Reliable Deniable Communication: Hiding Messages in Noise

Mayank Bakshi Mahdi Jafari Siavoshani

ME

Sidharth Jaggi

The Chinese University of Hong Kong

The Institute of Network Coding

Pak Hou (Howard) Che

Alice

Reliability

Bob

Willie(the Warden)

Reliability

Deniability

AliceBob

Willie-sky

Reliability

Deniability

AliceBob

M

T

t

�⃑�

Alice’s Encoder

𝑁=2𝜃 (√𝑛)

M

T

Message Trans. Status

BSC(pb) �̂�=𝐷𝑒𝑐 (�⃑�𝑏)�⃑�𝑏�⃑�

Alice’s Encoder

Bob’s Decoder

𝑁=2𝜃 (√𝑛)

�̂�

M

T

Message Trans. Status

BSC(pb) �̂�=𝐷𝑒𝑐 (�⃑�𝑏)�⃑�𝑏�⃑�

Alice’s Encoder

Bob’s Decoder

BSC(pw)

�̂�=𝐷𝑒𝑐 (�⃑�𝑤)

�⃑�𝑤

𝑁=2𝜃 (√𝑛)

Willie’s (Best) Estimator

�̂�

�̂�

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.

€

O n .log(n)( ) bits

AWGN channels

But capacity only

€

O n( ) bits!

This workNo shared secret

BSC(pb)

BSC(pw)

pb < pw

Wicked Willie(s) Base-station Bob

Aerial Alice

Directional antenna

Steganography: Other work

Steganography: Other work

Other work: “Common” modelShared secret key

Capacity O(n) message bitsInformation-theoretically tight characterization(Gel’fand-Pinsker/Dirty paper coding)

O(n.log(n)) bits (not optimized)

[2] Y. Wang and P. Moulin, "Perfectly Secure Steganography: Capacity, Error Exponents, and Code Constructions," IEEE Trans. on Information Theory, special issue on Information Theoretic Security, June 2008

Stegotext(covertext,message,key)

Message,Covertext

No noise

d(stegotext,covertext) “small”

Other work: Square-root “law”(“empirical”)

•“Steganographic capacity is a loosely-defined concept, indicating the size of payload whichmay securely be embedded in a cover object using a particular embedding method. What constitutes “secure” embedding is a matter for debate, but we will argue that capacity should grow only as the square root of the cover size under a wide range of definitions of security.” [3]

•“Thanks to the Central Limit Theorem, the more covertext we give the warden, the better he will be able to estimate its statistics, and so the smaller the rate at which [the steganographer] will be able to tweak bits safely.” [4]

[3] A. Ker, T. Pevny`, J. Kodovsky`, and J. Fridrich, “The square root law of steganographic capacity,” in Proceedings of the 10th ACM workshop on Multimedia and security. ACM, 2008, pp. 107–116.[4] R. Anderson, “Stretching the limits of steganography,” in Information Hiding, 1996, pp. 39–48.

•“[T]he reference to the Central Limit Theorem... suggests that a square root relationship should be considered. “ [3]

M

T

Message Trans. Status

BSC(pb) �̂�=𝐷𝑒𝑐 (�⃑�𝑏)�⃑�𝑏�⃑�

Alice’s Encoder

Bob’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

Hypothesis Testing Willie’s Estimate

Alice’s Transmission

Status

Hypothesis Testing Willie’s Estimate

Alice’s Transmission

Status

Intuition

𝐓=0 , 𝐲𝑤=�⃑�𝑤 Binomial(𝑛 ,𝑝𝑤)

Intuition

Theorem 1 (Wt(c.w.))(high deniability => low weight codewords)

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

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

Theorems 2 & 3

𝑝𝑏

𝑝𝑤

0 1/2

1/2

pb>pw

Theorems 2 & 3

𝑝𝑏

𝑝𝑤

0 1/2

1/2

𝑁=0

(Symmetrizability)

Theorems 2 & 3

𝑝𝑏

𝑝𝑤

0 1/2

1/2pw=1/2

Theorems 2 & 3

𝑝𝑏

𝑝𝑤

0 1/2

1/2

€

N ≈ 2(1−H (pb ))n

(BSC(pb))

