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

Filter Design with Secrecy Constraints

Hugo Reboredohugoreboredo@dcc.fc.up.pt

Thank You