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Square Root Law for Communication with Low Probability of Detection on AWGN Channels. Boulat A. Bash Dennis Goeckel Don Towsley. Introduction. Problem: communicate so that adversary’s detection capability is limited to tolerable level Low probability of detection (LPD) communication - PowerPoint PPT Presentation
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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
3Department of Computer Science 3
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
4Department of Computer Science 4
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
5Department of Computer Science 5
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
or ?
Thanks!
6Department of Computer Science 6
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
7Department of Computer Science 7
Outline
Introduction
Channel model
Hypothesis testing
Achievability
Converse
Conclusion
8Department of Computer Science 8
Channel Model
decode
transmit
Decide:
is
or something else?
i.i.d.
i.i.d.
9Department of Computer Science 9
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)
10Department of Computer Science 10
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
11Department of Computer Science 11
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
13Department of Computer Science 13
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.
14Department of Computer Science 14
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
15Department of Computer Science 15
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
16Department of Computer Science 16
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”
17Department of Computer Science 17
Outline
Introduction
Channel model
Hypothesis testing
Achievability
Converse
Conclusion
18Department of Computer Science 18
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
19Department of Computer Science 19
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
20Department of Computer Science 20
Converse: Analysis
Probability of false alarm To obtain set
Probability of a missed detection When ,
Alice transmitsAlice doesn’t transmit
21Department of Computer Science 21
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
22Department of Computer Science 22
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