Secrecy Codes for Wireless Control...

Secrecy Codes for Wireless Control Systems

Anastasios Tsiamis, Konstantinos Gatsis, George J. Pappas

University of Pennsylvania

July 13, 2018

Secrecy Issues in the Internet of Things

I Interconnected sensors and actuators.

I Communication over wireless networks.

The wireless medium may be compromised.

I Broadcast nature.

I Eavesdroppers may intercept sensitive data about the physicalsystem.

Eavesdropping attacks in Dynamical Systems

Dynamical System

Sensor

ChannelDEC

Eavesdropper

GoalI Protect dynamical state information, e.g. position, velocity.

I Design secrecy codes.

ChallengesI Tradeoff between security and code complexity.

ApproachI Can we exploit model knowledge/physics for secrecy?

Dynamical System

Sensor

ChannelDEC

Eavesdropper

Dynamical System

Sensor

ChannelDEC

Eavesdropper

Dynamical System

Sensor

ChannelDEC

Eavesdropper

Previous work

Encryption

I Generic tool applied in practice. J. Katz and Y. Lindell. 2014

I Adversary should have bounded computational capabilities.

I Communication/computation overheads.

Physical layer security

I Information theoretic tools. A. D. Wyner. 1975, M. Wiese et al. 2016

I Provable guarantees.

I Requires eavesdropper’s channel model.

Our approach: State-Secrecy Codes

I Exploit the system dynamics.

I New tradeoff between code complexity and security.

I Simple and fast.

I Strong security guarantees about the current state.

Related work

I Codes for several types of linear systems

A. Tsiamis et al. CDC 2017, ACC 2018, CDC 2018

I Non-coding approach

A. S. Leong et al. IFAC 2017, CDC 2017

A. Tsiamis et al. IFAC 2017

Previous work

Encryption

I Simple and fast.

Related work

Previous work

Encryption

I Simple and fast.

Related work

Previous work

I Simple and fast.

Related work

System

Sensor &Encoder

Eavesdropper

x̂u,k

x̂e,k

Linear System

I state xk; e.g. position and velocity at time k

xk+1 = Axk + wk+1, wk : Gaussian process noise

Sensor and encoder

I zk encoded version of state xk.

System

Sensor &Encoder

Eavesdropper

x̂u,k

x̂e,k

Linear System

I state xk; e.g. position and velocity at time k

xk+1 = Axk + wk+1, wk : Gaussian process noise

Sensor and encoder

I zk encoded version of state xk.

System

Sensor &Encoder

Eavesdropper

x̂u,k

x̂e,k

ChannelI Packet drop channels:

I Message either received intact or dropped.

I γu,k = 1: message zk received.

I γu,k = 0: message zk dropped.

I Acknowledgment signals available.

System

Sensor &Encoder

Eavesdropper

x̂u,k

x̂e,k

Decoders: Minimum Mean Square Error Estimators

I Information sets:

Iu,k = {Received messages zk}Ie,k = {Intercepted messages zk,User’s outcomes γu,k}

I Estimation:

x̂u,k minimizes E(‖xk − x̂u,k‖2 |Iu,k)

Pu,k user mmse covariance given Iu,k

System

Sensor &Encoder

Eavesdropper

x̂u,k

x̂e,k

Decoders: Minimum Mean Square Error Estimators

I Information sets:

Iu,k = {Received messages zk}Ie,k = {Intercepted messages zk,User’s outcomes γu,k}

I Estimation:

x̂u,k minimizes E(‖xk − x̂u,k‖2 |Iu,k)

Pu,k user mmse covariance given Iu,k

Problem: Secrecy

Secrecy

Find a coding scheme such that:

I Eavesdropper’s minimum mean square error (mmse) forthe current state is maximum asymptotically.

I User’s mmse is optimal.

Assumptions

I Public A.

I Passive eavesdropper.

I Eavesdropper knows model, coding scheme, acknowledgments.

State-Secrecy Code

Coding Scheme

zk = xk − Lk−tkxtk

tk: the most recent message received at the user

I Matrix L depends on the dynamical system’s model.

I Choice of L: makes zk less correlated to xk for the eavesdropper.

I ACKs: user and sensor agree on tk.

I Fast and simple.

I Can be used along with encryption.

Example

Encoder User Eavesdropper

x1 − Lx0 −x2 − L2x0 −x3 − Lx2 − x3 − Lx2x4 − L2x2 x4 − L2x2

I User decodes by adding.

I Eavesdropper decodes initially.

I Eavesdropper cannot decode for k > 2.

