Fisher Information Determines Capacity of ε-secure...

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Fisher Information Determines Capacity of ε -secure Steganography Tomáš Filler and Jessica Fridrich Dept. of Electrical and Computer Engineering SUNY Binghamton, New York 11th Information Hiding, Darmstadt, Germany, 2009

Transcript of Fisher Information Determines Capacity of ε-secure...

Page 1: Fisher Information Determines Capacity of ε-secure Steganographydde.binghamton.edu/filler/pdf/Fill09ihw-slides.pdf · 2009. 6. 15. · Presentation Outline 1 ASSUMPTIONS model of

Fisher Information Determines Capacityof ε-secure Steganography

Tomáš Filler and Jessica Fridrich

Dept. of Electrical and Computer Engineering

SUNY Binghamton, New York

11th Information Hiding, Darmstadt, Germany, 2009

Page 2: Fisher Information Determines Capacity of ε-secure Steganographydde.binghamton.edu/filler/pdf/Fill09ihw-slides.pdf · 2009. 6. 15. · Presentation Outline 1 ASSUMPTIONS model of

Perfect vs. Imperfect Steganography

Stegosystems can be divided into two classes:

Perfectly secure stegosyst.

KL diverg. cover & stegoDKL(P ||Q) = 0

Detector does NOT exist.

Secure capacity is linear.

Communication rate isPOSITIVE.

Imperfect stegosyst.

DKL(P ||Q) = ε

Detector DOES exist.

Sec. capacity is SUBlinear.

Communication rate isZERO.

perfectly secure stegosystems exist for artificial coversources

all known stegosystems for digital media are imperfect

Filler, Fridrich Fisher Information Determines Capacity of ε-secure Steganography 2 of 18

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Secure Capacity of ε-secure Stegosystems

From Taylor expansion ... (nβ = # of changes)

DKL

(

P (n)||Q(n)β

)

=1

2nβ 2

I+O(β 3) = ε

1

2nβ 2

I≈ ε ⇒ nβ ≈√

1

I

√2εn

I ... Fisher information (rate) at β = 0.

Capacity of ε-secure stegosystems scales as r√

n.

Root rate: (more refined measure of capacity)

r ≈ 1√I

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Is Root Rate Useful?

Root rate: (more refined measure of capacity)

r ≈ 1√I

Fisher information rate can be expressed in a closed-form.

Applications:

BENCHMARKING - compare stegosystems by their rootrates.

STEGANOGRAPHY DESIGN - maximize root ratew.r.t. embedding operation for fixed cover source.

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Presentation Outline

1 ASSUMPTIONSmodel of cover objects, model of embedding impact

2 STEGANOGRAPHIC CAPACITY & ROOT RATEformal connection between detectability and root rate

3 APPLICATIONcomparison of LSB and ±1 embedding in spatial domain

4 CONCLUSION

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ASSUMPTIONS

Page 7: Fisher Information Determines Capacity of ε-secure Steganographydde.binghamton.edu/filler/pdf/Fill09ihw-slides.pdf · 2009. 6. 15. · Presentation Outline 1 ASSUMPTIONS model of

Assumptions Summary

1 COVER SOURCE - Markov Chainfirst order Markov Chain with tran. prob. mat. A = (aij)

2 EMBEDDING ALGORITHM - MI embeddingindependent substitution of states (next slide)

3 STEGOSYSTEM - ε-secure (imperfect)stegosystem is ε-SECURE

MC

HMC

i

i ′

j

j ′

k

k ′

l

l ′

aij ajk akl ...cover

stego

embedding

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Mutually Independent Embedding Operation

Impact of embedding is modeled as probabilistic mappingacting on each cover element independently - MI embedding.

Pr(Yl = j |Xl = i) = bij(β )Xl ... l-th cover elementYl ... l-th stego elementβ ... change rate (rel. payload)

Matrix B = (bij ) is “transition probability matrix” for β ≥ 0.

LSB embedding:

= 1−β = β

B =

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Mutually Independent Embedding Operation

Impact of embedding is modeled as probabilistic mappingacting on each cover element independently - MI embedding.

Pr(Yl = j |Xl = i) = bij(β )Xl ... l-th cover elementYl ... l-th stego elementβ ... change rate (rel. payload)

Matrix B = (bij ) is “transition probability matrix” for β ≥ 0.

LSB embedding:

= 1−β = β

B = = I+ βC = + β ·

= 1 = −1

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Page 10: Fisher Information Determines Capacity of ε-secure Steganographydde.binghamton.edu/filler/pdf/Fill09ihw-slides.pdf · 2009. 6. 15. · Presentation Outline 1 ASSUMPTIONS model of

Mutually Independent Embedding Operation

Impact of embedding is modeled as probabilistic mappingacting on each cover element independently - MI embedding.

Pr(Yl = j |Xl = i) = bij(β )Xl ... l-th cover elementYl ... l-th stego elementβ ... change rate (rel. payload)

Matrix B = (bij ) is “transition probability matrix” for β ≥ 0.

±1 embedding:

= 1−β = β2

= β

B = = I+ βC = + β ·

= 1 = 0.5 = −1

Filler, Fridrich Fisher Information Determines Capacity of ε-secure Steganography 8 of 18

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Mutually Independent Embedding Operation

Impact of embedding is modeled as probabilistic mappingacting on each cover element independently - MI embedding.

