MIMO Gaussian Broadcast Channels with Common, Private …

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.



Motivation


Susceptibility of wireless communication to eavesdropping.



Motivation



Eavesdroppers are not always a malicious entity:



Motivation




◮ Legitimate recipient of some messages.



Motivation





◮ Eavesdropper of other.



Motivation





◮ Eavesdropper of other.

Modern BC scenario - Common, Private and Confidential

messages.



Motivation - Banking Site




Common

Common - Advertisement.




CommonPrivate


Private - On-demand Public info (programs, reports, forecasts).




CommonPrivate

Confidential


Private - On-demand Public info (programs, reports, forecasts).

Confidential - Online banking (access account, transactions).Z. Goldfeld Ben Gurion University


MIMO Gaussian BC - Problem Setup

X

G1

G2

Z1

Z2

Y1

Y2

(M0,M1,M2)

(

M(1)0

,M1

)

(

M(2)0

,M2

)

M1




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 .




X

G1

G2

Z1

Z2

Y1

Y2

(M0,M1,M2)

(

M(1)0

,M1

)

(

M(2)0

,M2

)

M1


Dimensions: X, Y1, Y2, Z1, Z2 ∈ Rt ; G1, G2 ∈ R

t×t.




X

G1

G2

Z1

Z2

Y1

Y2

(M0,M1,M2)

(

M(1)0

,M1

)

(

M(2)0

,M2

)

M1



t×t.

Noise Processes: i.i.d. samples of Zj ∼ N (0, It), j = 1, 2.




X

G1

G2

Z1

Z2

Y1

Y2

(M0,M1,M2)

(

M(1)0

,M1

)

(

M(2)0

,M2

)

M1



t×t.


Input Covariance Constraint: 1n

∑ni=1 E

[

X(i)X⊤(i)]

� K.




X

G1

G2

Z1

Z2

Y1

Y2

(M0,M1,M2)

(

M(1)0

,M1

)

(

M(2)0

,M2

)

M1



t×t.


Input Covariance Constraint: 1n

∑ni=1 E

[

X(i)X⊤(i)]

� K.

Security Criterion: 1n

I(M1; Yn2 ) −−−→

n→∞0.



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.




X

G1

G2

Z1

Z2

Y1

Y2

(M0,M1,M2)

(

M(1)0

,M1

)

(

M(2)0

,M2

)

M1


Q: Do they match for the MIMO Gaussian case?




X

G1

G2

Z1

Z2

Y1

Y2

(M0,M1,M2)

(

M(1)0

,M1

)

(

M(2)0

,M2

)

M1


Q: Do they match for the MIMO Gaussian case?

Q: Do Gaussian inputs achieve boundary points?



MIMO Gaussian BC - Some Literature

MIMO Gaussian BCs with Eavesdropping Receivers:





M0 M1 M2 Solution

— Private Private Weingarten-Steinberg-Shamai 2006





M0 M1 M2 Solution


Public Private Private Geng-Nair 2014





M0 M1 M2 Solution



Public Secret — Ly-Liu-Liang 2010





M0 M1 M2 Solution




— Secret Secret Liu-Liu-Poor-Shamai 2010





M0 M1 M2 Solution





Public Secret Secret Ekrem-Ulukus 2012





M0 M1 M2 Solution






— Secret Private





M0 M1 M2 Solution






— Secret Private

Public Secret Private





M0 M1 M2 Solution






— Secret Private This work

Public Secret Private This work





M0 M1 M2 Solution






— Secret Private This work

Public Secret Private This work

⋆ Solution for two last unsolved cases via Upper Concave Envelopes ⋆



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

∣

∣

∣

∣

∣

.




Theorem (ZG 2016)


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.




Theorem (ZG 2016)


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 .



MIMO Gaussian BC - Secrecy-Capacity ResultsM0 - Common ; M1 - Confidential ; M2 - Private

Theorem (ZG 2016)


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

∣

∣

∣

∣

∣

.




Theorem (ZG 2016)


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




Theorem (ZG 2016)


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

∣

∣

∣

∣

∣

.


R2 Bound: MIMO Gaussian PTP with input K2 (K1 is noise).




Theorem (ZG 2016)


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

∣

∣

∣

∣

∣

.


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).



Secrecy-Capacity without M0 - Proof Outline

Outer Bound:




Outer Bound: Fix a covariance constraint K� 0.





1 [ZG-Kramer-Permuter 2016]





1 [ZG-Kramer-Permuter 2016] =⇒ IK ⊆ CK ⊆ OK






2 OK bounded & convex






2 OK bounded & convex =⇒ Supporting hyperplanesmax

(R1,R2)∈OK

λ1R1 + λ2R2







(R1,R2)∈OK

λ1R1 + λ2R2

3 max(R1,R2)∈OK

λ1R1 + λ2R2







(R1,R2)∈OK

λ1R1 + λ2R2

3 max(R1,R2)∈OK

λ1R1 + λ2R2 ≤ Upper Concave Envelope







(R1,R2)∈OK

λ1R1 + λ2R2

3 max(R1,R2)∈OK


4 UCE maximized by Gaussian inputs







(R1,R2)∈OK

λ1R1 + λ2R2

3 max(R1,R2)∈OK


4 UCE maximized by Gaussian inputs

=⇒ OK ⊆ Region from Theorem




Achievability:




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







⇓

OK ⊆ Region from Theorem ⊆ IK







⇓

OK ⊆ Region from Theorem ⊆ IK

⇓

OK = Region from Theorem = IK

�



Secrecy-Capacity without M0 - Visualization

Secrecy-Capacity Region under Average Power Constraint:





Corollary:





Corollary: 1n

n∑

i=1E

[

∣

∣

∣

∣X(i)∣

∣

∣

∣

2]

≤ P





Corollary: 1n

n∑

i=1E

[

∣

∣

∣

∣X(i)∣

∣

∣

∣

2]

≤ P =⇒ CP =⋃

0�K:Tr(K)≤P

CK





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



Summary

MIMO Gaussian BC - Common, Private and Conf. Messages:



Summary


◮ Practical: Natural broadcasting scenario.



Summary



◮ Theoretical: Last two unsolved cases.



Summary




Secrecy-Sapacity Results:



Summary





◮ Characterization & Optimality of Gaussian inputs.



Summary






◮ Proof via Upper Concave Envelopes & Dirty-Paper Coding.



Summary






◮ Proof via Upper Concave Envelopes & Dirty-Paper Coding.

Thank You!



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