Concave Envelope
4 UCE maximized by Gaussian inputs
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 9
Secrecy-Capacity without M0 - Proof Outline
Outer Bound: Fix a covariance constraint K� 0.
1 [ZG-Kramer-Permuter 2016] =⇒ IK ⊆ CK ⊆ OK
2 OK bounded & convex =⇒ Supporting hyperplanesmax
(R1,R2)∈OK
λ1R1 + λ2R2
3 max(R1,R2)∈OK
λ1R1 + λ2R2 ≤ Upper Concave Envelope
4 UCE maximized by Gaussian inputs
=⇒ OK ⊆ Region from Theorem
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 9
Secrecy-Capacity without M0 - Proof Outline
Achievability:
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 10
Secrecy-Capacity without M0 - Proof Outline
Achievability: Substituting Gaussian inputs into IK.
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 10
Secrecy-Capacity without M0 - Proof Outline
Achievability: Substituting Gaussian inputs into IK.
Dirty Paper Coding to cancel M2 signal at Receiver 1.
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 10
Secrecy-Capacity without M0 - Proof Outline
Achievability: Substituting Gaussian inputs into IK.
Dirty Paper Coding to cancel M2 signal at Receiver 1.
⋆ Other variant of DPC (cancel M1 at Rec. 2) not necessary.
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 10
Secrecy-Capacity without M0 - Proof Outline
Achievability: Substituting Gaussian inputs into IK.
Dirty Paper Coding to cancel M2 signal at Receiver 1.
⋆ Other variant of DPC (cancel M1 at Rec. 2) not necessary.
⇓
Region from Theorem ⊆ IK
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 10
Secrecy-Capacity without M0 - Proof Outline
Achievability: Substituting Gaussian inputs into IK.
Dirty Paper Coding to cancel M2 signal at Receiver 1.
⋆ Other variant of DPC (cancel M1 at Rec. 2) not necessary.
⇓
OK ⊆ Region from Theorem ⊆ IK
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 10
Secrecy-Capacity without M0 - Proof Outline
Achievability: Substituting Gaussian inputs into IK.
Dirty Paper Coding to cancel M2 signal at Receiver 1.
⋆ Other variant of DPC (cancel M1 at Rec. 2) not necessary.
⇓
OK ⊆ Region from Theorem ⊆ IK
⇓
OK = Region from Theorem = IK
�
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 10
Secrecy-Capacity without M0 - Visualization
Secrecy-Capacity Region under Average Power Constraint:
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 11
Secrecy-Capacity without M0 - Visualization
Secrecy-Capacity Region under Average Power Constraint:
Corollary:
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 11
Secrecy-Capacity without M0 - Visualization
Secrecy-Capacity Region under Average Power Constraint:
Corollary: 1n
n∑
i=1E
[
∣
∣
∣
∣X(i)∣
∣
∣
∣
2]
≤ P
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 11
Secrecy-Capacity without M0 - Visualization
Secrecy-Capacity Region under Average Power Constraint:
Corollary: 1n
n∑
i=1E
[
∣
∣
∣
∣X(i)∣
∣
∣
∣
2]
≤ P =⇒ CP =⋃
0�K:Tr(K)≤P
CK
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 11
Secrecy-Capacity without M0 - Visualization
Secrecy-Capacity Region under Average Power Constraint:
Corollary: 1n
n∑
i=1E
[
∣
∣
∣
∣X(i)∣
∣
∣
∣
2]
≤ P =⇒ CP =⋃
0�K:Tr(K)≤P
CK
0.6 0.8 10.2 0.4
1
2
3
4
5
0
0
M1 Confidential
Both Confidential
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 11
Summary
MIMO Gaussian BC - Common, Private and Conf. Messages:
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 12
Summary
MIMO Gaussian BC - Common, Private and Conf. Messages:
◮ Practical: Natural broadcasting scenario.
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 12
Summary
MIMO Gaussian BC - Common, Private and Conf. Messages:
◮ Practical: Natural broadcasting scenario.
◮ Theoretical: Last two unsolved cases.
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 12
Summary
MIMO Gaussian BC - Common, Private and Conf. Messages:
◮ Practical: Natural broadcasting scenario.
◮ Theoretical: Last two unsolved cases.
Secrecy-Sapacity Results:
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 12
Summary
MIMO Gaussian BC - Common, Private and Conf. Messages:
◮ Practical: Natural broadcasting scenario.
◮ Theoretical: Last two unsolved cases.
Secrecy-Sapacity Results:
◮ Characterization & Optimality of Gaussian inputs.
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 12
Summary
MIMO Gaussian BC - Common, Private and Conf. Messages:
◮ Practical: Natural broadcasting scenario.
◮ Theoretical: Last two unsolved cases.
Secrecy-Sapacity Results:
◮ Characterization & Optimality of Gaussian inputs.
◮ Proof via Upper Concave Envelopes & Dirty-Paper Coding.
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 12
Summary
MIMO Gaussian BC - Common, Private and Conf. Messages:
◮ Practical: Natural broadcasting scenario.
◮ Theoretical: Last two unsolved cases.
Secrecy-Sapacity Results:
◮ Characterization & Optimality of Gaussian inputs.
◮ Proof via Upper Concave Envelopes & Dirty-Paper Coding.
Thank You!
Z. Goldfeld Ben Gurion University
MIMO Gaussian BCs with Common, Private and Confidential Messages 12