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 Alice’s Encoder

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

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

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

Hypothesis Testing Willie’s Estimate Alice’s 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. 448–452. 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