LSB detection by Pairs Analysis CSM25 Secure Information Hiding Dr - - PowerPoint PPT Presentation

LSB detection by Pairs Analysis

CSM25 Secure Information Hiding Dr Hans Georg Schaathun

University of Surrey

Spring 2007

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 1 / 41

Outcomes

Learn how to implement pairs analysis Understand strengths and limitations of pairs analysis

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 2 / 41

Background

Where χ2 falls short

The χ2 test we have seen

Analyses histogram only. Detects embedding in consecutive pixels

What if the message is randomly spread across the image?

Generalised χ2 analysis.

Yes/No answer; cannot estimate message length Can be fooled if the message is biased (more 0-s than 1-s or v.v.)

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 3 / 41

Background

Higher-order statistics

Pixels in neighbourhoods

Pairs of Values counts single pixels

→ first-order statistic

Higher-order statistics

Count pairs of (neighbour) pixels (2nd order) Pixel triplets (3rd order)

Study relations between pixels in a neighbourhood

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 4 / 41

Pairs analysis

Reading

Core Reading «Quantitative steganalysis of digital images: estimating the secret message length» by Jessica Fridrich, Miroslav Goljan, Dorin Hogea, David Soukal, in Multimedia Systems 2003 Suggested Reading «Higher-order statistical steganalysis of palette images» by Jessica Fridrich, Miroslav Goljan, David Soukal

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 5 / 41

Pairs analysis

Pairs Analysis

Pairs Analysis is quantitative

i.e. estimates the message length

Originally designed for GIF. We present it for spatial, grayscale images.

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 6 / 41

Pairs analysis The characteristic sequence

Outline

1

Background

2

Pairs analysis The characteristic sequence Homogenous pairs

3

Where Pairs Analysis fails Dithered backgrounds

4

RS steganalysis The idea The result Counter-measures

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 7 / 41

Pairs analysis The characteristic sequence

The characteristic sequence

Let c, c′ be two colours (grayscales). Read image row by row (left to right and top down). Assign 0 to c and 1 to c′. Ignore all other colours. Resulting sequence is denoted Z(c, c′). Definition Z = Z(0, 1)|Z(2, 3)|Z(4, 5)| . . . |Z(254, 255), (1) Z ′ = Z(1, 2)|Z(3, 4)|Z(5, 6)| . . . |Z(255, 0). (2)

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 8 / 41

Pairs analysis The characteristic sequence

The colour cut

Z( , ) extracted from an image Extracted column-wise (Matlab-style) Row-wise extraction is equally valid.

• 001111000000111111001101110010111001111110

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 9 / 41

Pairs analysis Homogenous pairs

Outline

1

Background

2

Pairs analysis The characteristic sequence Homogenous pairs

3

Where Pairs Analysis fails Dithered backgrounds

4

RS steganalysis The idea The result Counter-measures

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 10 / 41

Pairs analysis Homogenous pairs

Second-order structure

Second-order structure (of Z and of Z ′)

count pairs of consecutive bits four possible pairs 00,01,10,11

Homogenous pairs: 00, 11 Let F be frequency of Homogenous pairs in Z. Let R = F/n be the relative frequency.

where n = N · M is the number of pixels.

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 11 / 41

Pairs analysis Homogenous pairs

Exercise

Make a function producing Z from an image X. Make a function producing Z ′ from an image X. Make a function counting homogenous pairs in a sequence Z. Test the functions on stego-image and cover-images you have used before. How many homogenous pairs are there in Z? How many in Z ′?

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 12 / 41

Pairs analysis Homogenous pairs

Expected structure of Z

Let R(p) = E(R) be

expected, relative frequency of homogenous pairs in Z when a fraction p of pixel LSB-s have been flipped. (e.g. if a random unbiased bit string of length 2p has been embedded)

Theorem R(p) is a parabola with minimum at R(1/2) = 1/2. R(p) = ap2 + bp + c for some constants a, b, and c.

