Secure Steganography: Statistical
Restoration of the Second Order
Dependencies for Improved Security
A. Sarkar, K. Solanki, U. Madhow, S. Chandrasekaran and B. S.
Manjuanth
Presented by:
Anindya Sarkar
Vision Research Lab,
Department of Electrical & Computer Engg,
University of California, Santa Barbara
April 16, 2007
April 13, 2020
information bits to be
embedded
BASIC DATA HIDING SYSTEM
011100111001
Retrieve the embedded
information bits
110011001111
Encode information bits by
adding redundancy
011100111001
110011001111
11010101010010111111
100111…….1111111110
Embed bits
Noisy
channel
(coding or
benign
attacks)
Original image (host
signal or cover)
April 13, 2020
Stego image (with
embedded data)
Image, after
coding or benign
attacks
Ways to detect hiding
PERCEPTUAL
TRANSPARENCY
YES
Focus of
steganography and
steganalysis work
STATISTICAL
TRANSPARENCY
To ensure
undetectability based
on this setup
Received Image
Model for0.4
0.3
Cover
Image’s 0.2
0.1
PMF
0.4
Image 0.3
PMF 0.2
0
0.1
In DCT 0
domain
-4 -3 -2 -1 0 1 2 3 4
April 13, 2020
NOT
SURE
-4 -3 -2 -1 0 1 2 3 4
Model for0.3
0.2
Stego
Image’s 0.1
0
PMF
-4 -3 -2 -1 0 1 2 3 4
Make stego
pmf very
close to
cover pmf
model
Make stego
pmf very
close to
original
cover pmf
Steganography – main contributions
• Restoration of nth order co-occurrence statistics,
based on correspondence between Earth
Mover’s Distance and Statistical Restoration
framework
• Locally optimal method that restores intra and
inter-block correlation
• Result – resisting detection using 1D, 2D and
joint intra and inter block based statistics –
robust steganography method
April 13, 2020
Random
Random
detector
detector
Steganalysis performed using the 4
different features: the statistical
compensation was done using 2-D
EMD on non-overlapping pairs,
belonging to the same row of a 8x8
block
April 13, 2020
Steganalysis performed using the 4
different features: the statistical
compensation was done using joint
intra & inter- block compensation
bincount
Histogram
Original dataset X
X=H U C: H for hiding, C for compensation
H
Bin indices (values taken by X)
C
^
When H is used for hiding, H changes to H: pmf changes
H,hidden
^
(H)
C
^
Using C for compensation, C is changed to C: we restore original pmf
H,hidden C,compensated
^
^
(H)
(C)
April 13, 2020
STATISTICAL RESTORATION SETUP
Histogram difference before compensation
Histogram difference after compensation
1200
25
1000
20
15
800
10
600
400
Image
200
:10%
hiding
0
5
0
-5
-10
-200
-15
-400
-600
-40
-20
-30
-20
-10
0
10
20
30
40
-25
-40
-30
-20
-10
0
10
20
30
40
Histogram difference, considering joint intra-inter correlation, before and after
compensation: Max value in LHS=1200, in RHS=25: 1D pmf is compensated
After
compensation
After
compensation
Intra-block (left) and inter-block (right) histogram (absolute) difference
matrix, plotted in log scale (white: very high value, black: very low value)
April 13, 2020
Histogram difference before compensation
Histogram difference after compensation
2500
40
30
2000
20
1500
10
image:
20%
1000
0
-10
500
hiding
0
-20
-30
-40
-500
-50
-1000
-40
-30
-20
-10
0
10
20
30
40
-60
-40
-30
-20
-10
0
10
20
30
40
Histogram difference, considering joint intra-inter correlation, before and after
compensation: Max value in LHS=2500, in RHS=50: 1D pmf is compensated
After
compensation
After
compensation
Intra-block (left) and inter-block (right) histogram (absolute) difference
matrix, plotted in log scale (white: very high value, black: very low value)
April 13, 2020
Histogram
bincount
EMD based flow helps to convert a given pmf to another
with minimum changes
After
compensation
a
b
c
d
e
bin index
pmf of C (part of X to be
used for compensation)
f
a
b
c
d
e
f
Target pmf of C (that
ensures pmf matching)
EMD solution: it gives the redistribution of weights (flows) among the bins
such that ∑ ∑ D.F is minimized (D: inter cluster distance, F: inter cluster flow)
Problems – cannot restore both intra and inter-block statistics, handle overlapping
pairs and determine the optimal perturbations per coefficient,
April 13, 2020
Clockwise from top left:
zigzag intra-block scanning,
alternate scanning across rows of an
image to obtain inter-correlated blocks,
matrix A with Nc columns, Nc being no. of
AC DCT terms chosen per block and Nb
rows (no. of 8x8 blocks in image) –
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C-compensation point, Hnon-compensation point
(may be used for hiding)
H and C locations
alternate
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If N is a C-point, then the points
T,B,L and R are all H-points. Given
a C-point, say N, we compute the
bin-counts in the pairs (L,N), (N,R)
(T,N) and (N,B)
(Nc terms)
4 AC DCT coefficients per block
1
1
L
2
2
1
1
1
With respect to
current point N,
T
N
B1
1
2
R
3
3
2
1
1
2
2
2
inter-block histogram,
computed using
T=top, B1=bottom
overlapping pairs,
column-wise
L=left, R=right
neighbor
4
2
(1,1) (1,2) (2,1) (2,2)
5
Bin-count
4 8x8 blocks
(Nr terms)
2
intra-block histogram,
computed using
overlapping pairs, row-wise
5
1
1
(1,1) (1,2) (2,1) (2,2)
Histogram bins
Intra and inter-block histogram computation for the Nr x Nc joint
correlation matrix A (here Nc=Nr=4)
April 13, 2020
Details of the Joint Intra-Inter Block based Scheme
L et B i n t r a and B i n t er denot e t he int r a-block and int er -block bincount s obt ained
P using t he m at r ix A.
