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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.
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Reliable Deniable Communication: Hiding Messages in Noise Pak HouChe MayankBakshi SidharthJaggi 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 t
Alice’s Encoder M Bob’s Decoder BSC(pb) T Message Trans. Status
Alice’s Encoder M Bob’s Decoder BSC(pb) T Message Trans. Status BSC(pw) Willie’s (Best) Estimator
Hypothesis Testing • Want:
Hypothesis Testing • Want: • Known: for opt. estimator
Hypothesis Testing • Want: • Known: for opt. estimator • , (w.h.p.)
Bash, Goeckel & Towsley [1] Shared secret bits AWGN channels Capacity = 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)
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) • , if • if
Theorem 3 – Reliability • Random codebook ( i.i.d. ) ) • minimum distance decoder • For ,
logarithm of # binary vectors 0 n
log(# vectors) n 0
log(# vectors) n 0
Theorem 3 – Deniability proof sketch • Recall: want to show
Theorem 3 – Deniability proof sketch • Recall: want to show
Theorem 3 – Deniability proof sketch log(# vectors) n 0
Theorem 3 – Deniability proof sketch with high probability
Theorem 3 – Deniability proof sketch logarithm of # vectors 0 n
Theorem 3 – Deniability proof sketch logarithm of # vectors 0 n
Theorem 3 – Deniability proof sketch # codewords of “type”
Theorem 3 – Deniability proof sketch • w.p. • close to w.p. • , w.h.p.
Theorem 4 logarithm of # codewords 0 n
Theorem 4 Too fewcodewords => Not deniable logarithm of # codewords 0 n
Summary 1/2 • Thm 1 & 2 Converse • (Upper Bound) • Thm 3 Achievability • Thm 4 Lower Bound 0 1/2