Reliable Deniable Communication: Hiding Messages in Noise

The Chinese University of Hong Kong

The Institute of Network Coding

Pak Hou CheMayank BakshiSidharth Jaggi

Alice

Reliability

Bob

Willie(the Warden)

Reliability

Deniability

AliceBob

M

T

t

�⃑�

Alice’s Encoder

𝑁=2𝜃 (√𝑛)

M

T

Message Trans. Status

BSC(pb) �̂�=𝐷𝑒𝑐 (�⃑�𝑏)�⃑�𝑏�⃑�

Alice’s EncoderBob’s Decoder

𝑁=2𝜃 (√𝑛)

�̂�

M

T

Message Trans. Status

BSC(pb) �̂�=𝐷𝑒𝑐 (�⃑�𝑏)�⃑�𝑏�⃑�

Alice’s EncoderBob’s Decoder

BSC(pw)

�̂�=𝐷𝑒𝑐 (�⃑�𝑤)

�⃑�𝑤

𝑁=2𝜃 (√𝑛)

Willie’s (Best) Estimator

�̂�

�̂�

Hypothesis Testing Willie’s Estimate

Alice’s Transmission

Status

𝛼=Pr ( �̂�=1|𝐓=0 ) , 𝛽=Pr ( �̂�=0|𝐓=1 )

Hypothesis Testing Willie’s Estimate

Alice’s Transmission

Status

• Want:

Hypothesis Testing Willie’s Estimate

Alice’s Transmission

Status

• Want: • Known: for opt. estimator

Hypothesis Testing Willie’s Estimate

Alice’s Transmission

Status

• Want: • Known: for opt. estimator• , ( w.h.p.)

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

Capacity = bits

bits

This workNo shared secret

BSC(pb)

BSC(pw)

pb < pw

Intuition

𝐓=0 , 𝐲𝑤=�⃑�𝑤 Binomial(𝑛 ,𝑝𝑤)

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)

Too  many   codewords  with   weight  “much ” greater   than𝑐 √𝑛 , then   the   system   is  “ not   very ” deniable

Theorem 2 (Converse)

• , if • if

Theorem 3 – Reliability

• Random codebook ( i.i.d. ) )• minimum distance decoder• For ,

𝑤𝑡𝐻 (𝒚𝑤 )

0 n

logarithm of# binary vectors

log ( 𝑛𝑛/2)≈𝑛

𝑤𝑡𝐻 (𝐲𝑤)0 n𝑝𝑤𝑛+𝑂 (√𝑛)𝑝𝑤𝑛

log(# vectors)

Pr�⃑�𝑤

(𝑤𝑡𝐻 (𝐲𝑤 ))

𝑂 (1/√𝑛)

𝑛𝐻 (𝑝𝑤 )

𝐱=0⃗

log(# vectors)

𝑛𝐻 (𝑝𝑤∗𝜌 )

𝑐 √𝑛

log(# codewords)

𝑛𝐻 (𝑝𝑤∗𝜌 )

𝑐 √𝑛

𝑤𝑡𝐻 (𝐲𝑤)0 n

(𝑝¿¿𝑤∗𝜌)𝑛+𝑂(√𝑛)¿(𝑝¿¿𝑤∗𝜌 )𝑛¿(𝑝¿¿𝑤∗𝜌 )𝑛−𝑂(√𝑛)¿

log(# vectors)

Pr𝐌 ,𝐙𝑤

(𝑤𝑡𝐻 (𝐲𝑤 ))

𝑛𝐻 (𝑝𝑤∗𝜌 )

𝑐 √𝑛

𝑂 (1/√𝑛)

• Recall: want to show

Theorem 3 – Deniability proof sketch

• Recall: want to show

𝐏0 𝐏1

Theorem 3 – Deniability proof sketch

0 n

log(# vectors)

Theorem 3 – Deniability proof sketch

𝐏0 𝐏1

!!!

Theorem 3 – Deniability proof sketch

𝐏0 𝐏1

!!!

Theorem 3 – Deniability proof sketch

𝐏1𝑬𝑪(𝐏¿¿1)¿

Theorem 3 – Deniability proof sketch

with high probability

𝑤𝑡𝐻 (𝒚𝑤 )

0 n𝑝𝑤𝑛+𝑂 (√𝑛)𝑝𝑤𝑛

logarithm of# vectors

Theorem 3 – Deniability proof sketch

𝑤𝑡𝐻 (𝒚𝑤 )

0 n𝑝𝑤𝑛+𝑂 (√𝑛)𝑝𝑤𝑛

logarithm of# vectors

Theorem 3 – Deniability proof sketch

# codewords of “type”

𝑇

Theorem 3 – Deniability proof sketch

Theorem 3 – Deniability proof sketch

𝑇

Theorem 3 – Deniability proof sketch

𝑇

Theorem 3 – Deniability proof sketch

𝑇

Theorem 3 – Deniability proof sketch

𝑇

• w.p.

Theorem 3 – Deniability proof sketch

𝑇

• w.p.

• close to w.p. • , w.h.p.

Theorem 3 – Deniability proof sketch

𝑇

Theorem 4

𝑤𝑡𝐻 (𝒚𝑤 )

0 n

logarithm of# codewords

𝑤𝑡𝐻 (𝒚𝑤 )

0 n

logarithm of# codewords

Too few codewords=> Not deniable

Theorem 4

Summary

𝑝𝑏

𝑝𝑤

0 1/2

1/2 • Thm 1 & 2 Converse (Upper Bound)

• Thm 3 Achievability• Thm 4 Lower Bound