Writing on Wet Paper

IEEE Transactions on Signal Processing, Vol. 53, Issue 10, Part 2, Oct. 2005

by Jessica Fridrich, Miroslav Goljan, Petr Lisonek and David Soukal

Outlines

Introduction Wet Paper Code Practical Wet Paper Code Experiment

Introduction

Main building blocks of steganographic algorithm The choice of the cover work The embedding and extracting algorithm

Symbol assignment function The embedding modification The selection rule

Stego key management

Wet Paper Code

Scenario The sender wants to communicate q bits

m=(m1, …, mq)T. Both of sender and receiver know the sha

red key and the length of message q. A binary column vector and a set

of indices , of those bits that can be modified to embed. (cover media)

Wet Paper Code

“Dry” pixels: may be modified by the sender

“Wet” pixels: are not to be modified during embedding

Wet Paper Code - Encode

Use a shared stego key to generate a pseudo-random binary matrix D (q×n)

Modify some bj, j ∈ C, the modified binary column vector satisfies

Wet Paper Code - Decode

Use a shared stego key to generate a pseudo-random binary matrix D (q×n)

Obtain the message m from

Wet Paper Code - message length q

The sender can reserve the first bits of the message m for a header to inform the recipient of the number of rows in D

The recipient first generators the first rows of D, multiplies by the received vector b’ to get the message length q

Wet Paper Code– Average maximal payload

Use variable v=b’-b to rewrite (1) to

k dry pixels, unknown values, vi, i ∈ C

n-k wet pixels, zeros, vi, i ∉ C

Wet Paper Code– Average maximal payload

Remove from D all n-k columns i, i ∉ C Remove from v all n-k elements vi, i ∉ C

where H is a binary q×k matrix consisting of those columns of D corresponding to indices C, and v is an unknown k×1 binary vector

Wet Paper Code– Average maximal payload

The solution of (3) for arbitrary message m as long as rank(H)=q

The probability Pq,k(s) that the rank of a random q×k binary matrix is s, s≤min(q,k) is

Wet Paper Code– Average maximal payload

From (4), it shows that for a large fixed k, quickly approach 1 with decreasing q<k (Fig.1)

The expected number of bits (q) that can be communicated is approximately equal to k

Wet Paper Code– Average maximal payload

Practical Wet Paper Code

Assuming that the maximal length message q=k is sent, the complexity of Gaussian elimination for (3) is O(k3)

Practical Wet Paper Code

The best performance and most flexible method for (3) Divide bit-stream b into β disjoint pseudo-

random subsets Bi

Use Gaussian elimination on each subset separately

If the factor is β, improve O(k3) to O(k3/β2)

Practical Wet Paper Code

Some definitions The range of the rate of communicating r=k/

n, r1≤r≤r2

The changeable bits in each subset kavg~250 The number of sets, β= The size ni of each subset Bi will be

chosen so that

Practical Wet Paper Code

Some definitions The number of changeable bits ki varies f

or each subset Bi and follows the hypergeometric distribution with mean value k/β

b=(b(1), b(2), …, b(β)), b(i) is a vector of ni bits from Bi

r1, r2 and kavg are publicly known parameter by parties

Practical Wet Paper Code

Practical Wet Paper Code

Problem: The encoding process may fail in the last

subset because this is only subset in which the sender doesn’t have the freedom to decrease qβ

Solution Start dividing the message bits with q+10

rather than q

Practical Wet Paper Code – Encoder (1/4)

Practical Wet Paper Code – Encoder (2/4)

Practical Wet Paper Code – Encoder (3/4)

Practical Wet Paper Code – Encoder (4/4)

Practical Wet Paper Code - Decoder

Practical Wet Paper Code - Decoder

Experiment

A different example of a SR is given when the information-reducing transformation is recompression of the cover JPEG image using a lower JPEG quality factor.

Use the article, “Feature-Based Steganalysis for JPEG Images and its Implications for Future Design of Steganographic Schemes,” method to show the security of the proposed method.

Experiment

The detection accuracy ρ = 2A–1, where A is the area under the ROC curve, for a simple linear classifier trained on 1400 cover and 1400 stego (fully embedded) images and tested on 400 never seen image.

Double compression for different embedding rates expressed using bpc = bits per non-zero stego DCT coefficient (U = unachievable rate)

Experiment

Testing Methods F5: F5 F5_111: F5 with matrix embedding (1,1,1) OQ: OutGuess 0.2 MB1: model Based Steganography without debl

ocking MB2: model Based Steganography with deblocki

ng PQ: the proposed Perturbed Quantization

Experiment