joint work with miguel rodrigues, munnunjahan ara, vinay prabhu and joão xavier
DESCRIPTION
Filter Design with Secrecy Constraints. Hugo Reboredo Instituto de Telecomunicações Departamento de Ciências de Computadores Faculdade de Ciências da Universidade do Porto. Joint work with Miguel Rodrigues, Munnunjahan Ara, Vinay Prabhu and João Xavier. Outline. Motivation - PowerPoint PPT PresentationTRANSCRIPT
Joint work with Miguel Rodrigues, Munnunjahan Ara,
Vinay Prabhu and João Xavier
Filter Design with Secrecy Constraints
Hugo ReboredoInstituto de Telecomunicações
Departamento de Ciências de ComputadoresFaculdade de Ciências da Universidade do Porto
Outline
• Motivation
• Problem Statement
• Optimal Receive Filter
• Optimal Transmit Filter
• Algorithm
• Numerical Results
• Final Remarks
• Computational Security
– Alice sends a k-bit message M to Bob using an encryption scheme;
– Security schemes are based on assumptions of intractability of certain functions;
– Typically done at upper layers of the protocol stack
Alice
Eve
Bobk-bit message M k-bit decoded message
Mb
• Information-Theoretic Security
– strictest notion of security, no computability assumption
H(M|X)=H(M) or I(X;M)=0
– e.g. One-time pad
– Shannon, 1949: H(K)≥H(M)
– Suggests a physical-layer approach to security
key K
X X
X key K
Why? Some security notions…
Alice Bob
Eve
Xn
p(y|x)
p(z|y)
Yn
Zn
message M mesg. estimate Mb
mesg. estimate Me
RELIABILITY CRITERION:
SECURITY CRITERION:
Pr(M=Mb)→1
H(M|Zn)→H(M)[Wyner’75]
Why? Wiretap Channel
Transmission rate
H(M)
CS CM
D
equivocation rate
Alice Bob
Eve
X Y
Z
NM
NW
Secrecy Capacity: Cs=CM-CW=log2(1+P/NM)log2(1+P/NW)
[Leung and Hellman’78]
Why? Gaussian Wiretap Channel
Positive Secrecy Capacity -> degraded scenario
Filter design with secrecy constraints
s.t.
Optimal Receive FilterWiener Filter
Zero Forcing Filter
HT HM HRM
HE HRE
Alice
Bob
Eve
NM
YE
YMX
Optimal Transmit FilterWeiner filters
s.t. s.t.
s.t. s.t.
GEVD
Optimal Transmit FilterWeiner filters
HT HM HRM
HE HRE
Alice
Bob
Eve
NM
YE
YMX
Optimal Transmit FilterWeiner filters
HT HM HRM
HE HRE
Alice
Bob
Eve
NM
YE
YMX
Optimal Transmit FilterZF filters
s.t.s.t.
HT HM HRM
HE HRE
Alice
Bob
Eve
NM
YE
YMX
Optimal Transmit FilterZF filters
AlgorithmWiener Filters
:
AlgorithmZF Filters
:
Numerical ResultsWiener Filters
Gaussian MIMO 2x2 channel
Numerical ResultsWiener Filters
Gaussian MIMO 2x2 channel
Numerical ResultsZF Filters
Gaussian MIMO 2x2 channel
Main and eavesdropper MSE vs. secrecy constraint gamma and input power vs. secrecy constraint – Degraded Scenario
Numerical ResultsZF Filters
Gaussian MIMO 2x2 channel
Main and eavesdropper MSE vs. input power – gamma = 1 Degraded Scenario
Numerical ResultsZF Filters
Gaussian MIMO 2x2 channel
Main and eavesdropper MSE vs. secrecy constraint gamma and input power vs. secrecy constraint – Non-degraded Scenario
Final Remarks
Wiener Filters at the receiver:
• Optimization Problem
• Optimal Receive Filter
• Optimal Transmit Filter
• GEVD does not affect power
• Suitable Algorithm
• Minimum gamma for finite power
Final Remarks
ZF Filters at the receivers:
• Address a more general case•Non-degraded scenario
• Introducing a power constraint
• Optimal Transmit Filter
• Suitable Algorithm
• Straightforward Algorithm
• Need to solve a nonlinear equation