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 13 / 41

Pairs analysis Homogenous pairs

Why parabola?

k1 k2 k3 k4 · · · kr Z = 0000

• 111

00 . . . 0 11 . . . 1 · · · 11 . . . 1 nR(0) = r

i=0(ki − 1)

Homogenous pair remains homogenous: Pr = q2 + (1 − q)2

Both change + Neither changes

Heterougenous pair remains homogenous: Pr = 2q(1 − q) nR(q) =

r

• i=1

[q2 + (1 − q)2](ki − 1) + 2q(1 − q)(r − 1)

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 14 / 41

Pairs analysis Homogenous pairs

Why parabola?

k1 k2 k3 k4 · · · kr Z = 0000

• 111

00 . . . 0 11 . . . 1 · · · 11 . . . 1 nR(0) = r

i=0(ki − 1)

Homogenous pair remains homogenous: Pr = q2 + (1 − q)2

Both change + Neither changes

Heterougenous pair remains homogenous: Pr = 2q(1 − q) nR(q) =

r

• i=1

[q2 + (1 − q)2](ki − 1) + 2q(1 − q)(r − 1)

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 14 / 41

Pairs analysis Homogenous pairs

Why parabola?

k1 k2 k3 k4 · · · kr Z = 0000

• 111

00 . . . 0 11 . . . 1 · · · 11 . . . 1 nR(0) = r

i=0(ki − 1)

Homogenous pair remains homogenous: Pr = q2 + (1 − q)2

Both change + Neither changes

Heterougenous pair remains homogenous: Pr = 2q(1 − q) nR(q) =

r

• i=1

[q2 + (1 − q)2](ki − 1) + 2q(1 − q)(r − 1)

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 14 / 41

Pairs analysis Homogenous pairs

The R-function

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 15 / 41

Pairs analysis Homogenous pairs

Structure of the shifted pairs

Compare the pairs Z with shifted pairs Z ′ Two parabolas Assumption R′(p) are parabolic and symmetric around p = 1/2, i.e. R′(p) = a′p2 + b′p + c′. We will study D(p) = R(p) − R′(p). Difference between two parabolæ is a parabola

D(p) = ap2 + bp + c for unknown a, b, p.

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 16 / 41

Pairs analysis Homogenous pairs

Structure of the shifted pairs

Compare the pairs Z with shifted pairs Z ′ Two parabolas Assumption R′(p) are parabolic and symmetric around p = 1/2, i.e. R′(p) = a′p2 + b′p + c′. We will study D(p) = R(p) − R′(p). Difference between two parabolæ is a parabola

D(p) = ap2 + bp + c for unknown a, b, p.

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 16 / 41

Pairs analysis Homogenous pairs

Structure of the shifted pairs

Compare the pairs Z with shifted pairs Z ′ Two parabolas Assumption R′(p) are parabolic and symmetric around p = 1/2, i.e. R′(p) = a′p2 + b′p + c′. We will study D(p) = R(p) − R′(p). Difference between two parabolæ is a parabola

D(p) = ap2 + bp + c for unknown a, b, p.

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 16 / 41

Pairs analysis Homogenous pairs

Structure of the shifted pairs

Compare the pairs Z with shifted pairs Z ′ Two parabolas Assumption R′(p) are parabolic and symmetric around p = 1/2, i.e. R′(p) = a′p2 + b′p + c′. We will study D(p) = R(p) − R′(p). Difference between two parabolæ is a parabola

D(p) = ap2 + bp + c for unknown a, b, p.

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 16 / 41

Pairs analysis Homogenous pairs

Structure of the shifted pairs

Compare the pairs Z with shifted pairs Z ′ Two parabolas Assumption R′(p) are parabolic and symmetric around p = 1/2, i.e. R′(p) = a′p2 + b′p + c′. We will study D(p) = R(p) − R′(p). Difference between two parabolæ is a parabola

D(p) = ap2 + bp + c for unknown a, b, p.

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 16 / 41

Pairs analysis Homogenous pairs

Structure of the shifted pairs

Compare the pairs Z with shifted pairs Z ′ Two parabolas Assumption R′(p) are parabolic and symmetric around p = 1/2, i.e. R′(p) = a′p2 + b′p + c′. We will study D(p) = R(p) − R′(p). Difference between two parabolæ is a parabola

D(p) = ap2 + bp + c for unknown a, b, p.

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 16 / 41

Pairs analysis Homogenous pairs

The R′-function

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 17 / 41

Pairs analysis Homogenous pairs

The R′-function

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 17 / 41

Pairs analysis Homogenous pairs

Structure of the shifted pairs (II)

Write Z ′ = b1, b2, b3, . . . , bn Theorem nR′(1/2) =

n−1

• k=1

2−khk, where hk is number of homogenous pairs among (b1, bk+1), (b2, bk+2), . . . , (bn−k, bn).

