Reliable Deniable Communication: Hiding Messages from Noise

Pak Hou CheJoint Work with Sidharth Jaggi,

Mayank Bakshi and Madhi Jafari Siavoshani

Institute of Network CodingThe Chinese University of Hong Kong

Introduction

Is Alice talking to someone?

Alice

Willie

Bob

Introduction

Is Alice talking to someone?

Alice

Willie

Bob

Goal: decode message

Goal: detect Alice’s status

Goal: transmitreliably & deniably

Model

M

T

t

�⃑�

Alice’s Encoder

𝑁=2𝜃 (√𝑛)

Model

M

T

Message Trans. Status

BSC(pb) �̂�=𝐷𝑒𝑐 (�⃑�𝑏)�⃑�𝑏�⃑�

Alice’s EncoderBob’s Decoder

𝑁=2𝜃 (√𝑛)

�̂�

Model

M

T

Message Trans. Status

BSC(pb) �̂�=𝐷𝑒𝑐 (�⃑�𝑏)�⃑�𝑏�⃑�

Alice’s EncoderBob’s Decoder

𝑁=2𝜃 (√𝑛)

�̂�

Model

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 Estimator

�̂�

�̂�

Model

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 Estimator

�̂�

�̂�

Model

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 Estimator

�̂�

�̂�

Asymmetry pb < pw

Prior Work

Alice Bob

Willie

Shared secret ([1] Bash, Goeckel & Towsley)

[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.

Our Case

Alice Bob

Willie

Asymmetry pb < pw

Hypothesis TestingWillie’s Estimation

Alice’s Transmit

StatusSilent

Transmit

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

Hypothesis TestingWillie’s Estimation

Alice’s Transmit

StatusSilent

Transmit

Hypothesis TestingWillie’s Estimation

Alice’s Transmit

StatusSilent

Transmit

Hypothesis TestingWillie’s Estimation

Alice’s Transmit

StatusSilent

Transmit

Intuition

𝐓=0 , 𝐲𝑤=�⃑�𝑤 Binomial(𝑛 ,𝑝𝑤)

Intuition

Theorem 1(high deniability => low weight codewords)

Too  many   codewords  with   weight  “much ” greater   than𝑐 √𝑛 , h𝑡 𝑒𝑛 h𝑡 𝑒𝑠𝑦𝑠𝑡𝑒𝑚𝑖𝑠   “not   very”   deniable

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

Theorem 2 & 3

𝑝𝑏

𝑝𝑤

0 1/2

1/2

pb>pw

Theorem 2 & 3

𝑝𝑏

𝑝𝑤

0 1/2

1/2

𝑁=0

Theorem 2 & 3

𝑝𝑏

𝑝𝑤

0 1/2

1/2pw=1/2

Theorem 2 & 3

𝑝𝑏

𝑝𝑤

0 1/2

1/2

𝑁 ≤2(1−𝐻 (𝑝𝑤 )+𝜖)𝑛

Theorem 2 & 3

𝑝𝑏

𝑝𝑤

0 1/2

1/2

𝑁 ≥2(1−𝐻 (𝑝𝑤 )−𝜖)𝑛

Theorem 2 & 3

𝑝𝑏

𝑝𝑤

0 1/2

1/2

pb=1/2

Theorem 2 & 3

𝑝𝑏

𝑝𝑤

0 1/2

1/2𝑁=2𝑂 (√𝑛 log𝑛)

Theorem 2 & 3

𝑝𝑏

𝑝𝑤

0 1/2

1/2

𝑁=2𝑂 (√𝑛 log𝑛) ,( 𝑛√𝑛)=2𝑂 (√𝑛 log𝑛)

Theorem 2 & 3

𝑝𝑏

𝑝𝑤

0 1/2

1/2

𝑁=2Ω(√𝑛 log𝑛)

Theorem 2 & 3

𝑝𝑏

𝑝𝑤

0 1/2

1/2

pw>pb

Theorem 2 & 3

𝑝𝑏

𝑝𝑤

0 1/2

1/2𝑁=2𝑂 (√𝑛)

Theorem 2 & 3

𝑝𝑏

𝑝𝑤

0 1/2

1/2

Achievable region

Theorem 3 – Proof Idea

• Recall: want to show

Theorem 3 – Proof Idea

𝑤𝑡𝐻 (𝒚𝑤 )

0 n𝑝𝑤𝑛+𝑂 (√𝑛)𝑝𝑤𝑛

logarithm of# codewords

log ( 𝑛𝑛/2)≈𝑛

Theorem 3 – Proof Idea

𝑤𝑡𝐻 (𝒚𝑤 )

0 n

logarithm of# codewords

Theorem 3 – Proof Idea

𝑤𝑡𝐻 (𝒚𝑤 )

0 n

logarithm of# codewords

Too few codewords=> Not deniable (Thm4)

Theorem 3 – Proof Idea

𝑤𝑡𝐻 (𝒚𝑤 )

0 n

logarithm of# codewords

𝑂 (√𝑛)

Theorem 3 – Proof Idea

𝑤𝑡𝐻 (𝒚𝑤 )

0 n

logarithm of# codewords

Theorem 3 – Proof Idea

𝑝 (𝐲𝑤)

Logarithm of# codewords

Theorem 3 – Proof Idea

• Recall: want to show

𝐏0 𝐏1

Theorem 3 – Proof Idea

𝐏0 𝐏1

!!!

Theorem 3 – Proof Idea

𝐏0 𝐏1

!!!

Theorem 3 – Proof Idea

• Chernoff bound is weak• Other concentration inequality

𝐏1𝑬𝑪(𝐏¿¿1)¿

Theorem 3 – Proof Idea

Theorem 3 – Proof Idea

𝑤𝑡𝐻 (𝒚𝑤 )

0 n

logarithm of# codewords

Theorem 3 – Proof Idea

𝑤𝑡𝐻 (𝒚𝑤 )

0 n𝑝𝑤𝑛+𝑂 (√𝑛)𝑝𝑤𝑛

logarithm of# codewords

Theorem 3 – Proof Idea

𝑤𝑡𝐻 (𝒚𝑤 )

0 n𝑝𝑤𝑛+𝑂 (√𝑛)𝑝𝑤𝑛

logarithm of# codewords

Theorem 3 – Sketch Proof

# codewords of “type”

𝑇 1𝑇 2

𝑇 3

Theorem 3 – Sketch Proof

Theorem 3 – Sketch Proof

Theorem 3 – Sketch Proof

• w.p.

Theorem 3 – Sketch Proof

• w.p.

Theorem 3 – Sketch Proof

• w.p. • close to w.p.

Theorem 3 – Sketch Proof

• w.p. • close to w.p. • , w.h.p.

Summary

𝑝𝑏

𝑝𝑤

0 1/2

1/2

Summary

𝑝𝑏

𝑝𝑤

0 1/2

1/2