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Minimizing A dditive Disto rtion F untions with Non-bina ry Emb edding Op eration in Steganography T om Filler and Jessia F ridrih Dept. of Eletrial and Computer Engineering SUNY Binghamton, New Y o rk

slide-1
SLIDE 1 Minimizing A dditive Disto rtion F un tions with Non-bina ry Emb edding Op eration in Steganography T
  • m
Filler and Jessi a F ridri h Dept.
  • f
Ele tri al and Computer Engineering SUNY Binghamton, New Y
  • rk
Se ond IEEE International W
  • rkshop
  • n
Info rmation F
  • rensi s
and Se urit y De emb er 15, 2010
slide-2
SLIDE 2 Steganography Steganography is a mo de
  • f
  • vert
  • mmuni ation.
Emb(·) message m k ey k
  • ver
X Ext(·) k ey k message m hannel with passive w a rden stego Y X and Y a re r.v.
  • n X
n
  • digital
images fo r example Emb(·) , Ext(·) ... emb edding, extra tion fun tions P erfe tly se ure steganography: Probabilit y distribution
  • f
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
  • f
16
slide-3
SLIDE 3 Can w e
  • nstru t
p erfe tly se ure stegosystems? Y es, but ...
  • nly
fo r a rti ial
  • ver
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
  • f
16
slide-4
SLIDE 4 Can w e
  • nstru t
p erfe tly se ure stegosystems? Y es, but ...
  • nly
fo r a rti ial
  • ver
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
  • ver
mo di ation: Stego
  • bje t
Y is p ro du ed b y slightly mo difying some
  • f
the elements (pixels, DCT
  • e ients,
...) in X . Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 3
  • f
16
slide-5
SLIDE 5 Whi h pixels an b e hanged? Pixels in ha rd-to-mo del
  • ntent.
Do not hange saturated pixels! Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 4
  • f
16
slide-6
SLIDE 6 Minimal-disto rtion Emb edding Pixels in textured a reas an b e hanged mo re frequently than those in smo
  • th
a reas. Emb edding
  • p
eration I i ⊂ I : Set
  • f
stego pixels into whi h i th
  • ver
pixel an b e hanged. Bina ry if |I i| = 2 fo r all pixels. A dditive disto rtion fun t.: ρ i( y i, x) =
  • st
  • f
hanging x i → y i
  • st
  • f
hanging
  • ver
x to stego y D( x, y) = n

∑

i=1

ρ

i( y i, x) Example:

ρ

i( x i, x) = and ρ i( x i − 1, x) = ρ i( x i + 1, x) = 1 #
  • f
hanges

ρ

i( y i, x) ≫ 1 if y i should almost never b e used fo r pixel i Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 5
  • f
16
slide-7
SLIDE 7 Problem F
  • mulation
& Optimal Solution THEORY Emb edding algo rithm fo r FIXED
  • ver
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
  • se 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.
  • nstr.
Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 6
  • f
16
slide-8
SLIDE 8 Problem F
  • mulation
& Optimal Solution THEORY Emb edding algo rithm fo r FIXED
  • ver
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
  • se 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.
  • nstr.
PRA CTICE: Send m bits in stego y with D( x, y) as small as p
  • ssible.
Re eiver do es not kno w
  • ver
x and
  • sts ρ
i , just msg. size! Problem ba res strong relationship with the sour e
  • ding
with a delit y riterion (Shannon 1959). Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 6
  • f
16
slide-9
SLIDE 9 Problem F
  • mulation
& Optimal Solution THEORY Emb edding algo rithm fo r FIXED
  • ver
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
  • se 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.
  • nstr.
PRA CTICE: Send m bits in stego y with D( x, y) as small as p
  • ssible.
Re eiver do es not kno w
  • ver
x and
  • sts ρ
i , just msg. size! 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
  • f
16
slide-10
SLIDE 10 (1) Bina ry emb edding
  • p
eration. Cover and stego pixels ∈ {0, 1} Review
  • f
kno wn fa ts and algo rithms. Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 7
  • f
16
slide-11
SLIDE 11 Syndrome Co ding Common to
  • l
fo r solving the sour e- o ding p roblem.

H ∈ {

0, 1} m× n ... sha red pa rit y- he k matrix Extra tion fun tion:

= H

y m m = Ext( y) = H y Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 8
  • f
16
slide-12
SLIDE 12 Syndrome Co ding Common to
  • l
fo r solving the sour e- o ding p roblem.

