Minimizing A dditive Disto rtion F un tions with Non-bina ry Emb edding Op eration in Steganography T om�� Filler and Jessi a F ridri h Dept. of Ele tri al and Computer Engineering SUNY Binghamton, New Y o rk Se ond IEEE International W o rkshop on Info rmation F o rensi s and Se urit y De emb er 15, 2010
Steganography Steganography is a mo de of overt ommuni ation. message m message m Emb ( · ) stego Y Ext ( · ) over X hannel with k ey k k ey k passive w a rden n X and Y a re r.v. on X � digital images fo r example Emb ( · ) , Ext ( · ) ... emb edding, extra tion fun tions P erfe tly se ure steganography: Probabilit y distribution of X and Y a re exa tly the same. No statisti al test (w a rden) an dete t steganography . Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 2 of 16
Can w e onstru t p erfe tly se ure stegosystems? Y es, but ... only fo r a rti� ial over sour es fo r whi h w e kno w the exa t p robabilit y distribution (Gaussian). No p erfe tly se ure stegosystem exists fo r real digital media. Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 3 of 16
Can w e onstru t p erfe tly se ure stegosystems? Y es, but ... only fo r a rti� ial over sour es fo r whi h w e kno w the exa t p robabilit y distribution (Gaussian). No p erfe tly se ure stegosystem exists fo r real digital media. In p ra ti e, w e have to do... Steganography b y over mo di� ation: Stego obje t Y is p ro du ed b y slightly mo difying some of the elements (pixels, DCT o e� ients, ...) in X . Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 3 of 16
Whi h pixels an b e hanged? Pixels in ha rd-to-mo del ontent. Do not hange saturated pixels! Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 4 of 16
Minimal-disto rtion Emb edding Pixels in textured a reas an b e hanged mo re frequently than those in smo oth a reas. Emb edding op eration I : i ⊂ I Set of stego pixels into whi h i th over pixel an b e hanged. Bina ry if | I 2 fo r all pixels. i | = A dditive disto rtion fun t.: ρ y x ) = ost of hanging x y i ( i , i → i n ost of hanging D ( x , y ) = y x ) i ( i , over x to stego y i = 1 Example: x x ) = 0 and ρ x 1 , x ) = ρ x 1 , x ) = 1 # of hanges i ( i , i ( i − i ( i + y x ) ≫ 1 if y should almost never b e used fo r pixel i i ( i , i Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 5 of 16 ∑ ρ ρ ρ
Problem F omulation & Optimal Solution Emb edding algo rithm fo r FIXED over x : Sele t stego y with p robabilit y Pr ( y | x ) = π ( y | x ) . What is the b est distribution π ? P a yload-limited sender: ho ose π su h that minimize exp e ted disto rtion while Entrop y [ π ] = m bits Solution: π ( y | x ) ∝ exp ( − λ D ( x , y )) and λ solves pa yl. onstr. THEORY Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 6 of 16
Problem F omulation & Optimal Solution Emb edding algo rithm fo r FIXED over x : Sele t stego y with p robabilit y Pr ( y | x ) = π ( y | x ) . What is the b est distribution π ? P a yload-limited sender: ho ose π su h that minimize exp e ted disto rtion while Entrop y [ π ] = m bits Solution: π ( y | x ) ∝ exp ( − λ D ( x , y )) and λ solves pa yl. onstr. PRA CTICE: Send m bits in stego y with D ( x , y ) as small as p ossible. THEORY Re eiver do es not kno w over x and osts ρ , just msg. size! i Problem ba res strong relationship with the �sour e o ding with a �delit y riterion� (Shannon 1959). Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 6 of 16
