Quantitative Steganalysis of LSB Embedding in JPEG Domain

Transcription

1 Quantitative Steganalysis of LSB Embedding in JPEG Domain Jan Kodovský, Jessica Fridrich September 10, 2010 / ACM MM&Sec 10 1 / 17

2 Motivation Least Significant Bit (LSB) embedding Simplicity, high embedding capacity Used in Jsteg, JP Hide&Seek, and other commercial stego software Steganalysis of LSB embedding in spatial domain is mature area [Dumitrescu-2002], [Ker-2008] Our focus Transform domain JPEG format Quantitative steganalysis Outputs the estimate of the message length 2 / 17

3 Jsteg Jsteg: [Upham-1993] LSB replacement Embedding along a pseudo-random path DCT histogram Skipping 0 and / 17

4 Jsteg Jsteg: [Upham-1993] LSB replacement Embedding along a pseudo-random path Full embedding Skipping 0 and Embedding violates histogram symmetry 3 / 17

5 Selected Existing Attacks [Zhang,Ping-2003] the first quantitative attack Employed violation of histogram symmetry [Yu-2004] histogram-based attack Generalized Cauchy ML fit Chi-square test [Lee-2006], [Lee-2007] Category attack Technically not quantitative [Westfeld-2007], [Böhme-2008] adaptation of spatial domain attacks [Pevný-2009] support vector regression Feature-based non-structural attack Currently the most accurate quantitative attack 4 / 17

6 Our Goals / Challenges Improve the accuracy of existing quantitative attacks to Jsteg Achieve better performance than the feature-based machine learning approach (SVR) Focus on the structure of LSB embedding Deliver theoretically well-founded modular framework Explore the applicability of the proposed attacks to a different LSB embedding paradigms 5 / 17

7 Maximum Likelihood β... change rate P x (x) x Emb(β) Emb(β) P (y x,β) P (y x,β) cover feature vector x stego feature vector y P (y, β) = P (y, x, β)dx = ˆβ = arg max β 0 P (y x, β)p (x, β)dx = P (β) P (y β) = arg max β 0 P (y x, β)p x(x)dx Choice of the feature vector x is crucial P (y x, β)p x(x)dx 6 / 17

8 Features of Zhang & Ping x = [ x 1 x 2 x 3 ] [Zhang,Ping-2003] P (y x, β) Binomial distribution Gaussian approximation Embedding invariants: x 1 + x 2, x 3 P x (x) precover assumption [Ker-2007] 7 / 17

9 Features of Zhang & Ping x = [ x 1 x 2 x 3 ] [Zhang,Ping-2003] P (y x, β) Binomial distribution Gaussian approximation Embedding invariants: x 1 + x 2, x 3 P x (x) precover assumption [Ker-2007] 7 / 17

10 Features of Zhang & Ping x = [ x 1 x 2 x 3 ] [Zhang,Ping-2003] P (y x, β) Binomial distribution Gaussian approximation Embedding invariants: x 1 + x 2, x 3 P x (x) precover assumption [Ker-2007] 7 / 17

11 Features of Zhang & Ping x = [ x 1 x 2 x 3 ] [Zhang,Ping-2003] P (y x, β) Binomial distribution Gaussian approximation Embedding invariants: x 1 + x 2, x 3 P x (x) precover assumption [Ker-2007] 7 / 17

12 Features of Zhang & Ping Emb(β) x = [ x 1 x 2 x 3 ] β 1 β y = [ x β 1 x β 2 x β 3 ] [Zhang,Ping-2003] arg max β 0 P (y x, β)p x(x)dx P (y x, β) Binomial distribution Gaussian approximation Embedding invariants: x 1 + x 2, x 3 P x (x) precover assumption [Ker-2007] Precover 1/2 1/2 x 1 x 2 + x 3 7 / 17

13 Performance Evaluation Median absolute error 10 2 Jsteg ML - Zhang & Ping Change rate β 3,250 JPEG images resized and compressed to QF=75 Performance similar to [Zhang,Ping-2003] Assumption x β 1 = expected value Zhang & Ping s estimator 8 / 17

14 Performance Evaluation Median absolute error 10 2 Jsteg ML - Zhang & Ping SVR Cartesian-calibrated Pevný features (548) Additional 3,250 images for training Change rate β 3,250 JPEG images resized and compressed to QF=75 Performance similar to [Zhang,Ping-2003] Assumption x β 1 = expected value Zhang & Ping s estimator 8 / 17

15 First-Order Statistics x = [ x 2L, x 2L+1,..., x 2R, x 2R+1 ] ˆβ = arg max β 0 P (y x, β)p x(x)dx Embedding changes in individual LSB pairs are independent ( ) ( ) P (y x, β) = P x β 0 x 0, β P x β 1 x 1, β P Embedding invariants: x 0, x 1, x 2k + x 2k+1 Binomial distribution Gaussian approximation k ( ) x β 2k, xβ 2k+1 x 2k, x 2k+1, β 9 / 17

16 First-Order Statistics x = [ x 2L, x 2L+1,..., x 2R, x 2R+1 ] ˆβ = arg max β 0 P (y x, β)p x(x)dx p 1 2s ( x + 1 ) p s DCT coefficients are i.i.d. drawn from generalized Cauchy distribution Parameters p and s are ML estimates, given embedding invariants Precover assumption for every LSB pair Precover Embedding invariants: x 0, x 1, x 2k + x 2k+1 9 / 17