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 18 / 41

Pairs analysis Homogenous pairs

The estimate of R′(1/2)

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 19 / 41

Pairs analysis Homogenous pairs

Zero message assumption

D(p) = ap2 + bp + c Assumption Z and Z ′ have the same structure when no message is embedded. R(0) = R′(0) D(0) = 0 c = 0

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 20 / 41

Pairs analysis Homogenous pairs

Zero message assumption

D(p) = ap2 + bp + c Assumption Z and Z ′ have the same structure when no message is embedded. R(0) = R′(0) D(0) = 0 c = 0

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 20 / 41

Pairs analysis Homogenous pairs

Zero point

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 21 / 41

Pairs analysis Homogenous pairs

Symmetry

Swapping all bits does not change the statistic Swapping 1 − q random bits means

Swapping all bits, and then (re)swap q random bits

Embedding q or 1 − q bits is the same thing. We conclude, D(q) = D(1 − q)

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 22 / 41

Pairs analysis Homogenous pairs

Symmetry

Swapping all bits does not change the statistic Swapping 1 − q random bits means

Swapping all bits, and then (re)swap q random bits

Embedding q or 1 − q bits is the same thing. We conclude, D(q) = D(1 − q)

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 22 / 41

Pairs analysis Homogenous pairs

Symmetry

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 23 / 41

Pairs analysis Homogenous pairs

Solving a second-order equation

We can estimate R and R′ at q = 1

2

R( 1

2) − R′( 1 2) = D( 1 2) = a/4 + b/2 (left side known)

We exploit symmetry

0 = D(q) − D(1 − q) = (aq2 + bq) − (a(1 − q)2 + b(1 − q)).

We solve for a and b q is now determined by 4D(1 2)q − 4D(1 2)q2 = D(q) where D(1

2) and D(q) are known

Estimated message length is 2p

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 24 / 41

Where Pairs Analysis fails Dithered backgrounds

Outline

1

Background

2

Pairs analysis The characteristic sequence Homogenous pairs

3

Where Pairs Analysis fails Dithered backgrounds

4

RS steganalysis The idea The result Counter-measures

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 25 / 41

Where Pairs Analysis fails Dithered backgrounds

Dithered backgrounds

Dithering is used to simulate additional colours Two colours c1 and c2 alternate over an area. The appearance would be a uniform colour somewhere in between. Colour cut (c1, c2) has many heterogenous pairs. The result is that R(0) = R′(0) and our assumption fails. This can be fixed by clever choice of colour cut. (I did not find any good examples.)

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 26 / 41

Where Pairs Analysis fails Dithered backgrounds

Dithered backgrounds

Dithering is used to simulate additional colours Two colours c1 and c2 alternate over an area. The appearance would be a uniform colour somewhere in between. Colour cut (c1, c2) has many heterogenous pairs. The result is that R(0) = R′(0) and our assumption fails. This can be fixed by clever choice of colour cut. (I did not find any good examples.)

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 26 / 41

Where Pairs Analysis fails Dithered backgrounds

Dithered backgrounds

Dithering is used to simulate additional colours Two colours c1 and c2 alternate over an area. The appearance would be a uniform colour somewhere in between. Colour cut (c1, c2) has many heterogenous pairs. The result is that R(0) = R′(0) and our assumption fails. This can be fixed by clever choice of colour cut. (I did not find any good examples.)

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 26 / 41

Where Pairs Analysis fails Dithered backgrounds

Dithered backgrounds

Dithering is used to simulate additional colours Two colours c1 and c2 alternate over an area. The appearance would be a uniform colour somewhere in between. Colour cut (c1, c2) has many heterogenous pairs. The result is that R(0) = R′(0) and our assumption fails. This can be fixed by clever choice of colour cut. (I did not find any good examples.)

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 26 / 41

Where Pairs Analysis fails Dithered backgrounds

Dithered backgrounds

Dithering is used to simulate additional colours Two colours c1 and c2 alternate over an area. The appearance would be a uniform colour somewhere in between. Colour cut (c1, c2) has many heterogenous pairs. The result is that R(0) = R′(0) and our assumption fails. This can be fixed by clever choice of colour cut. (I did not find any good examples.)

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 26 / 41

RS steganalysis The idea

Outline

1

Background

2

Pairs analysis The characteristic sequence Homogenous pairs

3

Where Pairs Analysis fails Dithered backgrounds

4

RS steganalysis The idea The result Counter-measures

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 27 / 41

RS steganalysis The idea

References

Core Reading «Quantitative steganalysis of digital images: estimating the secret message length» by Jessica Fridrich, Miroslav Goljan, Dorin Hogea, David Soukal, in Multimedia Systems 2003 Suggested Reading Jessica Fridrich, Miroslav Goljan, and Rui Du (State University of New York, Binghamton) ‘Detecting LSB Steganography in Color and Gray-Scale Images’ in Multimedia and Security 2001

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 28 / 41

RS steganalysis The idea

RS steganalysis

Proposed for true colour images Use information from all 8 bits of a pixel Linked to so-called lossless capacity

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 29 / 41

RS steganalysis The idea

Pixel groups and smoothness

Divide image into pixel groups G1, G2, . . .

