Reliable Deniable Communication:
Hiding Messages in Noise
Pak Hou Che
Mayank Bakshi
Sidharth Jaggi
The Chinese University
of Hong Kong
The Institute of
Network Coding
Alice
Bob
Reliability
Alice
Bob
Reliability
Deniability
Willie
(the Warden)
Alice’s Encoder
M
T
𝐼𝑓 𝐓 = 0, 𝐗 = 𝟎
𝐼𝑓 𝐓 = 1, 𝐗 = 𝐸𝑛𝑐(𝐌)
𝑚𝑒𝑠𝑠𝑎𝑔𝑒 𝐌 ∈ {1, … , 𝑁}
t𝑟𝑎𝑛𝑠. 𝑠𝑡𝑎𝑡𝑢𝑠 𝐓 ∈ {0, 1}
𝑁 = 2𝜃( 𝑛)
𝑇ℎ𝑟𝑜𝑢𝑔ℎ𝑝𝑢𝑡 𝜏 = log 𝑁
log 𝑁
𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑡ℎ𝑟𝑜𝑢𝑔ℎ𝑝𝑢𝑡 𝑟 =
𝑛
𝐗
Alice’s Encoder
M
T
𝐼𝑓 𝐓 = 0, 𝐗 = 𝟎
𝐼𝑓 𝐓 = 1, 𝐗 = 𝐸𝑛𝑐(𝐌)
Message 𝐌 ∈ {1, … , 𝑁}
Trans. Status 𝐓 ∈ {0, 1}
𝑁 = 2𝜃(
𝑛)
Bob’s Decoder
𝐗
BSC(pb)
𝐘𝑏
𝐌 = 𝐷𝑒𝑐(𝐘𝑏 )
𝐌
1 − 𝜖 𝑟𝑒𝑙𝑖𝑎𝑏𝑙𝑒
Pr 𝐌 = 𝐌 > 1 − 𝜖
Alice’s Encoder
M
T
Bob’s Decoder
𝐼𝑓 𝐓 = 0, 𝐗 = 𝟎
𝐗
BSC(pb)
𝐼𝑓 𝐓 = 1, 𝐗 = 𝐸𝑛𝑐(𝐌)
𝐌 = 𝐷𝑒𝑐(𝐘𝑏 )
𝐌
1 − 𝜖 𝑟𝑒𝑙𝑖𝑎𝑏𝑙𝑒
Pr 𝐌 = 𝐌 > 1 − 𝜖
Message 𝐌 ∈ {1, … , 𝑁}
Trans. Status 𝐓 ∈ {0, 1}
𝑁 = 2𝜃(
𝐘𝑏
𝑛)
BSC(pw)
𝐘𝑤
𝐓 = 𝐷𝑒𝑐(𝐘𝑤 )
𝐓
Willie’s (Best) Estimator
Bash, Goeckel & Towsley [1]
Shared secret
𝑂( 𝑛 log 𝑛) bits
AWGN channels
But capacity only 𝑂
𝑛 bits
[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.
This work
No shared secret
BSC(pb)
pb < pw
BSC(pw)
Alice’s Encoder
M
T
Bob’s Decoder
𝐼𝑓 𝐓 = 0, 𝐗 = 𝟎
𝐗
BSC(pb)
𝐼𝑓 𝐓 = 1, 𝐗 = 𝐸𝑛𝑐(𝐌)
𝐌 = 𝐷𝑒𝑐(𝐘𝑏 )
𝐌
1 − 𝜖 𝑟𝑒𝑙𝑖𝑎𝑏𝑙𝑒
Pr 𝐌 = 𝐌 > 1 − 𝜖
Message 𝐌 ∈ {1, … , 𝑁}
Trans. Status 𝐓 ∈ {0, 1}
𝑁 = 2𝜃(
𝐘𝑏
𝑛)
BSC(pw)
𝐘𝑤
𝐓 = 𝐷𝑒𝑐(𝐘𝑤 )
𝐓
Willie’s (Best) Estimator
Hypothesis Testing
Willie’s Estimate
Alice’s
Transmission
Status
• 𝛼 = Pr 𝐓 = 1 𝐓 = 0 , 𝛽 = Pr 𝐓 = 0 𝐓 = 1
Hypothesis Testing
Willie’s Estimate
Alice’s
Transmission
Status
Hypothesis Testing
Willie’s Estimate
Alice’s
Transmission
Status
Hypothesis Testing
Willie’s Estimate
Alice’s
Transmission
Status
Intuition
• 𝐓 = 0, 𝐲𝑤 = 𝐳𝑤 ~Binomial(𝑛, 𝑝𝑤 )
Intuition
• 𝐓 = 0, 𝐲𝑤 = 𝐳𝑤 ~Binomial 𝑛, 𝑝𝑤
• 𝑊ℎ𝑒𝑛 𝐓 = 1,
Theorem 1 (Wt(c.w.))
(high deniability => low weight codewords)
• Too many codewords with weight “much” greater
than 𝑐 𝑛, then the system is “not very” deniable
Theorems 2 & 3
(Converse & achievability for reliable & deniable comm.)