Pr(Yl = j |Xl = i) = bij(β )Xl ... l-th cover elementYl ... l-th stego elementβ ... change rate (rel. payload)

Matrix B = (bij ) is “transition probability matrix” for β ≥ 0.

F5 embedding:

= 1−β = β = 1

B = = I+ βC = + β ·

= −1

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Page 12: Fisher Information Determines Capacity of ε-secure Steganographydde.binghamton.edu/filler/pdf/Fill09ihw-slides.pdf · 2009. 6. 15. · Presentation Outline 1 ASSUMPTIONS model of

Mutually Independent Embedding Operation

Impact of embedding is modeled as probabilistic mappingacting on each cover element independently - MI embedding.

Pr(Yl = j |Xl = i) = bij(β )Xl ... l-th cover elementYl ... l-th stego elementβ ... change rate (rel. payload)

Matrix B = (bij ) is “transition probability matrix” for β ≥ 0.

Assumption:

B(β ) = I+βC

C describes the inner workings of the embedding algorithm.

Filler, Fridrich Fisher Information Determines Capacity of ε-secure Steganography 8 of 18

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STEGANOGRAPHIC

CAPACITY

&√

RATE

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Square Root Law

1 If nβn√n→ 0 then the stegosyst. are asymptotically secure.

2 If nβn√n→ +∞ then arbitrarily accurate stego detectors

exist.

For fixed level of security,

nβn√n

< C ⇒ βn → 0 as n → ∞.

[Filler, Ker, Fridrich, “The Square Root Law of Steganographic Capacity forMarkov Covers”, Proc. SPIE, 2009]

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The Best Possible Steganalyzer

Hypothesis test:

H0 : β = 0 decide cover

H1 : β > 0 (known) decide stego.

Detection statistics (log-likelihood ratio):

L(n)β (X ) =

1√n

lnQ

(n)β (X )

P (n)(X )

stego distribution withn elements

cover distribution withn elementschange rate

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The Root Rate

For small β , L(n)β (X ) is Gaussian (Local Asymptotic Normality).

−µ µ0

σ2 σ2

cover stegoµ =

√nβ 2

I/2

σ2 = β 2I

Fisher information rate I = limn→∞1n

d2

dβ2 DKL

(

P (n)||Q(n)β

)∣

β=0

Deflection coefficient: d2 = (−µ −µ)2/σ2 = nβ 2I < ε

nβ < r√

εn r =

1

IROOT RATE

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Closed Form for Fisher Information Rate

Theorem (Fisher Information Rate)

Let A = (aij) define the MC cover model and B = I+ βC represent

MI embedding. Then, the Fisher information rate I exists and can

be written as

I = cTFc.

c = (c11, . . . ,cNN)T is column vector obtained directly from C.

Matrix F ∈ RN2×N2

, F = f (A) and does not depend on B.

Maximizing root rate ⇒ minimizing I w.r.t. C.

Closed form for I enables us to

compare stegosystems

maximize capacity w.r.t. embedding operation C.

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APPLICATIONS

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Markov Model of Spatial Domain Images

Image databases: CAMRAW, NRCS, NRCS-JPEG70.

Spatial domain 8-bit grayscale images: A = (aij) ∈ R256×256

aij =1

Zie−(|i−j |/τ)γ

0

255

0 255

−5

−10

−15

CAMRAW - log aij

−10

−5

0

0 127 255

j

log a127,j

data

model

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Convex Combination of LSB and ±1 Embedding

Use ±1 embedding in first λn pixels and LSB embedding inthe rest.

1

2

0 - LSB 1 - ±1

NRCS

NRCS-JPEG

CAMRAW

λ

r(λ)

Root rate r(λ ):

cλ = λc±1 +(1−λ )cLSB

r(λ ) =

1

cTλ Fcλ

Higher root rate = better method.

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Convex Combination of LSB and ±1 Embedding

Use ±1 embedding in first λn pixels and LSB embedding inthe rest.

1

2

0 - LSB 1 - ±1

NRCS

NRCS-JPEG

CAMRAW

λ

r(λ)

Pevný, Bass, Fridrich:Steganalysis bySubtractive PixelAdjacency Matrix,submitted to ACMMM&SEC 2009,Princeton, NJ

Higher root rate = better method.

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±1 is Asymptotically Optimal

What embedding method minimizes Fisher information rateand modifies cover by at most 1?

0.2

0.4

0.6

0.8

1 254

±1 emb.

NRCS

i

viClass of embeddingoperations:

1− v1

v1

1− v4

v4

±1 embedding is asymptotically optimal asnumber of grayscales N → ∞.

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Conclusion and Future Directions

Steganographic capacity of ε-SECURE stegosystems ≈ r√

n.

We coin a new term for constant r - the Root Rate.

Root rate

is determined by Fisher information rate.

can be expressed in a closed-form amenable tooptimization (under mentioned assumptions).

was used to compare spatial domain stegosystems.

Future work: use this framework for JPEG images.

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Conclusion and Future Directions

Steganographic capacity of ε-SECURE stegosystems ≈ r√

n.

We coin a new term for constant r - the Root Rate.

Root rate

is determined by Fisher information rate.

can be expressed in a closed-form amenable tooptimization (under mentioned assumptions).

was used to compare spatial domain stegosystems.

Future work: use this framework for JPEG images.

Thank you!

[email protected], Fridrich Fisher Information Determines Capacity of ε-secure Steganography 18 of 18