Using High-Dimensional Image Models to Perform Highly Undetectable Steganography

(1)

HAL Id: hal-00541353

https://hal.archives-ouvertes.fr/hal-00541353

Submitted on 30 Nov 2010

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Using High-Dimensional Image Models to Perform

Highly Undetectable Steganography

Tomas Pevny, Tomas Filler, Patrick Bas

To cite this version:

Tomas Pevny, Tomas Filler, Patrick Bas. Using High-Dimensional Image Models to Perform Highly

Undetectable Steganography.

Information Hiding, Jun 2010, Calgary, Canada.

pp.2010.

�hal-00541353�

(2)

1 2 3 1 2 3 107 . 7×

(3)

±1 ±1

(4)

X _{= (x}_ij_{) ∈ X = {0, . . . , 255}}n1×n2 _Y _{= (y} ij) ∈ X n = n1n2 X_{= (x}_i₎n i=1 Y= (yi)ni=1 D : X × X → [0, ∞]. Y D(X, Y) |xi−yi| ≤ 1, e + 1

(5)

i ∈ {1, . . . , n}, D(X, Y) = n ! i=1 ρi|xi− yi|. 0 ≤ ρi≤ ∞ ρi = ∞ D m n ρ = (ρi)n_i=1 0 ≤ ρi < ∞ i ∈ {1, . . . , n} 0 ≤ m ≤ n D (m, n, ρ) = n ! i=1 piρi, pi= e−λρi 1 + e−λρi i λ − n ! i=1 " pilog2pi+ (1 − pi) log2(1 − pi) # = m. ρi) pi ρi

(6)

ρi

8 {←, →, ↓, ↑, *, +, ,, -}

I_{∈ X} _n₁_{× n}₂

D•_,

(7)

i ∈ {1, . . . , n1}, j ∈ {1, . . . , n2− 1}. M→_d 1d2= P r(D → i,j+1= d1|D→ij = d2), M→_d 1d2d3 = P r(D → i,j+2 = d1|D→i,j+1 = d2, D→ij = d3), di ∈ {−T, . . . , T }. F•_1,...,k₌1 4$M → • + M←• + M↓•+ M↑•% , F•_k+1,...,2k ₌1 4$M ' • + M(• + M)• + M*• % , k = (2T + 1)2 _{k = (2T + 1)}3 T = 4 162 T = 3 686 C→_d 1d2 = P r(D → ij = d1, D→i,j+1= d2), C→_d 1d2d3 = P r(D → ij = d1, D→i,j+1= d2, D→i,j+2 = d3). T = 3 {Ck d1d2, C k d1d2d3|k ∈ {→, ↑, *, -}, −3 ≤ di≤ 3} 4×(343+49) = 1568 ρi C→ d1d2d3= C ← −d3,−d2,−d1 M → d1d2d3= C → d3d2d1/C → d2d1

(8)

T. T ρi. ρi ρi ρi

(9)

−6 −2 0 2 6 −6 −2 0 2 6 0 1 d2 d1 FLD criteria between CX,→ d1d2and C Y,→ d1d2 features −6 0 6 −6 0 6 0 5 · 10−2 0.1 d2 d1 Mean of CX,→ d1d2 feature C→ d1d2 C→ d1d2 50% ρi, C→ d1d2 d1 d2 &E[CX,→ d1d2] − E[C Y,→ d1d2] '2 E$CX,→ d1d2 − E[C X,→ d1d2] %2 + E$CY,→ d1d2 − E[C Y,→ d1d2] %2, E[·] CX,→_d 1d2 C Y,→ d1d2 C→_d 1d2 C→_−2,2 C→_2,−2

(10)

C→ 0,0, C→d,d, C→_−2,2 C→_2,−2 C•_d 1d2d3 ρi. Cover Distortion computation Coding Model correction Stego High dimensional model

10800 512 × 512.

PE= min1

2&PFp+ PFn', PFp PFn

(11)

T = 4 T = 3 k(x, y) = exp(−γ .x − y.2) γ C (C, γ) ∈ ((10k_{, 2}j_{)|k ∈ {−3, . . . , 4}, j ∈ {−d − 3, −d + 3}}) d D ρi D D(X, Y) = T ! d1,d2,d3=−T  w(d1, d2, d3) , , , , , , ! k∈{→,←,↑,↓} CX,k d1d2d3− C Y,k d1d2d3 , , , , , , + +w(d1, d2, d3) , , , , , , ! k∈{',(,),*} CX,k d1,d2,d3− C Y,k d1,d2,d3 , , , , , ,  , w(d1, d2, d3) w(d1, d2, d3) w(d1, d2, d3) = 1 / 0d2 1+ d22+ d23+ σ 1γ, σ, γ > 0 C•_d 1d2d3 T = 3 w(d1, d2, d3) = 1 di T = 255 ρi

(12)

d1, d2, d3 w(d1, d2, d3) ρi,j= D(X, Yi,j), Yi,j _{(i, j)} X_. ±1 (ρi,j)

(13)

D D(X, Yi_), _Yi X _i ρi,j ρi,j ρi,j ρi,j T = 90, 512 × 512 T, γ σ, T, σ γ T T = 90 107 99% T

(14)

0.5 1 2 4 10−4 10−3 10−2 10−1 100 101 10−4 10−2 γ σ MMD 0.5 1 2 4 10−4 10−3 10−2 10−1 100 101 10−5 10−3 10−1 γ σ MMD γ σ (σ, γ) ∈ ((10k_{, 2}j_{)|k ∈ {−3, . . . , 1}, j ∈ {−1, 2}}) 0.25 γ σ γ = 4 σ = 10. PE= 40% 0.25 0.4

(15)

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.1 0.3 0.5 ternary LSB match. MC S2 no Model Correct. MC S1 HUGO simulated STC h = 10 Relative payload (bpp) Er ro r PE

(a) 2nd _{order SPAM}

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.1 0.3 0.5 WAM SPAM 1st SPAM 2nd CDF ternary LSB matching HUGO MC S2 Relative payload (bpp) Er ro r PE (b) featureset comparison PE T = 3 h = 10 PE d l(d) = (αOPT− αACT)/αOPT αOPT

αACT d 0 ≤ l(d) ≤ 1 l(d) 3% − 7% ρ h 0.30 40%.

(16)

700% PE= 40%

107

ρ

7×

(17)

(18)