1. 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

2. Alice
Bob
Reliability

3. Alice
Bob
Reliability
Deniability
Willie
(the Warden)

4. Alice’s Encoder M
𝐼𝑓 𝐓=0, 𝐗 = 𝟎
𝐼𝑓 𝐓=1, 𝐗 =𝐸𝑛𝑐(𝐌) 𝐗 T
𝑚𝑒𝑠𝑠𝑎𝑔𝑒 𝐌∈{1, …, 𝑁}
t𝑟𝑎𝑛𝑠. 𝑠𝑡𝑎𝑡𝑢𝑠 𝐓∈{0, 1}
𝑁= 2 𝜃( 𝑛 )
𝑇ℎ𝑟𝑜𝑢𝑔ℎ𝑝𝑢𝑡 𝜏= log 𝑁
𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑡ℎ𝑟𝑜𝑢𝑔ℎ𝑝𝑢𝑡 𝑟= log 𝑁 𝑛

5. Alice’s Encoder M
𝐼𝑓 𝐓=0, 𝐗 = 𝟎
𝐼𝑓 𝐓=1, 𝐗 =𝐸𝑛𝑐(𝐌)
Bob’s Decoder 𝐗
𝐘 𝑏
BSC(pb)
𝐌 =𝐷𝑒𝑐( 𝐘 𝑏 ) 𝐌 T
1−𝜖 𝑟𝑒𝑙𝑖𝑎𝑏𝑙𝑒
Pr 𝐌 =𝐌 >1−𝜖
Message 𝐌∈{1, …, 𝑁}
Trans. Status 𝐓∈{0, 1}
𝑁= 2 𝜃( 𝑛 )

6. Alice’s Encoder M
𝐼𝑓 𝐓=0, 𝐗 = 𝟎
𝐼𝑓 𝐓=1, 𝐗 =𝐸𝑛𝑐(𝐌)
Bob’s Decoder 𝐗
𝐘 𝑏
BSC(pb)
𝐌 =𝐷𝑒𝑐( 𝐘 𝑏 ) 𝐌 T
1−𝜖 𝑟𝑒𝑙𝑖𝑎𝑏𝑙𝑒
Pr 𝐌 =𝐌 >1−𝜖
Message 𝐌∈{1, …, 𝑁}
Trans. Status 𝐓∈{0, 1}
𝑁= 2 𝜃( 𝑛 )
BSC(pw)
𝐘 𝑤
𝐓 =𝐷𝑒𝑐( 𝐘 𝑤 )
Willie’s (Best) Estimator 𝐓

7. 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.

8. This work
No shared secret
BSC(pb)
pb < pw
BSC(pw)

9. Alice’s Encoder M
𝐼𝑓 𝐓=0, 𝐗 = 𝟎
𝐼𝑓 𝐓=1, 𝐗 =𝐸𝑛𝑐(𝐌)
Bob’s Decoder 𝐗
𝐘 𝑏
BSC(pb)
𝐌 =𝐷𝑒𝑐( 𝐘 𝑏 ) 𝐌 T
1−𝜖 𝑟𝑒𝑙𝑖𝑎𝑏𝑙𝑒
Pr 𝐌 =𝐌 >1−𝜖
Message 𝐌∈{1, …, 𝑁}
Trans. Status 𝐓∈{0, 1}
𝑁= 2 𝜃( 𝑛 )
BSC(pw)
𝐘 𝑤
𝐓 =𝐷𝑒𝑐( 𝐘 𝑤 )
Willie’s (Best) Estimator 𝐓

10. Alice’s Transmission Status
Hypothesis Testing
Willie’s Estimate
Alice’s Transmission Status
𝛼=Pr 𝐓 =1 𝐓=0 , 𝛽=Pr 𝐓 =0 𝐓=1

11. Alice’s Transmission Status
Hypothesis Testing
Willie’s Estimate
Alice’s Transmission Status

12. Alice’s Transmission Status
Hypothesis Testing
Willie’s Estimate
Alice’s Transmission Status

13. Alice’s Transmission Status
Hypothesis Testing
Willie’s Estimate
Alice’s Transmission Status

14. Intuition
𝐓=0, 𝐲 𝑤 = 𝐳 𝑤 ~Binomial(𝑛, 𝑝 𝑤 )

15. Intuition
𝐓=0, 𝐲 𝑤 = 𝐳 𝑤 ~Binomial 𝑛, 𝑝 𝑤
𝑊ℎ𝑒𝑛 𝐓=1,

16. 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

17. Theorems 2 & 3 (Converse & achievability for reliable & deniable comm

18. Theorems 2 & 3
𝑝 𝑤
1/2
pb>pw
𝑝 𝑏
1/2

19. Theorems 2 & 3
𝑝 𝑤
1/2
𝑁=0
𝑝 𝑏
1/2

20. Theorems 2 & 3
𝑝 𝑤
pw=1/2
1/2
𝑝 𝑏
1/2

21. Theorems 2 & 3
𝑝 𝑤
1/2
(BSC(pb))
𝑝 𝑏
1/2

22. Theorems 2 & 3
𝑝 𝑤
1/2
pb=0
𝑝 𝑏
1/2

23. Theorems 2 & 3 𝑝 𝑤 𝑁 = 2 𝑂( 𝑛 log 𝑛 ) , 𝑛 𝑛 = 2 𝑂( 𝑛 log 𝑛 ) 1/2 𝑝 𝑏
𝑝 𝑏
1/2

24. Theorems 2 & 3
𝑝 𝑤
1/2
pw>pb
𝑝 𝑏
1/2

25. Theorems 2 & 3 𝑝 𝑤 𝑁 = 2 𝑂( 𝑛 ) 1/2 “Standard” IT inequalities +
𝑁 = 2 𝑂( 𝑛 )
1/2
“Standard” IT inequalities +
Wt(“most codewords”)<√n
(Thm 1)
𝑝 𝑏
1/2

