Reliable Deniable Communication: Hiding Messages in Noise
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Reliable Deniable Communication: Hiding Messages in Noise
Mayank Bakshi Mahdi Jafari Siavoshani
ME
Sidharth Jaggi
The Chinese University of Hong Kong
The Institute of Network Coding
Pak Hou (Howard) Che
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Alice
Reliability
Bob
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Willie(the Warden)
Reliability
Deniability
AliceBob
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Willie-sky
Reliability
Deniability
AliceBob
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M
T
t
๏ฟฝโ๏ฟฝ
Aliceโs Encoder
๐=2๐ (โ๐)
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M
T
Message Trans. Status
BSC(pb) ๏ฟฝฬ๏ฟฝ=๐ท๐๐ (๏ฟฝโ๏ฟฝ๐)๏ฟฝโ๏ฟฝ๐๏ฟฝโ๏ฟฝ
Aliceโs EncoderBobโs Decoder
๐=2๐ (โ๐)
๏ฟฝฬ๏ฟฝ
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M
T
Message Trans. Status
BSC(pb) ๏ฟฝฬ๏ฟฝ=๐ท๐๐ (๏ฟฝโ๏ฟฝ๐)๏ฟฝโ๏ฟฝ๐๏ฟฝโ๏ฟฝ
Aliceโs EncoderBobโs Decoder
BSC(pw)
๏ฟฝฬ๏ฟฝ=๐ท๐๐ (๏ฟฝโ๏ฟฝ๐ค)
๏ฟฝโ๏ฟฝ๐ค
๐=2๐ (โ๐)
Willieโs (Best) Estimator
๏ฟฝฬ๏ฟฝ
๏ฟฝฬ๏ฟฝ
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Bash, Goeckel & Towsley [1]Shared secret
[1] B. A. Bash, D. Goeckel and D. Towsley, โSquare root law for communication with low probability of detection on AWGN channels,โ in Proceedings of the IEEE International Symposium on Information Theory (ISIT), 2012, pp. 448โ452.
โฌ
O n .log(n)( ) bits
AWGN channels
But capacity only
โฌ
O n( ) bits!
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This workNo shared secret
BSC(pb)
BSC(pw)
pb < pw
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Wicked Willie(s) Base-station Bob
Aerial Alice
Directional antenna
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Steganography: Other work
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Steganography: Other work
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Other work: โCommonโ modelShared secret key
Capacity O(n) message bitsInformation-theoretically tight characterization(Gelโfand-Pinsker/Dirty paper coding)
O(n.log(n)) bits (not optimized)
[2] Y. Wang and P. Moulin, "Perfectly Secure Steganography: Capacity, Error Exponents, and Code Constructions," IEEE Trans. on Information Theory, special issue on Information Theoretic Security, June 2008
Stegotext(covertext,message,key)
Message,Covertext
No noise
d(stegotext,covertext) โsmallโ
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Other work: Square-root โlawโ(โempiricalโ)
โขโSteganographic capacity is a loosely-defined concept, indicating the size of payload whichmay securely be embedded in a cover object using a particular embedding method. What constitutes โsecureโ embedding is a matter for debate, but we will argue that capacity should grow only as the square root of the cover size under a wide range of definitions of security.โ [3]
โขโThanks to the Central Limit Theorem, the more covertext we give the warden, the better he will be able to estimate its statistics, and so the smaller the rate at which [the steganographer] will be able to tweak bits safely.โ [4]
[3] A. Ker, T. Pevny`, J. Kodovsky`, and J. Fridrich, โThe square root law of steganographic capacity,โ in Proceedings of the 10th ACM workshop on Multimedia and security. ACM, 2008, pp. 107โ116.[4] R. Anderson, โStretching the limits of steganography,โ in Information Hiding, 1996, pp. 39โ48.
โขโ[T]he reference to the Central Limit Theorem... suggests that a square root relationship should be considered. โ [3]
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M
T
Message Trans. Status
BSC(pb) ๏ฟฝฬ๏ฟฝ=๐ท๐๐ (๏ฟฝโ๏ฟฝ๐)๏ฟฝโ๏ฟฝ๐๏ฟฝโ๏ฟฝ
Aliceโs EncoderBobโs Decoder
BSC(pw)
๏ฟฝฬ๏ฟฝ=๐ท๐๐ (๏ฟฝโ๏ฟฝ๐ค)
๏ฟฝโ๏ฟฝ๐ค
๐=2๐ (โ๐)
Willieโs (Best) Estimator
๏ฟฝฬ๏ฟฝ
๏ฟฝฬ๏ฟฝ
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Hypothesis Testing Willieโs Estimate
Aliceโs Transmission
Status
๐ผ=Pr ( ๏ฟฝฬ๏ฟฝ=1|๐=0 ) , ๐ฝ=Pr ( ๏ฟฝฬ๏ฟฝ=0|๐=1 )
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Hypothesis Testing Willieโs Estimate
Aliceโs Transmission
Status
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Hypothesis Testing Willieโs Estimate
Aliceโs Transmission
Status
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Hypothesis Testing Willieโs Estimate
Aliceโs Transmission
Status
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Intuition
๐=0 , ๐ฒ๐ค=๏ฟฝโ๏ฟฝ๐ค Binomial(๐ ,๐๐ค)
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Intuition
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Theorem 1 (Wt(c.w.))(high deniability => low weight codewords)
Too many codewords with weight โmuch โ greater than๐ โ๐ , h๐ก ๐๐ h๐ก ๐๐ ๐ฆ๐ ๐ก๐๐๐๐ โnot veryโ deniable
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Theorems 2 & 3(Converse & achievability for reliable & deniable comm.)
