Square Root Law for Communication with Low Probability of Detection on AWGN Channels

University of Massachusetts Amherst · Department of Computer Science

Square Root Law for Communication with Low Probability of Detection on

AWGN Channels

Boulat A. Bash

Dennis GoeckelDon Towsley

2Department of Computer Science 2

Introduction

Problem: communicate so that adversary’s detection capability is limited to tolerable level• Low probability of detection (LPD) communication• As opposed to protecting message content (encryption)

Why? Lots of applications…• Communication looks suspicious• “Camouflage” military operations• etc…

Fundamental limits of LPD communication


Scenario

Alice uses radio to covertly communicate with Bob• They share a secret (codebook)

Willie attempts to detect if Alice is talking to Bob• Willie is passive, doesn’t actively jam Alice’s channel

Willie’s problem: detect Alice Alice’s problem: limit Willie’s detection schemes Bob’s problem: decode Alice’s message


Scenario





Scenario




or ?

Thanks!


Main Result: The Square Root Law

Given that Alice has to tolerate some risk of being detected, how many bits can Alice covertly send to Bob?

• Not many: bits per n channel uses

• If she sends bits in n channel uses, either Willie detects her, or Bob is subject to decoding errors

• Intuition: Alice has to “softly whisper” to reduce detection, which hurts how much she can send


Outline

Introduction

Channel model

Hypothesis testing

Achievability

Converse

Conclusion


Channel Model

decode

transmit

Decide:

is

or something else?

i.i.d.

i.i.d.


Statistical Hypothesis Testing

Willie has n observations of Alice’s channel and attempts to classify them as noise or covert data• Null hypothesis H0: observations are noise

• Alternate H1: Alice sending covert signals

1-

1-

Willie’s test decisionNoise (H0) Data (H1)

is quiet (H0)

x-mitting (H1)

Alice

P(false alarm)

P(miss) P(detection)


Willie’s Detector

Willie picks (confidence in his detector)• Willie uses a detector that maximizes

Alice can lower-bound• Picks appropriate distribution for covert symbols

1

1

0

Detector ROC

and


Achievability

Alice can send bits in n channel uses to Bob while maintaining at Willie’s detector for any• Willie’s channel to Alice

Three step proof1. Construction2. Analysis of Willie’s detector3. Analysis of Bob’s decoding error

-

1

1

0

Willie’s Detector ROC


Achievability: Construction

Random codebook with average symbol power

Codebook revealed to Bob, but not to Willie Willie knows how codebook is constructed, as well

as n and • System obeys Kerckhoffs’s Law: all security is in the key

used to construct codebook

0 0 0 0 0···W10 0 0 0 1···W2

1 1 1 1 1···W2M

⁞2M

M-bit messages

x11 x12 x13 x1n··· c(W1)x21 x22 x23 x2n··· c(W2 )

x2M1 ··· c(W2M )x2M2x2M3 x2Mn

⁞

n-symbol codewords

Each symbol i.i.d.


Achievability: Analysis of Willie’s Detector

Joint distributions for Willie’s n observations:• when Alice quiet, since AWGN is i.i.d• . when Alice transmitting, since Willie

does not know Alice and Bob’s codebook

Bounding Willie’s detection:Total variationor ½L1 norm

Relativeentropy

Taylor seriesexpansion


Achievability: Analysis of Bob’s Decoding Error

Bob uses ML decoding to decode from

Therefore, Bob gets bits per n channel uses

another codeword is closer

Error ifis not here


Relationship with Steganography

Steganography: embed messages into covertext• Bob and Willie then see noiseless stegotext

Square root law in steganography • Ker, Fridrich, et al

• symbols can safely be modified in covertext of size n• Similarity due to hypothesis testing math

• bits can be embedded• Due to noiseless “channel”


Outline

Introduction

Channel model

Hypothesis testing

Achievability

Converse

Conclusion


Converse

When Alice tries to transmit bits in n channel uses, using arbitrary codebook, either• Detected by Willie with arbitrarily low error probability• Bob’s decoding error probability bounded away from zero

Arbitrary codebook with codewords of length n• Willie oblivious to design of Alice’s system

Two step proof: 1. Willie detects arbitrary codewords with average symbol

power using a simple power detector2. Bob cannot decode codewords that carry bits with

average symbol power with arbitrary low error


Converse: Willie’s Hypothesis Test

Willie collects n independent readings of his channel to Alice:

Interested in hypothesis test:

Test statistic: average received symbol power

Test implementation: pick some threshold t• Accept H0 if

• Reject H0 if


Converse: Analysis

Probability of false alarm To obtain set

Probability of a missed detection When ,

Alice transmitsAlice doesn’t transmit


Converse: Alice Using Low Power Codewords

Suppose Alice uses positive fraction of codewords with average symbol power• Then Willie can’t drive detection errors to zero• Analyze Bob’s decoding error:

Converse of Shannon Theorem• By sending bits in n channel

uses rate at too low power

and, therefore, Bob’s decoding error

Alice’s codebook


Conclusion

We proved a square root law for LPD channel

Future work

• Key efficiency• Can show that length K of Alice and Bob’s shared

secret • Open problem: can it be linear ?

• Covert networks


Thank you!

[email protected]

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Square Root Law for Communication with Low Probability of Detection on AWGN Channels

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