Theorems 2 & 3

𝑝𝑏

𝑝𝑤

0 1/2

1/2

pb=0

Theorems 2 & 3

𝑝𝑏

𝑝𝑤

0 1/2

1/2

𝑁=2𝑂 (√𝑛 log𝑛) ,( 𝑛√𝑛)=2𝑂 (√𝑛 log𝑛)

Theorems 2 & 3

𝑝𝑏

𝑝𝑤

0 1/2

1/2

pw>pb

Theorems 2 & 3

𝑝𝑏

𝑝𝑤

0 1/2

1/2𝑁=2𝑂 (√𝑛)

“Standard” IT inequalities+

Wt(“most codewords”)<√n(Thm 1)

Theorems 2 & 3

𝑝𝑏

𝑝𝑤

0 1/2

1/2

Main thm:

𝑤𝑡𝐻 (𝒚𝑤 )

0 n

logarithm of# codewords

log ( 𝑛𝑛/2)≈𝑛

𝑤𝑡𝐻 (𝐲𝑤)0 n𝑝𝑤𝑛+𝑂 (√𝑛)𝑝𝑤𝑛

log(# codewords)

Pr�⃑�𝑤

(𝑤𝑡𝐻 (𝐲𝑤 ))

𝑂 (1/√𝑛)

𝑛𝐻 (𝑝𝑤 )

𝐱=0⃗

𝑤𝑡𝐻 (𝐲𝑤)0 n

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

log(# codewords)

Pr𝐌 ,𝐙𝑤

(𝑤𝑡𝐻 (𝐲𝑤 ))

𝑛𝐻 (𝑝𝑤∗𝜌)

𝑐 √𝑛

𝑂 (1/√𝑛)

Theorem 3 – Reliability proof sketch

0 n

Noise magnitude >> Codeword weight!!!

Theorem 3 – Reliability proof sketch

.

.

.

1000001000000000100100000010000000100

0001000000100000010000000010000000001

0010000100000001010010000000100010011

0000100000010000000000010000000010000

Random code

2O(√n) codewords

Weight O(√n)

Theorem 3 – Reliability proof sketch

.

.

.

1000001000010000100100000010000000100

0001000000100000010000000010000000001

0010000100000001010010000000100010011

0000100000010000000000010000000010000

•E(Intersection of 2 codewords) = O(1)

Weight O(√n)

•Pr(dmin(x) < c√n) < 2-O(√n)

•“Most” codewords “well-isolated”

Theorem 3 – dmin decoding

•Pr(x decoded to x’) < 2-O(√n)

+ O(√n)

x

x’

• Recall: want to show

Theorem 3 – Deniability proof sketch

Theorem 4 – unexpected detour

𝑤𝑡𝐻 (𝒚𝑤 )

0 n

logarithm of# codewords

𝑤𝑡𝐻 (𝒚𝑤 )

0 n

logarithm of# codewords

Too few codewords=> Not deniable

Theorem 4 – unexpected detour

𝑤𝑡𝐻 (𝐲𝑤)0 n

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

log(# codewords)

Pr𝐌 ,𝐙𝑤

(𝑤𝑡𝐻 (𝐲𝑤 ))

𝑛𝐻 (𝑝𝑤∗𝜌)

𝑐 √𝑛

𝑂 (1/√𝑛)

• Recall: want to show

𝐏0 𝐏1

Theorem 3 – Deniability proof sketch

0 n

log(# codewords)

𝑛𝐻 (𝑝𝑤 )

Theorem 3 – Deniability proof sketch

𝑤𝑡𝐻 (𝒚𝑤 )

0 n

logarithm of# codewords

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

𝑤𝑡𝐻 (𝒚𝑤 )

0 n𝑝𝑤𝑛+𝑂 (√𝑛)𝑝𝑤𝑛

logarithm of# codewords

Theorem 3 – Deniability proof sketch

# codewords of “type”

𝑇 1𝑇 2

𝑇 3

Theorem 3 – Deniability proof sketch

Theorem 3 – Deniability proof sketch

Theorem 3 – Deniability proof sketch

Theorem 3 – Deniability proof sketch

• w.p.

Theorem 3 – Deniability proof sketch

• w.p.

Theorem 3 – Deniability proof sketch

• w.p. • close to w.p.

Theorem 3 – Deniability proof sketch

• w.p. • close to w.p. • , w.h.p.

Theorem 3 – Deniability proof sketch

Summary