Disjoint groups Consecutive pixels

Define the smoothness of a group G = (x1, x2, . . . , xn) f(G) =

n−1

• i=1

|xi − xi+1|. High f(G) means sharp changes from pixel to pixel.

Unusual for neighbour pixels in natural images

Compare f(F(G)) and f(G)

where F is flips pixels.

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 30 / 41

RS steganalysis The idea

Pixel groups and smoothness

Divide image into pixel groups G1, G2, . . .

Disjoint groups Consecutive pixels

Define the smoothness of a group G = (x1, x2, . . . , xn) f(G) =

n−1

• i=1

|xi − xi+1|. High f(G) means sharp changes from pixel to pixel.

Unusual for neighbour pixels in natural images

Compare f(F(G)) and f(G)

where F is flips pixels.

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 30 / 41

RS steganalysis The idea

Pixel groups and smoothness

Divide image into pixel groups G1, G2, . . .

Disjoint groups Consecutive pixels

Define the smoothness of a group G = (x1, x2, . . . , xn) f(G) =

n−1

• i=1

|xi − xi+1|. High f(G) means sharp changes from pixel to pixel.

Unusual for neighbour pixels in natural images

Compare f(F(G)) and f(G)

where F is flips pixels.

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 30 / 41

RS steganalysis The idea

Pixel groups and smoothness

Divide image into pixel groups G1, G2, . . .

Disjoint groups Consecutive pixels

Define the smoothness of a group G = (x1, x2, . . . , xn) f(G) =

n−1

• i=1

|xi − xi+1|. High f(G) means sharp changes from pixel to pixel.

Unusual for neighbour pixels in natural images

Compare f(F(G)) and f(G)

where F is flips pixels.

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 30 / 41

RS steganalysis The idea

Bit flipping

Maps on a single pixel

F+ : 2i ↔ 2i + 1 F− : 2i ↔ 2i − 1 F0 is the identity.

Maps on a group, say of four, G = (x1, x2, x3, x4)

F = [F0F+F+F0] = [0110] F(G) = (F0(x1), F1(x2), F1(x3), F0(x4))

Shifted bit flip

−F = [F0F−F−F0] = [0 − 1 − 10]

We will use one map F and the shift −F.

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 31 / 41

RS steganalysis The idea

Bit flipping

Maps on a single pixel

F+ : 2i ↔ 2i + 1 F− : 2i ↔ 2i − 1 F0 is the identity.

Maps on a group, say of four, G = (x1, x2, x3, x4)

F = [F0F+F+F0] = [0110] F(G) = (F0(x1), F1(x2), F1(x3), F0(x4))

Shifted bit flip

−F = [F0F−F−F0] = [0 − 1 − 10]

We will use one map F and the shift −F.

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 31 / 41

RS steganalysis The idea

Bit flipping

Maps on a single pixel

F+ : 2i ↔ 2i + 1 F− : 2i ↔ 2i − 1 F0 is the identity.

Maps on a group, say of four, G = (x1, x2, x3, x4)

F = [F0F+F+F0] = [0110] F(G) = (F0(x1), F1(x2), F1(x3), F0(x4))

Shifted bit flip

−F = [F0F−F−F0] = [0 − 1 − 10]

We will use one map F and the shift −F.

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 31 / 41

RS steganalysis The idea

Characteristic Groups

The groups

Regular group: f(G) < f(F(G)) Singular group: f(G) > f(F(G)) Useless group: f(G) = f(F(G))

The statistics

RF : number of regular groups under F SF : number of singular groups under F R−F : number of regular groups under −F S−F : number of singular groups under −F

RF, R−F, SF, S−F as functions of p

p is number of pixels flipped by the embedding.

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 32 / 41

RS steganalysis The idea

Characteristic Groups

The groups

Regular group: f(G) < f(F(G)) Singular group: f(G) > f(F(G)) Useless group: f(G) = f(F(G))

The statistics

RF : number of regular groups under F SF : number of singular groups under F R−F : number of regular groups under −F S−F : number of singular groups under −F

RF, R−F, SF, S−F as functions of p

p is number of pixels flipped by the embedding.

