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
Perfect vs. Imperfect Steganography Stegosystems can be divided into two classes: Perfectly secure stegosyst. Imperfect stegosyst. KL diverg. cover & stego D KL ( P || Q ) = ε D KL ( P || Q ) = 0 Detector does NOT exist. Detector DOES exist. Secure capacity is linear. Sec. capacity is SUBlinear. Communication rate is Communication rate is POSITIVE. ZERO. perfectly secure stegosystems exist for artificial cover sources all known stegosystems for digital media are imperfect Fisher Information Determines Capacity of ε -secure Steganography Filler, Fridrich 2 of 18
Secure Capacity of ε -secure Stegosystems From Taylor expansion ... ( n β = # of changes) = 1 � � P ( n ) || Q ( n ) 2 n β 2 I + O ( β 3 ) = ε D KL β � √ 1 1 2 n β 2 I ≈ ε n β ≈ 2 ε n ⇒ I 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 Fisher Information Determines Capacity of ε -secure Steganography Filler, Fridrich 3 of 18
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 root rates. STEGANOGRAPHY DESIGN - maximize root rate w.r.t. embedding operation for fixed cover source. Fisher Information Determines Capacity of ε -secure Steganography Filler, Fridrich 4 of 18
Presentation Outline 1 ASSUMPTIONS model of cover objects, model of embedding impact 2 STEGANOGRAPHIC CAPACITY & ROOT RATE formal connection between detectability and root rate 3 APPLICATION comparison of LSB and ± 1 embedding in spatial domain 4 CONCLUSION Fisher Information Determines Capacity of ε -secure Steganography Filler, Fridrich 5 of 18
(1) ASSUMPTIONS
Assumptions Summary 1 COVER SOURCE - Markov Chain first order Markov Chain with tran. prob. mat. A = ( a ij ) 2 EMBEDDING ALGORITHM - MI embedding independent substitution of states (next slide) 3 STEGOSYSTEM - ε -secure (imperfect) stegosystem is ε -SECURE j ′ stego i ′ k ′ l ′ HMC embedding a ij a jk a kl ... cover j i MC k l Fisher Information Determines Capacity of ε -secure Steganography Filler, Fridrich 7 of 18
Mutually Independent Embedding Operation Impact of embedding is modeled as probabilistic mapping acting on each cover element independently - MI embedding. X l ... l -th cover element Pr ( Y l = j | X l = i ) = b ij ( β ) Y l ... l -th stego element β ... change rate (rel. payload) Matrix B = ( b ij ) is “transition probability matrix” for β ≥ 0 . LSB embedding: B = = 1 − β = β Fisher Information Determines Capacity of ε -secure Steganography Filler, Fridrich 8 of 18
Mutually Independent Embedding Operation Impact of embedding is modeled as probabilistic mapping acting on each cover element independently - MI embedding. X l ... l -th cover element Pr ( Y l = j | X l = i ) = b ij ( β ) Y l ... l -th stego element β ... change rate (rel. payload) Matrix B = ( b ij ) is “transition probability matrix” for β ≥ 0 . LSB embedding: = I + β C = + β · B = = 1 − β = β = − 1 = 1 Fisher Information Determines Capacity of ε -secure Steganography Filler, Fridrich 8 of 18
Mutually Independent Embedding Operation Impact of embedding is modeled as probabilistic mapping acting on each cover element independently - MI embedding. X l ... l -th cover element Pr ( Y l = j | X l = i ) = b ij ( β ) Y l ... l -th stego element β ... change rate (rel. payload) Matrix B = ( b ij ) is “transition probability matrix” for β ≥ 0 . ± 1 embedding: = I + β C = + β · B = = β = 1 − β = β = 1 = 0 . 5 = − 1 2 Fisher Information Determines Capacity of ε -secure Steganography Filler, Fridrich 8 of 18
Mutually Independent Embedding Operation Impact of embedding is modeled as probabilistic mapping acting on each cover element independently - MI embedding. X l ... l -th cover element Pr ( Y l = j | X l = i ) = b ij ( β ) Y l ... l -th stego element β ... change rate (rel. payload) Matrix B = ( b ij ) is “transition probability matrix” for β ≥ 0 . F5 embedding: = I + β C = + β · B = = 1 − β = β = − 1 = 1 Fisher Information Determines Capacity of ε -secure Steganography Filler, Fridrich 8 of 18