Theorems 2 & 3
𝑝𝑤
1/2
pb>pw
0
1/2
𝑝𝑏
Theorems 2 & 3
𝑝𝑤
1/2
𝑁=0
0
1/2
𝑝𝑏
Theorems 2 & 3
𝑝𝑤
pw=1/2
1/2
0
1/2
𝑝𝑏
Theorems 2 & 3
𝑝𝑤
1/2
N  2(1 H ( p b ))n
(BSC(pb))

0
1/2
𝑝𝑏
Theorems 2 & 3
𝑝𝑤
1/2
pb=0
0
1/2
𝑝𝑏
Theorems 2 & 3
𝑝𝑤
𝑁 = 2𝑂(
1/2
0
𝑛 log 𝑛)
,
𝑛
= 2𝑂(
𝑛
1/2
𝑛 log 𝑛)
𝑝𝑏
Theorems 2 & 3
𝑝𝑤
1/2
pw>pb
0
1/2
𝑝𝑏
Theorems 2 & 3
𝑝𝑤
𝑁 = 2𝑂(
𝑛)
1/2
“Standard” IT inequalities
+
Wt(“most codewords”)<√n
(Thm 1)
0
1/2
𝑝𝑏
Theorems 2 & 3
𝑝𝑤
1/2
Main thm:
Achievable region
𝑁 = 2Ω ( 𝑛)
0
1/2
𝑝𝑏
log
𝑛
≈𝑛
𝑛/2
logarithm of
# codewords
0
n
𝑤𝑡𝐻 (𝒚𝑤 )
log(# codewords)
𝑛𝐻(𝑝𝑤 )
𝐱=0
Pr(𝑤𝑡𝐻 𝐲𝑤 )
𝐙𝑤
𝑂(1
𝑛)
0
𝑝𝑤 𝑛 − 𝑂( 𝑛)
𝑝𝑤 𝑛 𝑝𝑤 𝑛 + 𝑂( 𝑛)
n
𝑤𝑡𝐻 (𝐲𝑤 )
log(# codewords)
𝑛𝐻(𝑝𝑤 ∗ 𝜌)
Pr (𝑤𝑡𝐻 𝐲𝑤 )
𝑐 𝑛
𝐌,𝐙𝑤
𝑂(1
𝑛)
n
0
(𝑝𝑤 ∗ 𝜌)𝑛 − 𝑂( 𝑛)
(𝑝𝑤 ∗ 𝜌)𝑛
(𝑝𝑤 ∗ 𝜌)𝑛 + 𝑂( 𝑛)
𝑤𝑡𝐻 (𝐲𝑤 )
Theorem 3 – Reliability proof sketch
Random code
Weight 𝑂( 𝑛)
1000001000000000100100000010000000100
0001000000100000010000000010000000001
0010000100000001010010000000100010011
2𝑂(
𝑛) codewords
.
.
.
0000100000010000000000010000000010000
Theorem 3 – Reliability proof sketch
Weight 𝑂( 𝑛)
• E(Intersection of 2 codewords) = O(1)
• “Most” codewords “well-isolated”
1000001000010000100100000010000000100
0001000000100000010000000010000000001
0010000100000001010010000000100010011
.
.
.
0000100000010000000000010000000010000
Theorem 3 – dmin decoding
x
+
𝑂( 𝑛)
x’
•Pr(x decoded to x’) < 2−𝑂(
𝑛)
Theorem 3 – Deniability proof sketch
• Recall: want to show 𝑉 𝐏0 , 𝐏1 < 𝜖
log(# codewords)
𝑛𝐻(𝑝𝑤 ∗ 𝜌)
Pr (𝑤𝑡𝐻 𝐲𝑤 )
𝑐 𝑛
𝐌,𝐙𝑤
𝑂(1
𝑛)
n
0
(𝑝𝑤 ∗ 𝜌)𝑛 − 𝑂( 𝑛)
(𝑝𝑤 ∗ 𝜌)𝑛
(𝑝𝑤 ∗ 𝜌)𝑛 + 𝑂( 𝑛)
𝑤𝑡𝐻 (𝐲𝑤 )
Theorem 3 – Deniability proof sketch
• Recall: want to show 𝑉 𝐏0 , 𝐏1 < 𝜖
𝐏0
𝐏1
Theorem 3 – Deniability proof sketch
log(# codewords)
0
n
Theorem 3 – Deniability proof sketch
logarithm of
# codewords
0
n
𝑤𝑡𝐻 (𝒚𝑤 )
Theorem 3 – Deniability proof sketch
𝑬𝑪 (𝐏1 )!!!
𝐏0
𝐏1
Theorem 3 – Deniability proof sketch
• 𝑉 𝐏0 , 𝐏1 ≤ 𝑉 𝐏0 , 𝑬𝑪 (𝐏1 ) + 𝑉 𝑬𝑪 (𝐏1 ), 𝐏1
𝑬𝑪 (𝐏1 )!!!
𝐏0
𝐏1
Theorem 3 – Deniability proof sketch
𝑬𝑪 (𝐏1 )
𝐏1
Theorem 3 – Deniability proof sketch
logarithm of
# codewords
0
𝑝𝑤 𝑛 − 𝑂( 𝑛)
𝑝𝑤 𝑛 𝑝𝑤 𝑛 + 𝑂( 𝑛)
𝑤𝑡𝐻 (𝒚𝑤 )
n
Theorem 4
logarithm of
# codewords
0
n
𝑤𝑡𝐻 (𝒚𝑤 )
Theorem 4
Too few codewords
=> Not deniable
logarithm of
# codewords
0
n
𝑤𝑡𝐻 (𝒚𝑤 )
Summary
Summary