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 32 / 41

RS steganalysis The idea

Characteristic Groups

The groups

Regular group: f(G) < f(F(G)) Singular group: f(G) > f(F(G)) Useless group: f(G) = f(F(G))

The statistics

RF : number of regular groups under F SF : number of singular groups under F R−F : number of regular groups under −F S−F : number of singular groups under −F

RF, R−F, SF, S−F as functions of p

p is number of pixels flipped by the embedding.

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 32 / 41

RS steganalysis The idea

Statistics plot

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 33 / 41

RS steganalysis The result

Outline

1

Background

2

Pairs analysis The characteristic sequence Homogenous pairs

3

Where Pairs Analysis fails Dithered backgrounds

4

RS steganalysis The idea The result Counter-measures

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 34 / 41

RS steganalysis The result

Approximations

Approximations based on experimental investigation

SF and RF are parabolic S−F and R−F are linear

RF(p) = a1p2 + b1p + c1, (3) SF(p) = a2p2 + b2p + c2, (4) R−F(p) = a3p + b3, (5) S−F(p) = a4p + b4. (6) 11 unknowns (10 coefficients and p)

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 35 / 41

RS steganalysis The result

Approximations

Approximations based on experimental investigation

SF and RF are parabolic S−F and R−F are linear

RF(p) = a1p2 + b1p + c1, (3) SF(p) = a2p2 + b2p + c2, (4) R−F(p) = a3p + b3, (5) S−F(p) = a4p + b4. (6) 11 unknowns (10 coefficients and p)

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 35 / 41

RS steganalysis The result

Approximations

Approximations based on experimental investigation

SF and RF are parabolic S−F and R−F are linear

RF(p) = a1p2 + b1p + c1, (3) SF(p) = a2p2 + b2p + c2, (4) R−F(p) = a3p + b3, (5) S−F(p) = a4p + b4. (6) 11 unknowns (10 coefficients and p)

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 35 / 41

RS steganalysis The result

Equations

Observations

RF(p), SF(p), R−F(p), S−F(p) from stegogramme RF(1 − p), SF(1 − p), R−F(1 − p), S−F(1 − p) by flipping all LSB-s.

Assumptions

RF(1/2) = SF(1/2) RF(0) = R−F(0) SF(0) = S−F(0).

11 equations

With 11 unknowns, this can be solved

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 36 / 41

RS steganalysis The result

Equations

Observations

RF(p), SF(p), R−F(p), S−F(p) from stegogramme RF(1 − p), SF(1 − p), R−F(1 − p), S−F(1 − p) by flipping all LSB-s.

Assumptions

RF(1/2) = SF(1/2) RF(0) = R−F(0) SF(0) = S−F(0).

11 equations

With 11 unknowns, this can be solved

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 36 / 41

RS steganalysis The result

Equations

Observations

RF(p), SF(p), R−F(p), S−F(p) from stegogramme RF(1 − p), SF(1 − p), R−F(1 − p), S−F(1 − p) by flipping all LSB-s.

Assumptions

RF(1/2) = SF(1/2) RF(0) = R−F(0) SF(0) = S−F(0).

11 equations

With 11 unknowns, this can be solved

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 36 / 41

RS steganalysis The result

The message length

p = x x − 1/2, where x is the smaller root of s(d3 + d1)x2 + (d2 − d3 − d4 − 3d1)x + d1 − d2 = 0, where d1 = RF(p/2) − SF(p/2), d2 = R−F(p/2) − S−F(p/2), d3 = RF(1 − p/2) − SF(1 − p/2), d4 = R−F(1 − p/2) − S−F(1 − p/2).

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 37 / 41

RS steganalysis The result

Initial bias

Some experiments show estimates within ±1% of true length Some images have an initial bias

i.e. the cover image appear to have a short message. This must be taken into account Short messages cannot be detected with certainty

Gaussian distribution: µ = 0, σ = 0.5% Is it possible to estimate the initial bias?

Plot from Fridrich et al.

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 38 / 41

RS steganalysis The result

Example from Fridrich et al.

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 39 / 41

RS steganalysis Counter-measures

Outline

1

Background

2

Pairs analysis The characteristic sequence Homogenous pairs

3

Where Pairs Analysis fails Dithered backgrounds

4

RS steganalysis The idea The result Counter-measures

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 40 / 41

RS steganalysis Counter-measures

Good stego-systems?

How do we foil higher-order statistics?

Stegogramme should resemble cover-image Statistics-aware steganography

Designed for specific higher-order statistics Stegogramme resembles cover with respect to statistic Still ad hoc approach

Dr Hans Georg Schaathun LSB detection by Pairs Analysis Spring 2007 41 / 41