1. 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 Goeckel Don Towsley

2. 2 Department 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

3. 3 Department 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

4. 4 Department 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

5. 5 Department 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!

6. 6 Department 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

7. 7 Department of Computer Science 7 Outline  Introduction  Channel model  Hypothesis testing  Achievability  Converse  Conclusion

8. 8 Department of Computer Science 8 Channel Model decode transmit Decide: is or something else? i.i.d.

9. 9 Department 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 H 0 : observations are noise Alternate H 1 : Alice sending covert signals 1-   1-  Willie’s test decision Noise (H 0 )Data (H 1 ) is quiet (H 0 ) x-mitting (H 1 ) Alice P(false alarm) P(miss)P(detection)

10. 10 Department 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

11. 11 Department 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 proof 1.Construction 2.Analysis of Willie’s detector 3.Analysis of Bob’s decoding error  -- 1 1 0 Willie’s Detector ROC 

12. 12 Department of Computer Science 12 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 proof 1.Construction 2.Analysis of Willie’s detector 3.Analysis of Bob’s decoding error

13. 13 Department 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 00000··· W1W1 00001 W2W2 11111 W2MW2M 2M2M M-bit messages x 11 x 12 x 13 x 1n ··· c(W 1 ) x 21 x 22 x 23 x 2n ··· c(W 2 ) x2M1x2M1 ··· c(W 2 M ) x2M2x2M2 x2M3x2M3 x2Mnx2Mn n-symbol codewords Each symbol i.i.d.

14. 14 Department 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 variation or ½L 1 norm Relative entropy Taylor series expansion

15. 15 Department 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 if is not here

16. 16 Department 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”

17. 17 Department of Computer Science 17 Outline  Introduction  Channel model  Hypothesis testing  Achievability  Converse  Conclusion

18. 18 Department 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 detector 2.Bob cannot decode codewords that carry bits with average symbol power with arbitrary low error

19. 19 Department 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 H 0 if Reject H 0 if

20. 20 Department of Computer Science 20 Converse: Analysis  Probability of false alarm  To obtain set  Probability of a missed detection  When, Alice transmitsAlice doesn’t transmit

21. 21 Department 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

22. 22 Department 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

23. 23 Department of Computer Science 23 Thank you! boulat@cs.umass.edu