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 Alice’s status
Goal: transmitreliably & deniably
Model
M
T
t
�⃑�
Alice’s Encoder
𝑁=2𝜃 (√𝑛)
Model
M
T
Message Trans. Status
BSC(pb) �̂�=𝐷𝑒𝑐 (�⃑�𝑏)�⃑�𝑏�⃑�
Alice’s EncoderBob’s Decoder
𝑁=2𝜃 (√𝑛)
�̂�
Model
M
T
Message Trans. Status
BSC(pb) �̂�=𝐷𝑒𝑐 (�⃑�𝑏)�⃑�𝑏�⃑�
Alice’s EncoderBob’s Decoder
𝑁=2𝜃 (√𝑛)
�̂�
Model
M
T
Message Trans. Status
BSC(pb) �̂�=𝐷𝑒𝑐 (�⃑�𝑏)�⃑�𝑏�⃑�
Alice’s EncoderBob’s Decoder
BSC(pw)
�̂�=𝐷𝑒𝑐 (�⃑�𝑤)
�⃑�𝑤
𝑁=2𝜃 (√𝑛)
Willie’s Estimator
�̂�
�̂�
Model
M
T
Message Trans. Status
BSC(pb) �̂�=𝐷𝑒𝑐 (�⃑�𝑏)�⃑�𝑏�⃑�
Alice’s EncoderBob’s Decoder
BSC(pw)
�̂�=𝐷𝑒𝑐 (�⃑�𝑤)
�⃑�𝑤
𝑁=2𝜃 (√𝑛)
Willie’s Estimator
�̂�
�̂�
Model
M
T
Message Trans. Status
BSC(pb) �̂�=𝐷𝑒𝑐 (�⃑�𝑏)�⃑�𝑏�⃑�
Alice’s EncoderBob’s Decoder
BSC(pw)
�̂�=𝐷𝑒𝑐 (�⃑�𝑤)
�⃑�𝑤
𝑁=2𝜃 (√𝑛)
Willie’s Estimator
�̂�
�̂�
Asymmetry pb < pw
Prior Work
Alice Bob
Willie
Shared secret ([1] Bash, Goeckel & Towsley)
[1] B. A. Bash, D. Goeckel and D. Towsley, “Square root law for communication with low probability of detection on AWGN channels,” in Proceedings of the IEEE International Symposium on Information Theory (ISIT), 2012, pp. 448–452.
Our Case
Alice Bob
Willie
Asymmetry pb < pw
Hypothesis TestingWillie’s Estimation
Alice’s Transmit
StatusSilent
Transmit
𝛼=Pr ( �̂�=1|𝐓=0 ) , 𝛽=Pr ( �̂�=0|𝐓=1 )
Hypothesis TestingWillie’s Estimation
Alice’s Transmit
StatusSilent
Transmit
Hypothesis TestingWillie’s Estimation
Alice’s Transmit
StatusSilent
Transmit
Hypothesis TestingWillie’s Estimation
Alice’s Transmit
StatusSilent
Transmit
Intuition
𝐓=0 , 𝐲𝑤=�⃑�𝑤 Binomial(𝑛 ,𝑝𝑤)
Intuition
Theorem 1(high deniability => low weight codewords)
Too many codewords with weight “much ” greater than𝑐 √𝑛 , h𝑡 𝑒𝑛 h𝑡 𝑒𝑠𝑦𝑠𝑡𝑒𝑚𝑖𝑠 “not very” deniable
Theorem 2 & 3(Converse & achievability for reliable & deniable comm.)
