Post on 17-Dec-2015
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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
Alice
Reliability
Bob
Willie(the Warden)
Reliability
Deniability
AliceBob
Willie-sky
Reliability
Deniability
AliceBob
M
T
t
οΏ½βοΏ½
Aliceβs Encoder
π=2π (βπ)
M
T
Message Trans. Status
BSC(pb) οΏ½ΜοΏ½=π·ππ (οΏ½βοΏ½π)οΏ½βοΏ½ποΏ½βοΏ½
Aliceβs Encoder
Bobβs Decoder
π=2π (βπ)
οΏ½ΜοΏ½
M
T
Message Trans. Status
BSC(pb) οΏ½ΜοΏ½=π·ππ (οΏ½βοΏ½π)οΏ½βοΏ½ποΏ½βοΏ½
Aliceβs Encoder
Bobβs Decoder
BSC(pw)
οΏ½ΜοΏ½=π·ππ (οΏ½βοΏ½π€)
οΏ½βοΏ½π€
π=2π (βπ)
Willieβs (Best) Estimator
οΏ½ΜοΏ½
οΏ½ΜοΏ½
Bash, Goeckel & Towsley [1]Shared secret
[1] B. A. Bash, D. Goeckel and D. Towsley, βSquare root law for communication with low probability of detection on AWGN channels,β in Proceedings of the IEEE International Symposium on Information Theory (ISIT), 2012, pp. 448β452.
β¬
O n .log(n)( ) bits
AWGN channels
But capacity only
β¬
O n( ) bits!
This workNo shared secret
BSC(pb)
BSC(pw)
pb < pw
Wicked Willie(s) Base-station Bob
Aerial Alice
Directional antenna
Steganography: Other work
Steganography: Other work
Other work: βCommonβ modelShared secret key
Capacity O(n) message bitsInformation-theoretically tight characterization(Gelβfand-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) βsmallβ
Other work: Square-root βlawβ(βempiricalβ)
β’βSteganographic 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 βsecureβ 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.β [3]
β’βThanks 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.β [4]
[3] A. Ker, T. Pevny`, J. Kodovsky`, and J. Fridrich, βThe square root law of steganographic capacity,β in Proceedings of the 10th ACM workshop on Multimedia and security. ACM, 2008, pp. 107β116.[4] R. Anderson, βStretching the limits of steganography,β in Information Hiding, 1996, pp. 39β48.
β’β[T]he reference to the Central Limit Theorem... suggests that a square root relationship should be considered. β [3]
M
T
Message Trans. Status
BSC(pb) οΏ½ΜοΏ½=π·ππ (οΏ½βοΏ½π)οΏ½βοΏ½ποΏ½βοΏ½
Aliceβs Encoder
Bobβs Decoder
BSC(pw)
οΏ½ΜοΏ½=π·ππ (οΏ½βοΏ½π€)
οΏ½βοΏ½π€
π=2π (βπ)
Willieβs (Best) Estimator
οΏ½ΜοΏ½
οΏ½ΜοΏ½
Hypothesis Testing Willieβs Estimate
Aliceβs Transmission
Status
πΌ=Pr ( οΏ½ΜοΏ½=1|π=0 ) , π½=Pr ( οΏ½ΜοΏ½=0|π=1 )
Hypothesis Testing Willieβs Estimate
Aliceβs Transmission
Status
Hypothesis Testing Willieβs Estimate
Aliceβs Transmission
Status
Hypothesis Testing Willieβs Estimate
Aliceβs Transmission
Status
Intuition
π=0 , π²π€=οΏ½βοΏ½π€ Binomial(π ,ππ€)
Intuition
Theorem 1 (Wt(c.w.))(high deniability => low weight codewords)
Too many codewords with weight βmuch β greater thanπ βπ , hπ‘ ππ hπ‘ ππ π¦π π‘ππππ βnot veryβ deniable
Theorems 2 & 3(Converse & achievability for reliable & deniable comm.)
