Reliable Deniable Communication: Hiding Messages in Noise
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Transcript of Reliable Deniable Communication: Hiding Messages in Noise
Reliable Deniable Communication: Hiding Messages in Noise
The Chinese University of Hong Kong
The Institute of Network Coding
Pak Hou CheMayank BakshiSidharth Jaggi
Alice
Reliability
Bob
Willie(the Warden)
Reliability
Deniability
AliceBob
M
T
t
�⃑�
Alice’s Encoder
𝑁=2𝜃 (√𝑛)
M
T
Message Trans. Status
BSC(pb) �̂�=𝐷𝑒𝑐 (�⃑�𝑏)�⃑�𝑏�⃑�
Alice’s EncoderBob’s Decoder
𝑁=2𝜃 (√𝑛)
�̂�
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 (Best) Estimator
�̂�
�̂�
Hypothesis Testing Willie’s Estimate
Alice’s Transmission
Status
𝛼=Pr ( �̂�=1|𝐓=0 ) , 𝛽=Pr ( �̂�=0|𝐓=1 )
Hypothesis Testing Willie’s Estimate
Alice’s Transmission
Status
• Want:
Hypothesis Testing Willie’s Estimate
Alice’s Transmission
Status
• Want: • Known: for opt. estimator
Hypothesis Testing Willie’s Estimate
Alice’s Transmission
Status
• Want: • Known: for opt. estimator• , ( w.h.p.)
Bash, Goeckel & Towsley [1]Shared secret
[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.
AWGN channels
Capacity = bits
bits
This workNo shared secret
BSC(pb)
BSC(pw)
pb < pw
Intuition
𝐓=0 , 𝐲𝑤=�⃑�𝑤 Binomial(𝑛 ,𝑝𝑤)
Intuition
Main Theorems
• Theorem 1– Deniability low weight codewords
• Theorem 2 – Converse of reliability
• Theorem 3– Achievability (reliability & deniability)
• Theorem 4– Trade-off between deniability & size of codebook
Theorem 1 (wt(c.w.))(high deniability => low weight codewords)
Too many codewords with weight “much ” greater than𝑐 √𝑛 , then the system is “ not very ” deniable
Theorem 2 (Converse)
• , if • if
Theorem 3 – Reliability
• Random codebook ( i.i.d. ) )• minimum distance decoder• For ,
𝑤𝑡𝐻 (𝒚𝑤 )
0 n
logarithm of# binary vectors
log ( 𝑛𝑛/2)≈𝑛
𝑤𝑡𝐻 (𝐲𝑤)0 n𝑝𝑤𝑛+𝑂 (√𝑛)𝑝𝑤𝑛
log(# vectors)
Pr�⃑�𝑤
(𝑤𝑡𝐻 (𝐲𝑤 ))
𝑂 (1/√𝑛)
𝑛𝐻 (𝑝𝑤 )
𝐱=0⃗
log(# vectors)
𝑛𝐻 (𝑝𝑤∗𝜌 )
𝑐 √𝑛
log(# codewords)
𝑛𝐻 (𝑝𝑤∗𝜌 )
𝑐 √𝑛
𝑤𝑡𝐻 (𝐲𝑤)0 n
(𝑝¿¿𝑤∗𝜌)𝑛+𝑂(√𝑛)¿(𝑝¿¿𝑤∗𝜌 )𝑛¿(𝑝¿¿𝑤∗𝜌 )𝑛−𝑂(√𝑛)¿
log(# vectors)
Pr𝐌 ,𝐙𝑤
(𝑤𝑡𝐻 (𝐲𝑤 ))
𝑛𝐻 (𝑝𝑤∗𝜌 )
𝑐 √𝑛
𝑂 (1/√𝑛)
• Recall: want to show
Theorem 3 – Deniability proof sketch
• Recall: want to show
𝐏0 𝐏1
Theorem 3 – Deniability proof sketch
0 n
log(# vectors)
Theorem 3 – Deniability proof sketch
𝐏0 𝐏1
!!!
Theorem 3 – Deniability proof sketch
𝐏0 𝐏1
!!!
Theorem 3 – Deniability proof sketch
𝐏1𝑬𝑪(𝐏¿¿1)¿
Theorem 3 – Deniability proof sketch
with high probability
𝑤𝑡𝐻 (𝒚𝑤 )
0 n𝑝𝑤𝑛+𝑂 (√𝑛)𝑝𝑤𝑛
logarithm of# vectors
Theorem 3 – Deniability proof sketch
𝑤𝑡𝐻 (𝒚𝑤 )
0 n𝑝𝑤𝑛+𝑂 (√𝑛)𝑝𝑤𝑛
logarithm of# vectors
Theorem 3 – Deniability proof sketch
# codewords of “type”
𝑇
Theorem 3 – Deniability proof sketch
Theorem 3 – Deniability proof sketch
𝑇
Theorem 3 – Deniability proof sketch
𝑇
Theorem 3 – Deniability proof sketch
𝑇
Theorem 3 – Deniability proof sketch
𝑇
• w.p.
Theorem 3 – Deniability proof sketch
𝑇
• w.p.
• close to w.p. • , w.h.p.
Theorem 3 – Deniability proof sketch
𝑇
Theorem 4
𝑤𝑡𝐻 (𝒚𝑤 )
0 n
logarithm of# codewords
𝑤𝑡𝐻 (𝒚𝑤 )
0 n
logarithm of# codewords
Too few codewords=> Not deniable
Theorem 4
Summary
𝑝𝑏
𝑝𝑤
0 1/2
1/2 • Thm 1 & 2 Converse (Upper Bound)
• Thm 3 Achievability• Thm 4 Lower Bound