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 Alice Reliability Bob Willie (the Warden) Reliability Deniability Alice Bob M T Alices Encoder M T BSC(p b ) Alices Encoder Bobs Decoder M T BSC(p b ) Alices Encoder Bobs Decoder BSC(p w ) Willies (Best) Estimator Hypothesis Testing Willies Estimate Alices Transmission Status Hypothesis Testing Willies Estimate Alices Transmission Status Hypothesis Testing Willies Estimate Alices Transmission Status Hypothesis Testing Willies Estimate Alices Transmission Status 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. 448452. AWGN channels This work No shared secret BSC(p b ) BSC(p w ) p b < p w 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) Theorem 2 (Converse) Theorems 2 & 3 (Converse & achievability for reliable & deniable comm.) Theorems 2 & 3 0 1/2 p b >p w Theorems 2 & 3 0 1/2 Theorems 2 & 3 0 1/2 p w =1/2 Theorems 2 & 3 0 1/2 (BSC(p b )) Theorems 2 & 3 0 1/2 p b =0 Theorems 2 & 3 0 1/2 Theorems 2 & 3 0 1/2 p w >p b Theorems 2 & 3 0 1/2 Theorems 2 & 3 0 1/2 Main thm: Theorem 3 Reliability Theorem 3 Reliability proof sketch Random code Theorem 3 Reliability proof sketch Theorem 3 Reliability proof sketch E(Intersection of 2 codewords) = O(1) Most codewords well-isolated Theorem 3 d min decoding + x x 0n logarithm of # binary vectors 0 n log(# vectors) log(# codewords) 0 n log(# vectors) Theorem 3 Deniability proof sketch 0 n log(# vectors) Theorem 3 Deniability proof sketch 0n logarithm of # codewords Theorem 3 Deniability proof sketch 0n logarithm of # vectors Theorem 3 Deniability proof sketch 0n logarithm of # vectors Theorem 3 Deniability proof sketch Theorem 4 0n logarithm of # codewords 0n logarithm of # codewords Too few codewords => Not deniable Theorem 4 Summary 0 1/2 Thm 1 & 2 Converse (Upper Bound) Thm 3 Achievability Thm 4 Lower Bound Summary