Theorem 2 & 3
𝑝𝑏
𝑝𝑤
0 1/2
1/2
pb>pw
Theorem 2 & 3
𝑝𝑏
𝑝𝑤
0 1/2
1/2
𝑁=0
Theorem 2 & 3
𝑝𝑏
𝑝𝑤
0 1/2
1/2pw=1/2
Theorem 2 & 3
𝑝𝑏
𝑝𝑤
0 1/2
1/2
𝑁 ≤2(1−𝐻 (𝑝𝑤 )+𝜖)𝑛
Theorem 2 & 3
𝑝𝑏
𝑝𝑤
0 1/2
1/2
𝑁 ≥2(1−𝐻 (𝑝𝑤 )−𝜖)𝑛
Theorem 2 & 3
𝑝𝑏
𝑝𝑤
0 1/2
1/2
pb=1/2
Theorem 2 & 3
𝑝𝑏
𝑝𝑤
0 1/2
1/2𝑁=2𝑂 (√𝑛 log𝑛)
Theorem 2 & 3
𝑝𝑏
𝑝𝑤
0 1/2
1/2
𝑁=2𝑂 (√𝑛 log𝑛) ,( 𝑛√𝑛)=2𝑂 (√𝑛 log𝑛)
Theorem 2 & 3
𝑝𝑏
𝑝𝑤
0 1/2
1/2
𝑁=2Ω(√𝑛 log𝑛)
Theorem 2 & 3
𝑝𝑏
𝑝𝑤
0 1/2
1/2
pw>pb
Theorem 2 & 3
𝑝𝑏
𝑝𝑤
0 1/2
1/2𝑁=2𝑂 (√𝑛)
Theorem 2 & 3
𝑝𝑏
𝑝𝑤
0 1/2
1/2
Achievable region
Theorem 3 – Proof Idea
• Recall: want to show
Theorem 3 – Proof Idea
𝑤𝑡𝐻 (𝒚𝑤 )
0 n𝑝𝑤𝑛+𝑂 (√𝑛)𝑝𝑤𝑛
logarithm of# codewords
log ( 𝑛𝑛/2)≈𝑛
Theorem 3 – Proof Idea
𝑤𝑡𝐻 (𝒚𝑤 )
0 n
logarithm of# codewords
Theorem 3 – Proof Idea
𝑤𝑡𝐻 (𝒚𝑤 )
0 n
logarithm of# codewords
Too few codewords=> Not deniable (Thm4)
Theorem 3 – Proof Idea
𝑤𝑡𝐻 (𝒚𝑤 )
0 n
logarithm of# codewords
𝑂 (√𝑛)
Theorem 3 – Proof Idea
𝑤𝑡𝐻 (𝒚𝑤 )
0 n
logarithm of# codewords
Theorem 3 – Proof Idea
𝑝 (𝐲𝑤)
Logarithm of# codewords
Theorem 3 – Proof Idea
• Recall: want to show
𝐏0 𝐏1
Theorem 3 – Proof Idea
𝐏0 𝐏1
!!!
Theorem 3 – Proof Idea
𝐏0 𝐏1
!!!
Theorem 3 – Proof Idea
• Chernoff bound is weak• Other concentration inequality
𝐏1𝑬𝑪(𝐏¿¿1)¿
Theorem 3 – Proof Idea
Theorem 3 – Proof Idea
𝑤𝑡𝐻 (𝒚𝑤 )
0 n
logarithm of# codewords
Theorem 3 – Proof Idea
𝑤𝑡𝐻 (𝒚𝑤 )
0 n𝑝𝑤𝑛+𝑂 (√𝑛)𝑝𝑤𝑛
logarithm of# codewords
Theorem 3 – Proof Idea
𝑤𝑡𝐻 (𝒚𝑤 )
0 n𝑝𝑤𝑛+𝑂 (√𝑛)𝑝𝑤𝑛
logarithm of# codewords
Theorem 3 – Sketch Proof
# codewords of “type”
𝑇 1𝑇 2
𝑇 3
Theorem 3 – Sketch Proof
Theorem 3 – Sketch Proof
Theorem 3 – Sketch Proof
• w.p.
Theorem 3 – Sketch Proof
• w.p.
Theorem 3 – Sketch Proof
• w.p. • close to w.p.
Theorem 3 – Sketch Proof
• w.p. • close to w.p. • , w.h.p.
Summary
𝑝𝑏
𝑝𝑤
0 1/2
1/2
Summary
𝑝𝑏
𝑝𝑤
0 1/2
1/2
