+This section first presents the embedding scheme through its
+four main steps: the data encryption (Sect.~\ref{sub:bbs}),
+the cover pixel selection (Sect.~\ref{sub:edge}),
+the adaptive payload considerations (Sect.~\ref{sub:adaptive}),
+and how the distortion has been minimized (Sect.~\ref{sub:stc}).
+The message extraction is then presented (Sect.~\ref{sub:extract}) while a running example ends this section (Sect.~\ref{sub:xpl}).
+
+
The flowcharts given in Fig.~\ref{fig:sch}
summarize our steganography scheme denoted by
-STABYLO, which stands for STeganography with cAnny, Bbs, binarY embedding at LOw cost.
-What follows are successively details of the inner steps and flows inside
-both the embedding stage (Fig.~\ref{fig:sch:emb})
-and the extraction one (Fig.~\ref{fig:sch:ext}).
+STABYLO, which stands for STe\-ga\-no\-gra\-phy with
+Adaptive, Bbs, binarY embedding at LOw cost.
+What follows are successively some details of the inner steps and the flows both inside
+ the embedding stage (Fig.~\ref{fig:sch:emb})
+and inside the extraction one (Fig.~\ref{fig:sch:ext}).
Let us first focus on the data embedding.
-\begin{figure*}[t]
+\begin{figure*}%[t]
\begin{center}
- \subfloat[Data Embedding.]{
- \begin{minipage}{0.49\textwidth}
+ \subfloat[Data Embedding]{
+ \begin{minipage}{0.4\textwidth}
\begin{center}
- %\includegraphics[width=5cm]{emb.pdf}
- \includegraphics[scale=0.5]{emb.ps}
+ %\includegraphics[scale=0.45]{emb}
+ \includegraphics[scale=0.4]{emb}
\end{center}
\end{minipage}
\label{fig:sch:emb}
- }%\hfill
- \subfloat[Data Extraction.]{
+ }
+\hfill
+ \subfloat[Data Extraction]{
\begin{minipage}{0.49\textwidth}
\begin{center}
- %\includegraphics[width=5cm]{rec.pdf}
- \includegraphics[scale=0.5]{rec.ps}
+ \includegraphics[scale=0.4]{dec}
\end{center}
\end{minipage}
\label{fig:sch:ext}
}%\hfill
\end{center}
- \caption{The STABYLO Scheme.}
+ \caption{The STABYLO scheme}
\label{fig:sch}
\end{figure*}
-\subsection{Security Considerations}
-Among methods of message encryption/decryption
+\subsection{Security considerations}\label{sub:bbs}
+\JFC{To provide a self-contained article without any bias, we shortly
+pressent the retained encryption process.}
+Among the methods of message encryption/decryption
(see~\cite{DBLP:journals/ejisec/FontaineG07} for a survey)
-we implement the Blum-Goldwasser cryptosystem~\cite{Blum:1985:EPP:19478.19501}
+we implement the asymmetric
+Blum-Goldwasser cryptosystem~\cite{Blum:1985:EPP:19478.19501}
that is based on the Blum Blum Shub~\cite{DBLP:conf/crypto/ShubBB82}
pseudorandom number generator (PRNG) and the
XOR binary function.
-It has been indeed proven~\cite{DBLP:conf/crypto/ShubBB82} that this PRNG
+The main justification of this choice
+is that it has been proven~\cite{DBLP:conf/crypto/ShubBB82} that this PRNG
has the property of cryptographical security, \textit{i.e.},
for any sequence of $L$ output bits $x_i$, $x_{i+1}$, \ldots, $x_{i+L-1}$,
there is no algorithm, whose time complexity is polynomial in $L$, and
-which allows to find $x_{i-1}$ and $x_{i+L}$ with a probability greater
+which allows to find $x_{i-1}$ or $x_{i+L}$ with a probability greater
than $1/2$.
Equivalent formulations of such a property can
be found. They all lead to the fact that,
this step computes a message $m$, which is the encrypted version of \textit{mess}.
-\subsection{Edge-Based Image Steganography}
+\subsection{Edge-based image steganography}\label{sub:edge}
The edge-based image
edges in images (whose noise has been initially reduced).
They can be separated in two categories: first and second order detection
methods on the one hand, and fuzzy detectors on the other hand~\cite{KF11}.
-In first order methods like Sobel, Canny~\cite{Canny:1986:CAE:11274.11275}, \ldots,
+In first order methods like Sobel, Canny~\cite{Canny:1986:CAE:11274.11275}, and so on,
a first-order derivative (gradient magnitude, etc.) is computed
to search for local maxima, whereas in second order ones, zero crossings in a second-order derivative, like the Laplacian computed from the image,
are searched in order to find edges.
-As for as fuzzy edge methods are concerned, they are obviously based on fuzzy logic to highlight edges.
+As far as fuzzy edge methods are concerned, they are obviously based on fuzzy logic to highlight edges.
-Canny filters, on their parts, are an old family of algorithms still remaining a state-of-the-art edge detector. They can be well approximated by first-order derivatives of Gaussians.
-As the Canny algorithm is well known and studied, fast, and implementable
-on many kinds of architectures like FPGAs, smartphones, desktop machines, and
+Canny filters, on their parts, are an old family of algorithms still remaining a state of the art edge detector. They can be well-approximated by first-order derivatives of Gaussians.
+As the Canny algorithm is fast, well known, has been studied in depth, and is implementable
+on many kinds of architectures like FPGAs, smart phones, desktop machines, and
GPUs, we have chosen this edge detector for illustrative purpose.
+
+
+
This edge detection is applied on a filtered version of the image given
as input.
