Joint work with Miguel Rodrigues, Munnunjahan Ara, Vinay Prabhu and João Xavier Filter Design with Secrecy Constraints Hugo Reboredo Instituto de Telecomunicações Departamento de Ciências de Computadores Faculdade 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 Bob k-bit message M k-bit decoded message M b 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 XX X Why? Some security notions…

Alice Bob Eve XnXn p(y|x) p(z|y) YnYn ZnZn message M mesg. estimate M b mesg. estimate M e R ELIABILITY C RITERION : S ECURITY C RITERION : Pr(M=M b ) → 1 H(M|Z n ) → H(M) [Wyner’75] Why? Wiretap Channel Transmission rate H(M) CSCS CMCM D equivocation rate

Alice Bob Eve X Y Z NMNM NWNW Secrecy Capacity: C s =C M -C W =log 2 (1+P/N M )log 2 (1+P/N W ) [Leung and Hellman’78] Why? Gaussian Wiretap Channel Positive Secrecy Capacity -> degraded scenario

Filter design with secrecy constraints s.t.

Optimal Receive Filter Wiener Filter Zero Forcing Filter

HTHT HMHM H RM HEHE H RE Alic e Bob Eve NMNM YEYE YMYM X Optimal Transmit Filter Weiner filters s.t.

GEVD Optimal Transmit Filter Weiner filters

HTHT HMHM H RM HEHE H RE Alic e Bob Eve NMNM YEYE YMYM X Optimal Transmit Filter Weiner filters

HTHT HMHM H RM HEHE H RE Alic e Bob Eve NMNM YEYE YMYM X Optimal Transmit Filter ZF filters s.t.

HTHT HMHM H RM HEHE H RE Alic e Bob Eve NMNM YEYE YMYM X Optimal Transmit Filter ZF filters

Algorithm Wiener Filters :

Algorithm ZF Filters :

Numerical Results Wiener Filters Gaussian MIMO 2x2 channel

Numerical Results Wiener Filters Gaussian MIMO 2x2 channel

Numerical Results ZF Filters Gaussian MIMO 2x2 channel Main and eavesdropper MSE vs. secrecy constraint gamma and input power vs. secrecy constraint – Degraded Scenario

Numerical Results ZF Filters Gaussian MIMO 2x2 channel Main and eavesdropper MSE vs. input power – gamma = 1 Degraded Scenario

Numerical Results ZF 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 Reboredo Thank You