Example

x1 − Lx0 −x2 − L2x0 −x3 − Lx2 − x3 − Lx2x4 − L2x2 x4 − L2x2

Example

x1 − Lx0

−x2 − L2x0 −x3 − Lx2 − x3 − Lx2x4 − L2x2 x4 − L2x2

Example

x1 − Lx0 −

x2 − L2x0 −x3 − Lx2 − x3 − Lx2x4 − L2x2 x4 − L2x2

Example

x1 − Lx0 −x2 − L2x0

−x3 − Lx2 − x3 − Lx2x4 − L2x2 x4 − L2x2

Example

x1 − Lx0 −x2 − L2x0 x2 − L2x0

−x3 − Lx2 − x3 − Lx2x4 − L2x2 x4 − L2x2

Example

x1 − Lx0 −x2 − L2x0 x2 − L2x0

x3 − Lx2

− x3 − Lx2x4 − L2x2 x4 − L2x2

Example

x1 − Lx0 −x2 − L2x0 x2 − L2x0

x3 − Lx2 −

x3 − Lx2x4 − L2x2 x4 − L2x2

Example

x1 − Lx0 −x2 − L2x0 x2 − L2x0

x3 − Lx2 −

x3 − Lx2

x4 − L2x2

Example

x1 − Lx0 −x2 − L2x0 x2 − L2x0

x3 − Lx2 −

x3 − Lx2

x4 − L2x2 x4 − L2x2

x4 − L2x2

Example

x0 x0 x0

x1 − Lx0 − x1 − Lx0x2 − L2x0 x2 − L2x0 −x3 − Lx2 − x3 − Lx2x4 − L2x2 x4 − L2x2 x4 − L2x2

Example

x0 x0 x0

x1 − Lx0 − x1 − Lx0x2 − L2x0 x2 − L2x0 −x3 − Lx2 − x3 − Lx2x4 − L2x2 x4 − L2x2 x4 − L2x2

Example

x0 x0 x0

x1 − Lx0 − x1 − Lx0x2 − L2x0 x2 −x3 − Lx2 − x3 − Lx2x4 − L2x2 x4 − L2x2 x4 − L2x2

Example

x0 x0 x0

x1 − Lx0 − x1 − Lx0x2 − L2x0 x2 −x3 − Lx2 − x3 − Lx2x4 − L2x2 x4 x4 − L2x2

Example

x0 x0 x0

x1 − Lx0 − x1

x2 − L2x0 x2 −x3 − Lx2 − x3 − Lx2x4 − L2x2 x4 x4 − L2x2

Example

x0 x0 x0

x1 − Lx0 − x1

x2 − L2x0 x2 −x3 − Lx2 − x3 − Lx2, x3 =?

x4 − L2x2 x4 x4 − L2x2, x4 =?

Example

x0 x0 x0

x1 − Lx0 − x1

x2 − L2x0 x2 −x3 − Lx2 − x3 − Lx2, x3 =?

x4 − L2x2 x4 x4 − L2x2, x4 =?

Critical eventWhen the user receives while the eavesdropper misses:

γu,k = 1, γe,k = 0, for some k

Damages the eavesdropper (here for k = 2).

Secrecy guarantees

Theorem 1: Secrecy

If the critical event occurs at time k0 secrecy is achieved:

I Eavesdropper’s mmse is asymptotically maximum:

Pe,k → maximum value

I User’s mmse remains optimal:

Pu,k = 0, when user receives. (γu,k = 1)

I One occurrence of the critical event is sufficient.(user receives and eavesdropper fails to intercept at some time k.)

I Condition holds for most packet dropping channels.

Simulation scenario

I Linearized dynamics, planar quadrotor

I Hovers around target point (0, 0, 0)

I State is position and velocity

Simulation scenario

0 50 100 150 200 250 300 350 400

UserEavesdropperTrue state

I Height estimation

I Critical event occurs at time k = 47

I Eavesdropper gets confused about the state; trivial estimatexe = 0

I User’s estimation remains accurate

Simulation scenario

0 50 100 150 200 250 300 350 400

I Height estimation

Simulation scenario

0 50 100 150 200 250 300 350 400

I Height estimation

Simulation scenario

0 50 100 150 200 250 300 350 400

I Height estimation

Conclusion

Summary

I We can exploit the dynamical system model/physics for secrecy.

I Codes specialized for dynamical systems.

I Simple and fast, strong guarantees for the current state.

I Secrecy guarantees for several types of linear systems.

Future Work

I How to defend against active eavesdroppers?

I How can we combine codes with encryption efficiently?

I How can we apply the codes in closed-loop control systems?

I How can we protect past states?

Conclusion

Summary

I We can exploit the dynamical system model/physics for secrecy.

I Codes specialized for dynamical systems.

I Simple and fast, strong guarantees for the current state.

I Secrecy guarantees for several types of linear systems.

Future Work

I How to defend against active eavesdroppers?

I How can we combine codes with encryption efficiently?

I How can we apply the codes in closed-loop control systems?

I How can we protect past states?

Thank you!

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