Reliable Deniable Communication: Hiding Messages from Noise Pak Hou Che Joint Work with Sidharth...
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Reliable Deniable Communication: Hiding Messages from Noise
Pak Hou CheJoint Work with Sidharth Jaggi,
Mayank Bakshi and Madhi Jafari Siavoshani
Institute of Network CodingThe Chinese University of Hong Kong
Introduction
Is Alice talking to someone?
Alice
Willie
Bob
Introduction
Is Alice talking to someone?
Alice
Willie
Bob
Goal: decode message
Goal: detect Aliceβs status
Goal: transmitreliably & deniably
Model
M
T
t
οΏ½βοΏ½
Aliceβs Encoder
π=2π (βπ)
Model
M
T
Message Trans. Status
BSC(pb) οΏ½ΜοΏ½=π·ππ (οΏ½βοΏ½π)οΏ½βοΏ½ποΏ½βοΏ½
Aliceβs EncoderBobβs Decoder
π=2π (βπ)
οΏ½ΜοΏ½
Model
M
T
Message Trans. Status
BSC(pb) οΏ½ΜοΏ½=π·ππ (οΏ½βοΏ½π)οΏ½βοΏ½ποΏ½βοΏ½
Aliceβs EncoderBobβs Decoder
π=2π (βπ)
οΏ½ΜοΏ½
Model
M
T
Message Trans. Status
BSC(pb) οΏ½ΜοΏ½=π·ππ (οΏ½βοΏ½π)οΏ½βοΏ½ποΏ½βοΏ½
Aliceβs EncoderBobβs Decoder
BSC(pw)
οΏ½ΜοΏ½=π·ππ (οΏ½βοΏ½π€)
οΏ½βοΏ½π€
π=2π (βπ)
Willieβs Estimator
οΏ½ΜοΏ½
οΏ½ΜοΏ½
Model
M
T
Message Trans. Status
BSC(pb) οΏ½ΜοΏ½=π·ππ (οΏ½βοΏ½π)οΏ½βοΏ½ποΏ½βοΏ½
Aliceβs EncoderBobβs Decoder
BSC(pw)
οΏ½ΜοΏ½=π·ππ (οΏ½βοΏ½π€)
οΏ½βοΏ½π€
π=2π (βπ)
Willieβs Estimator
οΏ½ΜοΏ½
οΏ½ΜοΏ½
Model
M
T
Message Trans. Status
BSC(pb) οΏ½ΜοΏ½=π·ππ (οΏ½βοΏ½π)οΏ½βοΏ½ποΏ½βοΏ½
Aliceβs EncoderBobβs Decoder
BSC(pw)
οΏ½ΜοΏ½=π·ππ (οΏ½βοΏ½π€)
οΏ½βοΏ½π€
π=2π (βπ)
Willieβs Estimator
οΏ½ΜοΏ½
οΏ½ΜοΏ½
Asymmetry pb < pw
Prior Work
Alice Bob
Willie
Shared secret ([1] Bash, Goeckel & Towsley)
[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.
Our Case
Alice Bob
Willie
Asymmetry pb < pw
Hypothesis TestingWillieβs Estimation
Aliceβs Transmit
StatusSilent
Transmit
πΌ=Pr ( οΏ½ΜοΏ½=1|π=0 ) , π½=Pr ( οΏ½ΜοΏ½=0|π=1 )
Hypothesis TestingWillieβs Estimation
Aliceβs Transmit
StatusSilent
Transmit
Hypothesis TestingWillieβs Estimation
Aliceβs Transmit
StatusSilent
Transmit
Hypothesis TestingWillieβs Estimation
Aliceβs Transmit
StatusSilent
Transmit
Intuition
π=0 , π²π€=οΏ½βοΏ½π€ Binomial(π ,ππ€)
Intuition
Theorem 1(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
Theorem 2 & 3(Converse & achievability for reliable & deniable comm.)
Theorem 2 & 3
ππ
ππ€
0 1/2
1/2
pb>pw
Theorem 2 & 3
ππ
ππ€
0 1/2
1/2
π=0
Theorem 2 & 3
ππ
ππ€
0 1/2
1/2pw=1/2
Theorem 2 & 3
ππ
ππ€
0 1/2
1/2
π β€2(1βπ» (ππ€ )+π)π
Theorem 2 & 3
ππ
ππ€
0 1/2
1/2
π β₯2(1βπ» (ππ€ )βπ)π
Theorem 2 & 3
ππ
ππ€
0 1/2
1/2
pb=1/2
Theorem 2 & 3
ππ
ππ€
0 1/2
1/2π=2π (βπ logπ)
Theorem 2 & 3
ππ
ππ€
0 1/2
1/2
π=2π (βπ logπ) ,( πβπ)=2π (βπ logπ)
Theorem 2 & 3
ππ
ππ€
0 1/2
1/2
π=2Ξ©(βπ logπ)
Theorem 2 & 3
ππ
ππ€
0 1/2
1/2
pw>pb
Theorem 2 & 3
ππ
ππ€
0 1/2
1/2π=2π (βπ)
Theorem 2 & 3
ππ
ππ€
0 1/2
1/2
Achievable region
Theorem 3 β Proof Idea
β’ Recall: want to show
Theorem 3 β Proof Idea
π€π‘π» (ππ€ )
0 nππ€π+π (βπ)ππ€π
logarithm of# codewords
log ( ππ/2)βπ
Theorem 3 β Proof Idea
π€π‘π» (ππ€ )
0 n
logarithm of# codewords
Theorem 3 β Proof Idea
π€π‘π» (ππ€ )
0 n
logarithm of# codewords
Too few codewords=> Not deniable (Thm4)
Theorem 3 β Proof Idea
π€π‘π» (ππ€ )
0 n
logarithm of# codewords
π (βπ)
Theorem 3 β Proof Idea
π€π‘π» (ππ€ )
0 n
logarithm of# codewords
Theorem 3 β Proof Idea
π (π²π€)
Logarithm of# codewords
Theorem 3 β Proof Idea
β’ Recall: want to show
π0 π1
Theorem 3 β Proof Idea
π0 π1
!!!
Theorem 3 β Proof Idea
π0 π1
!!!
Theorem 3 β Proof Idea
β’ Chernoff bound is weakβ’ Other concentration inequality
π1π¬πͺ(πΒΏΒΏ1)ΒΏ
Theorem 3 β Proof Idea
Theorem 3 β Proof Idea
π€π‘π» (ππ€ )
0 n
logarithm of# codewords
Theorem 3 β Proof Idea
π€π‘π» (ππ€ )
0 nππ€π+π (βπ)ππ€π
logarithm of# codewords
Theorem 3 β Proof Idea
π€π‘π» (ππ€ )
0 nππ€π+π (βπ)ππ€π
logarithm of# codewords
Theorem 3 β Sketch Proof
# codewords of βtypeβ
π 1π 2
π 3
Theorem 3 β Sketch Proof
Theorem 3 β Sketch Proof
Theorem 3 β Sketch Proof
β’ w.p.
Theorem 3 β Sketch Proof
β’ w.p.
Theorem 3 β Sketch Proof
β’ w.p. β’ close to w.p.
Theorem 3 β Sketch Proof
β’ w.p. β’ close to w.p. β’ , w.h.p.
Summary
ππ
ππ€
0 1/2
1/2
Summary
ππ
ππ€
0 1/2
1/2