1. Reliable Deniable Communication: Hiding Messages in Noise The Chinese University of Hong Kong The Institute of Network Coding Pak Hou Che Mayank Bakshi Sidharth Jaggi

2. Alice Reliability Bob

3. Willie (the Warden) Reliability Deniability Alice Bob

4. M T Alice’s Encoder

5. M T BSC(p b ) Alice’s Encoder Bob’s Decoder

6. M T BSC(p b ) Alice’s Encoder Bob’s Decoder BSC(p w ) Willie’s (Best) Estimator

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

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

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

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

11. 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

12. This work No shared secret BSC(p b ) BSC(p w ) p b < p w

13. Intuition

15. 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

16. Theorem 1 (wt(c.w.)) (high deniability => low weight codewords)

17. Theorem 2 (Converse)

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

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

20. Theorems 2 & 3 0 1/2

21. Theorems 2 & 3 0 1/2 p w =1/2

22. Theorems 2 & 3 0 1/2 (BSC(p b ))

23. Theorems 2 & 3 0 1/2 p b =0

24. Theorems 2 & 3 0 1/2

25. Theorems 2 & 3 0 1/2 p w >p b

26. Theorems 2 & 3 0 1/2

27. Theorems 2 & 3 0 1/2 Main thm:

29. Theorem 3 – Reliability

30. Theorem 3 – Reliability proof sketch...... 1000001000000000100100000010000000100 0001000000100000010000000010000000001 0010000100000001010010000000100010011 0000100000010000000000010000000010000 Random code

31. Theorem 3 – Reliability proof sketch...... 1000001000010000100100000010000000100 0001000000100000010000000010000000001 0010000100000001010010000000100010011 0000100000010000000000010000000010000

32. Theorem 3 – Reliability proof sketch...... 1000001000010000100100000010000000100 0001000000100000010000000010000000001 0010000100000001010010000000100010011 0000100000010000000000010000000010000 E(Intersection of 2 codewords) = O(1) “Most” codewords “well-isolated”

33. Theorem 3 – d min decoding + x x’

34. 0n logarithm of # binary vectors

35. 0 n log(# vectors)

37. log(# codewords)

38. 0 n log(# vectors)

39. Theorem 3 – Deniability proof sketch

41. 0 n log(# vectors) Theorem 3 – Deniability proof sketch

42. 0n logarithm of # codewords Theorem 3 – Deniability proof sketch

47. 0n logarithm of # vectors Theorem 3 – Deniability proof sketch

48. 0n logarithm of # vectors Theorem 3 – Deniability proof sketch

56. Theorem 4 0n logarithm of # codewords

57. 0n logarithm of # codewords Too few codewords => Not deniable Theorem 4

58. Summary 0 1/2 Thm 1 & 2 Converse (Upper Bound) Thm 3 Achievability Thm 4 Lower Bound

59. Summary