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Theorems 2 & 3
๐๐
๐๐ค
0 1/2
1/2
pb>pw
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Theorems 2 & 3
๐๐
๐๐ค
0 1/2
1/2
๐=0(Symmetrizability)
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Theorems 2 & 3
๐๐
๐๐ค
0 1/2
1/2pw=1/2
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Theorems 2 & 3
๐๐
๐๐ค
0 1/2
1/2
โฌ
N โ 2(1โH (pb ))n
(BSC(pb))
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Theorems 2 & 3
๐๐
๐๐ค
0 1/2
1/2
pb=0
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Theorems 2 & 3
๐๐
๐๐ค
0 1/2
1/2
๐=2๐ (โ๐ log๐) ,( ๐โ๐)=2๐ (โ๐ log๐)
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Theorems 2 & 3
๐๐
๐๐ค
0 1/2
1/2
pw>pb
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Theorems 2 & 3
๐๐
๐๐ค
0 1/2
1/2๐=2๐ (โ๐)
โStandardโ IT inequalities+
Wt(โmost codewordsโ)<โn(Thm 1)
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Theorems 2 & 3
๐๐
๐๐ค
0 1/2
1/2
Main thm:
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๐ค๐ก๐ป (๐๐ค )
0 n
logarithm of# codewords
log ( ๐๐/2)โ๐
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๐ค๐ก๐ป (๐ฒ๐ค)0 n๐๐ค๐+๐ (โ๐)๐๐ค๐
log(# codewords)
Pr๏ฟฝโ๏ฟฝ๐ค
(๐ค๐ก๐ป (๐ฒ๐ค ))
๐ (1/โ๐)
๐๐ป (๐๐ค )
๐ฑ=0โ
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๐ค๐ก๐ป (๐ฒ๐ค)0 n
(๐ยฟยฟ๐คโ๐)๐+๐(โ๐)ยฟ(๐ยฟยฟ๐คโ๐ )๐ยฟ(๐ยฟยฟ๐คโ๐ )๐โ๐(โ๐)ยฟ
log(# codewords)
Pr๐ ,๐๐ค
(๐ค๐ก๐ป (๐ฒ๐ค ))
๐๐ป (๐๐คโ๐ )
๐ โ๐
๐ (1/โ๐)
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Theorem 3 โ Reliability proof sketch
0 n
Noise magnitude >> Codeword weight!!!
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Theorem 3 โ Reliability proof sketch
.
.
.
1000001000000000100100000010000000100
0001000000100000010000000010000000001
0010000100000001010010000000100010011
0000100000010000000000010000000010000
Random code
2O(โn) codewords
Weight O(โn)
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Theorem 3 โ Reliability proof sketch
.
.
.
1000001000010000100100000010000000100
0001000000100000010000000010000000001
0010000100000001010010000000100010011
0000100000010000000000010000000010000
โขE(Intersection of 2 codewords) = O(1)
Weight O(โn)
โขPr(dmin(x) < cโn) < 2-O(โn)
โขโMostโ codewords โwell-isolatedโ
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Theorem 3 โ dmin decoding
โขPr(x decoded to xโ) < 2-O(โn)
+ O(โn)
x
xโ
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โข Recall: want to show
Theorem 3 โ Deniability proof sketch
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Theorem 4 โ unexpected detour
๐ค๐ก๐ป (๐๐ค )
0 n
logarithm of# codewords
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๐ค๐ก๐ป (๐๐ค )
0 n
logarithm of# codewords
Too few codewords=> Not deniable
Theorem 4 โ unexpected detour
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๐ค๐ก๐ป (๐ฒ๐ค)0 n
(๐ยฟยฟ๐คโ๐)๐+๐(โ๐)ยฟ(๐ยฟยฟ๐คโ๐ )๐ยฟ(๐ยฟยฟ๐คโ๐ )๐โ๐(โ๐)ยฟ
log(# codewords)
Pr๐ ,๐๐ค
(๐ค๐ก๐ป (๐ฒ๐ค ))
๐๐ป (๐๐คโ๐ )
๐ โ๐
๐ (1/โ๐)
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โข Recall: want to show
๐0 ๐1
Theorem 3 โ Deniability proof sketch
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0 n
log(# codewords)
๐๐ป (๐๐ค )
Theorem 3 โ Deniability proof sketch
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๐ค๐ก๐ป (๐๐ค )
0 n
logarithm of# codewords
Theorem 3 โ Deniability proof sketch
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๐0 ๐1
!!!
Theorem 3 โ Deniability proof sketch
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๐0 ๐1
!!!
Theorem 3 โ Deniability proof sketch
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๐1๐ฌ๐ช(๐ยฟยฟ1)ยฟ
Theorem 3 โ Deniability proof sketch
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๐ค๐ก๐ป (๐๐ค )
0 n๐๐ค๐+๐ (โ๐)๐๐ค๐
logarithm of# codewords
Theorem 3 โ Deniability proof sketch
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# codewords of โtypeโ
๐ 1๐ 2
๐ 3
Theorem 3 โ Deniability proof sketch
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Theorem 3 โ Deniability proof sketch
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Theorem 3 โ Deniability proof sketch
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Theorem 3 โ Deniability proof sketch
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โข w.p.
Theorem 3 โ Deniability proof sketch
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โข w.p.
Theorem 3 โ Deniability proof sketch
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โข w.p. โข close to w.p.
Theorem 3 โ Deniability proof sketch
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โข w.p. โข close to w.p. โข , w.h.p.
Theorem 3 โ Deniability proof sketch
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Summary