H ∈ {

0, 1} m× n ... sha red pa rit y- he k matrix Extra tion fun tion:

= H

y m m = Ext( y) = H y Emb edding fun tion: y = Emb( x, m) = a rg min

H y=m

D( x, y) 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 matrix ⇒ w e need some stru ture in H . Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 8
  • f
16
slide-13
SLIDE 13 Syndrome-T rellis Co des (SPIE 2010) Pra ti al and very versatile lass
  • f
linea r
  • des.
P a rit y- he k matrix: banded matrix

H = ⇒

e ient graphi al rep resentation Emb edding a rg minH y=m D( x, y) is realized b y the Viterbi alg. STC en o der Viterbi STC de o der

H y

y m m x
  • sts{ρ
i} Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 9
  • f
16
slide-14
SLIDE 14 (2) Non-bina ry emb edding
  • p
eration. Main
  • ntribution
  • f
the pap er. Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 10
  • f
16
slide-15
SLIDE 15 Multi-la y ered Constru tion (1/2) Example (quaterna ry emb edding
  • p
eration): Pixels x i, y i ∈ { 0, 1, 2, 3} an b e rep resented as ( MSB, LSB)
  • 2
bits . Problem: Emb ed m bits into
  • ver
x su h that D( x, y) is minimal. Optimal
  • ding
s heme sends i th stego pixel a o rding to Pr( y i| x) ∝ exp(−λρ i( y i, x)). Use p ro du t rule Pr( MSB, LSB) = Pr( MSB)· Pr( LSB| MSB) . Entrop y[ MSB, LSB] = Entrop y[ MSB]
  • 1st
la y er
  • f
MSBs

+

Entrop y[ LSB| MSB]
  • 2nd
la y er
  • f
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
  • f
16
slide-16
SLIDE 16 Multi-la y ered Constru tion (2/2) Entrop y[ MSB, LSB] = Entrop y[ MSB]
  • 1st
la y er
  • f
MSBs

+

Entrop y[ LSB| MSB]
  • 2nd
la y er
  • f
LSBs 1st la y er
  • f
MSBs: Emb ed Entrop y[ MSB] bits into MSBs b y minimizing
  • sts

ρ

i( MSB = 0) = ρ i( 0, x)+ρ i( 1, x)

ρ

i( MSB = 1) = ρ i( 2, x)+ρ i( 3, x) 2nd la y er
  • f
LSBs: Emb ed Entrop y[ LSB| MSB] bits into LSBs with
  • sts

ρ

i( LSB = 0) = ρ i( 0, x)

ρ

i( LSB = 1) = ρ i( 1, x)

ρ

i( LSB = 0) = ρ i( 2, x)

ρ

i( LSB = 1) = ρ i( 3, x) MSB = 0 ⇒ MSB = 1 ⇒ This is
  • ptimal
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
  • f
16
slide-17
SLIDE 17 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]
  • MSBs
rst

+

Entrop y[ LSB| MSB]
  • then
LSBs

=

Entrop y[ LSB]
  • LSBs
rst

+

Entrop y[ MSB| LSB]
  • then
MSBs PRA CTICE: Order in whi h la y ers a re p ro essed DOES pla y a role. Dierent expansions lead to dierent
  • sts
assignments fo r whi h the p ra ti al
  • des
(STCs) ma y fail. Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 13
  • f
16
slide-18
SLIDE 18 Appli ation to Spatial-Domain Digital Images

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 LSB matching alg. ternary binary pentary (±2) ternary (±1) binary (±1) BOWS2 database Payload-limited sender simulated emb. STC/ML-STC coding loss Relative payload α (bits per pixel) Average error of SVM-based steganalyzer with SPAM features

detectable undetectable

Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 14
  • f
16
slide-19
SLIDE 19 Con lusion Prop
  • sed
Multi-la y ered
  • nstru tion
allo ws implementing the minimal-disto rtion emb edding pa radigm with non-bina ry emb edding
  • p
eration. Optimal if
  • ptimal
bina ry sour e- o ding exist. Nea r-optimal when realized with Syndrome-T rellis Co des No need to sha re the
  • sts
with the re eiver. F uture dire tions: Can w e minimize statisti al dete tabilit y b y lea rning
  • sts ρ
i( y i, x) ? ⇒ SPIE 2011. C++ and Matlab implementation available. Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 15
  • f
16
slide-20
SLIDE 20 Info rmation Hiding 2011, Ma y 18-20, Prague www.ih onferen e.o rg Submission deadline: Janua ry 17 (extension p
  • ssible)
IEEE ICASSP is also in Prague Ma y 22-27. See y
  • u
in Prague. Filler, F ridri h Minimizing A dditive Disto rtion F un tions ... in Steganography 16
  • f
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