Mutually Independent Embedding Operation Impact of embedding is modeled as probabilistic mapping acting on each cover element independently - MI embedding. X l ... l -th cover element Pr ( Y l = j | X l = i ) = b ij ( β ) Y l ... l -th stego element β ... change rate (rel. payload) Matrix B = ( b ij ) is “transition probability matrix” for β ≥ 0 . Assumption: B ( β ) = I + β C C describes the inner workings of the embedding algorithm. Fisher Information Determines Capacity of ε -secure Steganography Filler, Fridrich 8 of 18
(2) STEGANOGRAPHIC CAPACITY & √ RATE
Square Root Law 1 If n β n √ n → 0 then the stegosyst. are asymptotically secure. 2 If n β n √ n → + ∞ then arbitrarily accurate stego detectors exist. For fixed level of security, n β n β n → 0 as n → ∞ . √ n < C ⇒ [Filler, Ker, Fridrich, “The Square Root Law of Steganographic Capacity for Markov Covers”, Proc. SPIE, 2009] Fisher Information Determines Capacity of ε -secure Steganography Filler, Fridrich 10 of 18
The Best Possible Steganalyzer Hypothesis test: H 0 β = 0 : decide cover H 1 β > 0 ( known ) decide stego . : Detection statistics (log-likelihood ratio): stego distribution with Q ( n ) n elements β ( X ) β ( X ) = 1 L ( n ) √ n ln P ( n ) ( X ) cover distribution with n elements change rate Fisher Information Determines Capacity of ε -secure Steganography Filler, Fridrich 11 of 18
The Root Rate For small β , L ( n ) β ( X ) is Gaussian (Local Asymptotic Normality) . stego cover µ = √ n β 2 I / 2 σ 2 σ 2 σ 2 = β 2 I µ − µ 0 � �� P ( n ) || Q ( n ) d 2 Fisher information rate I = lim n → ∞ 1 d β 2 D KL � β n β = 0 � Deflection coefficient: d 2 = ( − µ − µ ) 2 / σ 2 = n β 2 I < ε � n β < r √ ε n 1 r = I ROOT RATE Fisher Information Determines Capacity of ε -secure Steganography Filler, Fridrich 12 of 18
Closed Form for Fisher Information Rate Theorem (Fisher Information Rate) Let A = ( a ij ) define the MC cover model and B = I + β C represent MI embedding. Then, the Fisher information rate I exists and can be written as I = c T F c . c = ( c 11 ,..., c NN ) T is column vector obtained directly from C . Matrix F ∈ R N 2 × N 2 , 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 . Fisher Information Determines Capacity of ε -secure Steganography Filler, Fridrich 13 of 18
(3) APPLICATIONS
Markov Model of Spatial Domain Images Image databases: CAMRAW, NRCS, NRCS-JPEG70. Spatial domain 8 -bit grayscale images: A = ( a ij ) ∈ R 256 × 256 a ij = 1 e − ( | i − j | / τ ) γ Z i CAMRAW - log a ij log a 127 ,j 0 0 data model − 5 − 5 − 10 − 10 − 15 0 127 255 255 0 255 j Fisher Information Determines Capacity of ε -secure Steganography Filler, Fridrich 15 of 18
Convex Combination of LSB and ± 1 Embedding Use ± 1 embedding in first λ n pixels and LSB embedding in the rest. r ( λ ) NRCS NRCS-JPEG Root rate r ( λ ) : CAMRAW 2 c λ = λ c ± 1 +( 1 − λ ) c LSB � 1 r ( λ ) = c T λ F c λ 1 0 - LSB λ 1 - ± 1 Higher root rate = better method. Fisher Information Determines Capacity of ε -secure Steganography Filler, Fridrich 16 of 18
Convex Combination of LSB and ± 1 Embedding Use ± 1 embedding in first λ n pixels and LSB embedding in the rest. r ( λ ) NRCS NRCS-JPEG Pevný, Bass, Fridrich: CAMRAW Steganalysis by 2 Subtractive Pixel Adjacency Matrix, submitted to ACM MM&SEC 2009, 1 Princeton, NJ 0 - LSB λ 1 - ± 1 Higher root rate = better method. Fisher Information Determines Capacity of ε -secure Steganography Filler, Fridrich 16 of 18
± 1 is Asymptotically Optimal What embedding method minimizes Fisher information rate and modifies cover by at most 1 ? v i Class of embedding NRCS operations: 0 . 8 1 − v 1 0 . 6 ± 1 emb. v 1 0 . 4 1 − v 4 0 . 2 v 4 1 i 254 ± 1 embedding is asymptotically optimal as number of grayscales N → ∞ . Fisher Information Determines Capacity of ε -secure Steganography Filler, Fridrich 17 of 18
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 to optimization (under mentioned assumptions) . was used to compare spatial domain stegosystems. Future work: use this framework for JPEG images. Fisher Information Determines Capacity of ε -secure Steganography Filler, Fridrich 18 of 18
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