26. Theorems 2 & 3 Main thm: 𝑝 𝑤 1/2 Achievable region 𝑁 = 2 Ω ( 𝑛 ) 𝑝 𝑏
𝑁 = 2 Ω ( 𝑛 )
𝑝 𝑏
1/2

27. log 𝑛 𝑛/2 ≈𝑛
logarithm of
# codewords n
𝑤 𝑡 𝐻 ( 𝒚 𝑤 )

28. log(# codewords)
𝑛𝐻( 𝑝 𝑤 )
𝐱 = 0
Pr 𝐙 𝑤 ⁡(𝑤 𝑡 𝐻 𝐲 𝑤 )
𝑂( 1 𝑛 )
𝑝 𝑤 𝑛
𝑝 𝑤 𝑛+𝑂( 𝑛 ) n
𝑤 𝑡 𝐻 ( 𝐲 𝑤 )
𝑝 𝑤 𝑛−𝑂( 𝑛 )

29. log(# codewords)
𝑛𝐻( 𝑝 𝑤 ∗𝜌)
𝑐 𝑛
Pr 𝐌, 𝐙 𝑤 ⁡(𝑤 𝑡 𝐻 𝐲 𝑤 )
𝑂( 1 𝑛 ) n
𝑤 𝑡 𝐻 ( 𝐲 𝑤 )
(𝑝 𝑤 ∗𝜌)𝑛−𝑂( 𝑛 )
(𝑝 𝑤 ∗𝜌)𝑛
(𝑝 𝑤 ∗𝜌)𝑛+𝑂( 𝑛 )

31. Theorem 3 – Reliability proof sketch
Weight 𝑂( 𝑛 )
Random code .
2 𝑂( 𝑛 ) codewords

32. Theorem 3 – Reliability proof sketch
Weight 𝑂( 𝑛 )
E(Intersection of 2 codewords) = O(1)
“Most” codewords “well-isolated” .

33. Theorem 3 – dmin decodingx +
𝑂( 𝑛 ) x’
Pr(x decoded to x’) < 2 −𝑂( 𝑛 )

34. Theorem 3 – Deniability proof sketch
Recall: want to show 𝑉 𝐏 0 , 𝐏 1 <𝜖

35. log(# codewords)
𝑛𝐻( 𝑝 𝑤 ∗𝜌)
𝑐 𝑛
Pr 𝐌, 𝐙 𝑤 ⁡(𝑤 𝑡 𝐻 𝐲 𝑤 )
𝑂( 1 𝑛 ) n
𝑤 𝑡 𝐻 ( 𝐲 𝑤 )
(𝑝 𝑤 ∗𝜌)𝑛−𝑂( 𝑛 )
(𝑝 𝑤 ∗𝜌)𝑛
(𝑝 𝑤 ∗𝜌)𝑛+𝑂( 𝑛 )

36. Theorem 3 – Deniability proof sketch
Recall: want to show 𝑉 𝐏 0 , 𝐏 1 <𝜖
𝐏 0
𝐏 1

37. Theorem 3 – Deniability proof sketch
log(# codewords) n
Pr 𝑪, 𝐙 𝑤 ⁡(𝑤 𝑡 𝐻 𝐲 𝑤 )
𝑂( 1 𝑛 )
𝑤 𝑡 𝐻 ( 𝐲 𝑤 )
(𝑝 𝑤 ∗𝜌)𝑛−𝑂( 𝑛 )
(𝑝 𝑤 ∗𝜌)𝑛
(𝑝 𝑤 ∗𝜌)𝑛+𝑂( 𝑛 )

38. Theorem 3 – Deniability proof sketch
logarithm of
# codewords n
𝑤 𝑡 𝐻 ( 𝒚 𝑤 )

39. Theorem 3 – Deniability proof sketch
𝑬 𝑪 (𝐏 1 )!!!
𝐏 0
𝐏 1

40. Theorem 3 – Deniability proof sketch
𝑉 𝐏 0 , 𝐏 1 ≤𝑉 𝐏 0 , 𝑬 𝑪 (𝐏 1 ) +𝑉 𝑬 𝑪 (𝐏 1 ), 𝐏 1
𝑬 𝑪 (𝐏 1 )!!!
𝐏 0
𝐏 1

41. Theorem 3 – Deniability proof sketch
𝑬 𝑪 (𝐏 1 )
𝐏 1

43. Theorem 3 – Deniability proof sketch
logarithm of
# codewords
𝑝 𝑤 𝑛
𝑝 𝑤 𝑛+𝑂( 𝑛 ) n
𝑤 𝑡 𝐻 ( 𝒚 𝑤 )
𝑝 𝑤 𝑛−𝑂( 𝑛 )

44. Theorem 4
logarithm of
# codewords n
𝑤 𝑡 𝐻 ( 𝒚 𝑤 )

45. Theorem 4 Too few codewords => Not deniable logarithm ofn
𝑤 𝑡 𝐻 ( 𝒚 𝑤 )

46. Summary

47. Summary