B i n t r a (a; b) = P
I ; I i j = 1 if f A i j = a; A i ;j + 1 = bg else 0
i ;j i j
B i n t er (a; b) =
J i j ; J i j = 1 if f A i j = a; A i + 1;j = bg else 0
i ;j
A ft er dat a hiding w it hout com p ensat ion, let t he m odi¯ed int r ablock and int er -block bin-count s of t he m at r ix A b e called B 0
and
intr a
B 0 , r esp ect ively. L et us consider t he elem ent A i j , w hich equals
i n t er
N , w hile it s D 4 neighb or s ar e A i ;j ¡ 1 = L ( left ) , A i ;j + 1 = R ( r ight ) ,
A i ¡ 1;j = T ( t op) and A i + 1;j = B 1 ( b ot t om )
Since we allow a p er t ur bat ion of only § 1, N can b e m app ed t o one
of f N ¡ 1; N ; N + 1g. W e com put e t he 4 bin-count di®er ence values for
N 0 2 f N ¡ 1; N ; N + 1g:
D(N 0; 1) = B i n t r a (L ; N 0) ¡ B 0
(L ; N 0)
intr a
D(N 0; 2) = B i n t r a (N 0; R) ¡ B 0
(N 0; R)
intr a
D(N 0; 3) = B i n t er (T; N 0) ¡ B 0 (T; N 0)
i n t er
D(N 0; 4) = B i n t er (N 0; B 1 ) ¡ B 0 (N 0; B 1 )
i n t er
April 13, 2020
Determining the Optimal perturbation for each point
T he squar ed di®er ence b et ween t he or iginal and m odi¯ed hist ogr am s for t he int r a and int er -block cases ar e consider ed. A m ax imum of 4£ 3= 12 D t er m s m ay var y dep ending on how N is changed, as
in t he expr ession for t he squar ed er r or cost funct ion J . N is conver t ed
t o t hat N opt for w hich t he squar ed di®er ence t er m J is m inim ized.
J (N + ±) =
X4
f D(N ; i ) + 1 ¡ I ±;0 g2 +
i= 1
X4
+
X4
f D(N ¡ 1; i ) ¡ I ±;¡ 1 g2
i= 1
f D(N + 1; i ) ¡ I ±;1 g2 ; ± = f ¡ 1; 0; 1g
i= 1
w her e t he indicat or funct ion I ±;k = 1 if ± = k and = 0 ot her w ise
N opt = ar g m in 0 0
J (N 0)
N ; N 2 f N ¡ 1;N ;N + 1g
W e r ep eat t his pr ocess t o obt ain a locally opt im al solut ion for each
com p ensat ion locat ion of A.
April 13, 2020
Random
Random
detector
detector
Steganalysis performed using the 4
different features: the statistical
compensation was done using 2-D
EMD on non-overlapping pairs,
belonging to the same row of a 8x8
block
April 13, 2020
Steganalysis performed using the 4
different features: the statistical
compensation was done using joint
intra & inter- block compensation
Conclusions
•
Correspondence between EMD – used for PMF
matching as an image similarity measure, and the
statistical restoration based steganography framework
•
Shortcomings of EMD framework have been tackled
using the joint correlation matrix locally optimum
method
•
We get close to perfect security for various 2D features
April 13, 2020
QUESTIONS??
April 13, 2020