Theorems 2 & 3
ππ
ππ€
0 1/2
1/2
pb>pw
Theorems 2 & 3
ππ
ππ€
0 1/2
1/2
π=0
(Symmetrizability)
Theorems 2 & 3
ππ
ππ€
0 1/2
1/2pw=1/2
Theorems 2 & 3
ππ
ππ€
0 1/2
1/2
β¬
N β 2(1βH (pb ))n
(BSC(pb))
Theorems 2 & 3
ππ
ππ€
0 1/2
1/2
pb=0
Theorems 2 & 3
ππ
ππ€
0 1/2
1/2
π=2π (βπ logπ) ,( πβπ)=2π (βπ logπ)
Theorems 2 & 3
ππ
ππ€
0 1/2
1/2
pw>pb
Theorems 2 & 3
ππ
ππ€
0 1/2
1/2π=2π (βπ)
βStandardβ IT inequalities+
Wt(βmost codewordsβ)<βn(Thm 1)
Theorems 2 & 3
ππ
ππ€
0 1/2
1/2
Main thm:
π€π‘π» (ππ€ )
0 n
logarithm of# codewords
log ( ππ/2)βπ
π€π‘π» (π²π€)0 nππ€π+π (βπ)ππ€π
log(# codewords)
PrοΏ½βοΏ½π€
(π€π‘π» (π²π€ ))
π (1/βπ)
ππ» (ππ€ )
π±=0β
π€π‘π» (π²π€)0 n
(πΒΏΒΏπ€βπ)π+π(βπ)ΒΏ(πΒΏΒΏπ€βπ)πΒΏ(πΒΏΒΏπ€βπ)πβπ(βπ)ΒΏ
log(# codewords)
Prπ ,ππ€
(π€π‘π» (π²π€ ))
ππ» (ππ€βπ)
π βπ
π (1/βπ)
Theorem 3 β Reliability proof sketch
0 n
Noise magnitude >> Codeword weight!!!
Theorem 3 β Reliability proof sketch
.
.
.
1000001000000000100100000010000000100
0001000000100000010000000010000000001
0010000100000001010010000000100010011
0000100000010000000000010000000010000
Random code
2O(βn) codewords
Weight O(βn)
Theorem 3 β Reliability proof sketch
.
.
.
1000001000010000100100000010000000100
0001000000100000010000000010000000001
0010000100000001010010000000100010011
0000100000010000000000010000000010000
β’E(Intersection of 2 codewords) = O(1)
Weight O(βn)
β’Pr(dmin(x) < cβn) < 2-O(βn)
β’βMostβ codewords βwell-isolatedβ
Theorem 3 β dmin decoding
β’Pr(x decoded to xβ) < 2-O(βn)
+ O(βn)
x
xβ
β’ Recall: want to show
Theorem 3 β Deniability proof sketch
Theorem 4 β unexpected detour
π€π‘π» (ππ€ )
0 n
logarithm of# codewords
π€π‘π» (ππ€ )
0 n
logarithm of# codewords
Too few codewords=> Not deniable
Theorem 4 β unexpected detour
π€π‘π» (π²π€)0 n
(πΒΏΒΏπ€βπ)π+π(βπ)ΒΏ(πΒΏΒΏπ€βπ)πΒΏ(πΒΏΒΏπ€βπ)πβπ(βπ)ΒΏ
log(# codewords)
Prπ ,ππ€
(π€π‘π» (π²π€ ))
ππ» (ππ€βπ)
π βπ
π (1/βπ)
β’ Recall: want to show
π0 π1
Theorem 3 β Deniability proof sketch
0 n
log(# codewords)
ππ» (ππ€ )
Theorem 3 β Deniability proof sketch
π€π‘π» (ππ€ )
0 n
logarithm of# codewords
Theorem 3 β Deniability proof sketch
π0 π1
!!!
Theorem 3 β Deniability proof sketch
π0 π1
!!!
Theorem 3 β Deniability proof sketch
π1π¬πͺ(πΒΏΒΏ1)ΒΏ
Theorem 3 β Deniability proof sketch
π€π‘π» (ππ€ )
0 nππ€π+π (βπ)ππ€π
logarithm of# codewords
Theorem 3 β Deniability proof sketch
# codewords of βtypeβ
π 1π 2
π 3
Theorem 3 β Deniability proof sketch
Theorem 3 β Deniability proof sketch
Theorem 3 β Deniability proof sketch
Theorem 3 β Deniability proof sketch
β’ w.p.
Theorem 3 β Deniability proof sketch
β’ w.p.
Theorem 3 β Deniability proof sketch
β’ w.p. β’ close to w.p.
Theorem 3 β Deniability proof sketch
β’ w.p. β’ close to w.p. β’ , w.h.p.
Theorem 3 β Deniability proof sketch
Summary