Problem F omulation & Optimal Solution Emb edding algo rithm fo r FIXED over x : Sele t stego y with p robabilit y Pr ( y | x ) = π ( y | x ) . What is the b est distribution π ? P a yload-limited sender: ho ose π su h that minimize exp e ted disto rtion while Entrop y [ π ] = m bits Solution: π ( y | x ) ∝ exp ( − λ D ( x , y )) and λ solves pa yl. onstr. PRA CTICE: Send m bits in stego y with D ( x , y ) as small as p ossible. THEORY Re eiver do es not kno w over x and osts ρ , just msg. size! i MAIN CONTRIBUTION: p ra ti al and nea r-optimal app roa h fo r solving non-bina ry emb edding p roblem. Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 6 of 16
(1) Bina ry emb edding op eration. Cover and stego pixels ∈ { 0 , 1 } Review of kno wn fa ts and algo rithms. Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 7 of 16
Syndrome Co ding Common to ol fo r solving the sour e- o ding p roblem. m × n 0 , 1 } ... sha red pa rit y- he k matrix y m Extra tion fun tion: m = Ext ( y ) = H y H ∈ { = H Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 8 of 16
Syndrome Co ding Common to ol fo r solving the sour e- o ding p roblem. m × n 0 , 1 } ... sha red pa rit y- he k matrix y m Extra tion fun tion: m = Ext ( y ) = H y Emb edding fun tion: y = Emb ( x , m ) = a rg min D ( x , y ) H y = m H ∈ { Repla e x with y , su h that D ( x , y ) is minimal and H y = m. Emb edding is NP ha rd p roblem fo r general pa rit y- he k = H matrix ⇒ w e need some stru ture in H . Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 8 of 16
Syndrome-T rellis Co des (SPIE 2010) Pra ti al and very versatile lass of linea r o des. P a rit y- he k matrix: banded matrix 0 e� ient graphi al 0 rep resentation Emb edding a rg min H y = m D ( x , y ) is realized b y the Viterbi alg. m y STC STC x m en o der de o der osts { ρ i } H = ⇒ H y Viterbi Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 9 of 16
(2) Non-bina ry emb edding op eration. Main ontribution of the pap er. Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 10 of 16
Multi-la y ered Constru tion (1/2) Example (quaterna ry emb edding op eration): Pixels x y 0 , 1 , 2 , 3 } an b e rep resented as ( MSB , LSB ) . i , i ∈ { 2 bits Problem: Emb ed m bits into over x su h that D ( x , y ) is minimal. Optimal o ding s heme sends i th stego pixel a o rding to Pr ( y x ) ∝ exp ( − λρ y x )) . i | i ( i , Use �p ro du t rule� Pr ( MSB , LSB ) = Pr ( MSB ) · Pr ( LSB | MSB ) . � �� � Entrop y [ MSB , LSB ] = Entrop y [ MSB ] Entrop y [ LSB | MSB ] 1 st la y er of MSBs 2 nd la y er of LSBs Ho w to implement this using STCs in p ra ti e? Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 11 of 16 + � �� � � �� �
Multi-la y ered Constru tion (2/2) Entrop y [ MSB , LSB ] = Entrop y [ MSB ] Entrop y [ LSB | MSB ] 1 st la y er of MSBs 2 nd la y er of LSBs 1 st la y er of MSBs: 2 nd la y er of LSBs: Emb ed Entrop y [ MSB ] bits into Emb ed Entrop y [ LSB | MSB ] bits MSBs b y minimizing osts into LSBs with osts LSB = 0 ) = ρ 0 , x ) i ( i ( MSB = 0 ⇒ MSB = 0 ) = ρ 0 , x )+ ρ 1 , x ) i ( i ( i ( LSB = 1 ) = ρ 1 , x ) i ( i ( + � �� � � �� � LSB = 0 ) = ρ 2 , x ) i ( i ( MSB = 1 ⇒ MSB = 1 ) = ρ 2 , x )+ ρ 3 , x ) i ( i ( i ( LSB = 1 ) = ρ 3 , x ) i ( i ( This is optimal if w e kno w ho w to solve the bina ry p roblems. Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 12 of 16 ρ ρ ρ ρ ρ ρ
Pra ti al Issues THEORY: Order in whi h la y ers a re p ro essed do es not matter. Entrop y [ MSB , LSB ] = Entrop y [ MSB ] Entrop y [ LSB | MSB ] MSBs �rst then LSBs Entrop y [ LSB ] Entrop y [ MSB | LSB ] LSBs �rst then MSBs PRA CTICE: Order in whi h la y ers a re p ro essed DOES pla y a role. + Di�erent expansions lead to di�erent osts assignments fo r � �� � � �� � whi h the p ra ti al o des (STCs) ma y fail. = + � �� � � �� � Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 13 of 16
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