17 Performance Evaluation Median absolute error 10 2 Jsteg ML - Zhang & Ping ML - First-order SVR Change rate β 10 / 17

18 Second-Order Statistics DCT coefficients are not i.i.d. We capture dependencies using adjacency matrix X Natural decomposition into k-nodes, k {1, 2, 4} Binomial / multinomial distributions Gaussian approximations [-2,3] [-1,3] [0,3] [1,3] [2,3] [3,3] [-2,2] [-1,2] [0,2] [1,2] [2,2] [3,2] arg max β 0 P (Y X, β)p x(x)dx [-2,1] [-1,1] [0,1] [1,1] [2,1] [3,1] [-2,0] [-1,0] [0,0] [1,0] [2,0] [3,0] Factorization of P (y x, β) [-2,-1] [-1,-1] [0,-1] [1,-1] [2,-1] [3,-1] Embedding invariants [-2,-2] [-1,-2] [0,-2] [1,-2] [2,-2] [3,-2] Analytic expression 11 / 17

19 Second-Order Statistics DCT coefficients are not i.i.d. We capture dependencies using adjacency matrix X Natural decomposition into k-nodes, k {1, 2, 4} Binomial / multinomial distributions Gaussian approximations [-2,3] [-1,3] [0,3] [1,3] [2,3] [3,3] [-2,2] [-1,2] [0,2] [1,2] [2,2] [3,2] arg max β 0 P (Y X, β)p x(x)dx [-2,1] [-1,1] [0,1] [1,1] [2,1] [3,1] [-2,0] [-1,0] [0,0] [1,0] [2,0] [3,0] Complications arise [-2,-1] [-1,-1] [0,-1] [1,-1] [2,-1] [3,-1] Good parametric model? [-2,-2] [-1,-2] [0,-2] [1,-2] [2,-2] [3,-2] High complexity 11 / 17

20 Zero Message Hypothesis (ZMH) Alternative heuristic approach Penalty function z(x) 0 satisfying z(x β ) 0 when β = 0 z(x β ) > 0 when β > 0 z(x) should be a quantitative description of a zero message hypothesis capturing a key cover property violated by embedding Assumption: y = E[x β ] = Emb(x, β) Assumption: mapping Emb is invertible x = Emb 1 (y, β) Comments ˆβ = arg min β 0 z(emb 1 (y, β)) Low computational complexity one-dimensional search over β ZMH-based steganalysis is not a new idea! [RS steganalysis,2001] 12 / 17

21 First-Order Statistics (ZMH) x = [x 2L, x 2L+1,..., x 2R 1, x 2R ] Penalty function z sym (x) = w k (x k x k ) 2 Weights w k chosen to minimize the estimator variance least squares steganalysis [Ker-2007] Final form of the penalty function: z sym (x) = k>0 (x k x k ) 2 x k + x k 13 / 17

22 Performance Evaluation Median absolute error 10 2 Jsteg ML - Zhang & Ping ML - First-order ZMH - First-order SVR Change rate β 14 / 17

23 Second-Order Statistics (ZMH) Feature vector: adjacency matrix X ZMH approach Decomposition into k-nodes Embedding is invertible provided 0 β < 1/2 Symmetry about D [-2,3] [-2,2] [-1,3] [-1,2] [0,3] [0,2] [1,3] [1,2] [2,3] [2,2] [3,3] [3,2] [-2,1] [-1,1] [0,1] [1,1] [2,1] [3,1] z adj (X) = i,j (x i,j x j, i ) 2 x i,j + x j, i [-2,0] [-1,0] [0,0] [1,0] [2,0] [3,0] [-2,-1] [-1,-1] [0,-1] [1,-1] [2,-1] [3,-1] [-2,-2] [-1,-2] [0,-2] [1,-2] [2,-2] [3,-2] D 15 / 17

24 Second-Order Statistics (ZMH) Feature vector: adjacency matrix X ZMH approach Decomposition into k-nodes Embedding is invertible provided 0 β < 1/2 Symmetry about D [-2,3] [-2,2] [-1,3] [-1,2] [0,3] [0,2] [1,3] [1,2] [2,3] [2,2] [3,3] [3,2] [-2,1] [-1,1] [0,1] [1,1] [2,1] [3,1] z adj (X) = i,j (x i,j x j, i ) 2 x i,j + x j, i [-2,0] [-1,0] [0,0] [1,0] [2,0] [3,0] [-2,-1] [-1,-1] [0,-1] [1,-1] [2,-1] [3,-1] [-2,-2] [-1,-2] [0,-2] [1,-2] [2,-2] [3,-2] D 15 / 17

25 Performance Evaluation Median absolute error 10 2 Jsteg ML - Zhang & Ping ML - First-order ZMH - First-order ZMH - Second-order SVR Change rate β 16 / 17

26 What Else Can You Find in the Paper / Journal Version Error analysis of between-image and within-image errors for selected attacks Verification of precover assumptions using two different statistical tests Discussion & experiments with the symmetrized version of Jsteg Conversion of the Category attack [Lee-2006] into a quantitative one through the proposed ZMH framework Experiments conducted on two different sources of images Results reported in terms of two more security measures: IQR, median bias 17 / 17