-More precisely, only $b$ most
-significant bits are concerned by this step, where
-the parameter $b$ is practically set with $6$ or $7$.
+More precisely, only $b$ most significant bits are concerned by this step,
+where the parameter $b$ is practically set with $6$ or $7$.
+Notice that only the 2 LSBs of pixels in the set of edges
+are returned if $b$ is 6, and the LSB of pixels if $b$ is 7.
If set with the same value $b$, the edge detection returns thus the same
set of pixels for both the cover and the stego image.
-In our flowcharts, this is represented by ``edgeDetection(b bits)''.
-Then only the 2 LSBs of pixels in the set of edges are returned if $b$ is 6,
-and the LSB of pixels if $b$ is 7.
-Let $x$ be the sequence of these bits.
+Moreover, to provide edge gradient value,
+the Canny algorithm computes derivatives
+in the two directions with respect to a mask of size $T$.
+The higher $T$ is, the coarse the approach is. Practically,
+$T$ is set with $3$, $5$, or $7$.
+In our flowcharts, this step is represented by ``Edge Detection(b, T, X)''.
+
+
+Let $x$ be the sequence of these bits.
+The next section presents how to adapt our scheme
+with respect to the size
+of the message $m$ to embed and the size of the cover $x$.
+
+
+
-\JFC{il faudrait comparer les complexites des algo fuzy and canny}
-% First of all, let us discuss about compexity of edge detetction methods.
-% Let then $M$ and $N$ be the dimension of the original image.
-% According to~\cite{Hu:2007:HPE:1282866.1282944},
-% even if the fuzzy logic based edge detection methods~\cite{Tyan1993}
-% have promising results, its complexity is in $C_3 \times O(M \times N)$
-% whereas the complexity on the Canny method~\cite{Canny:1986:CAE:11274.11275}
-% is in $C_1 \times O(M \times N)$ where $C_1 < C_3$.
-% \JFC{Verifier ceci...}
-% In experiments detailled in this article, the Canny method has been retained
-% but the whole approach can be updated to consider
-% the fuzzy logic edge detector.
+
+
+\subsection{Adaptive embedding rate}\label{sub:adaptive}
+Two strategies have been developed in our approach,
+depending on the embedding rate that is either \emph{Adaptive} or \emph{Fixed}.
+In the former the embedding rate depends on the number of edge pixels.
+The higher it is, the larger the message length that can be inserted is.
+Practically, a set of edge pixels is computed according to the
+Canny algorithm with parameters $b=7$ and $T=3$.
+The message length is thus defined to be lesser than
+half of this set cardinality.
+If $x$ is too short for $m$, the message is split into sufficient parts
+and a new cover image should be used for the remaining part of the message.
+
+In the latter, the embedding rate is defined as a percentage between the
+number of modified pixels and the length of the bit message.
+This is the classical approach adopted in steganography.
+Practically, the Canny algorithm generates
+a set of edge pixels related to increasing values of $T$ and
+until its cardinality
+is sufficient. Even in this situation, our scheme adapts
+its algorithm to meet all the user's requirements.
+
+
+Once the map of possibly modified pixels is computed,
+two methods may further be applied to extract bits that
+are really changed.
+The first one randomly chooses the subset of pixels to modify by
+applying the BBS PRNG again. This method is further denoted as a \emph{sample}.
+Once this set is selected, a classical LSB replacement is applied to embed the
+stego content.
+The second method considers the last significant bits of all the pixels
+inside the previous map. It next directly applies the STC
+algorithm~\cite{DBLP:journals/tifs/FillerJF11}.
+It is further referred to as \emph{STC} and is detailed in the next section.
-As argue in the introduction section, we do not adapt the parameters of the
-the edge detection as in~\cite{Luo:2010:EAI:1824719.1824720} but we modify
-the size of the embedding message. Practically, the lenght of $x$
-has to be at least twice as large
-as the size of the embedded encrypted message.
-Otherwise, a new image is used to hide the remaning part of the message.
-\subsection{Minimizing Distortion with Syndrome-Treillis Codes}
+
+\subsection{Minimizing distortion with Syndrome-Trellis Codes}\label{sub:stc}
\input{stc}
% but the whole approach can be updated to consider
% the fuzzy logic edge detector.
-% Next, following~\cite{Luo:2010:EAI:1824719.1824720}, our scheme automatically
-% modifies Canny parameters to get a sufficiently large set of edge bits: this
-% one is practically enlarged untill its size is at least twice as many larger
-% than the size of embedded message.
+For a given set of parameters,
+the Canny algorithm returns a numerical value and
+states whether a given pixel is an edge or not.
+In this article, in the Adaptive strategy
+we consider that all the edge pixels that
+have been selected by this algorithm have the same
+distortion cost \textit{i.e.} $\rho_X$ is always 1 for these bits.
+In the Fixed strategy, since pixels that are detected to be edge
+with small values of $T$ (e.g. when $T=3$)
+are more accurate than these with higher values of $T$,
+we give to STC the following distortion map of the corresponding bits
+$$
+\rho_X= \left\{
+\begin{array}{l}
+1 \textrm{ if an edge for $T=3$} \\
+10 \textrm{ if an edge for $T=5$} \\
+100 \textrm{ if an edge for $T=7$}
+\end{array}
+\right.
+$$.
+
+
+
+
+\subsection{Data extraction}\label{sub:extract}
+The message extraction summarized in Fig.~\ref{fig:sch:ext}
+follows the data embedding approach
+since there exists a reverse function for all its steps.
+
+More precisely, let $b$ be the most significant bits and
+$T$ be the size of the Canny mask, both be given as a key.
+Thus, the same edge detection is applied on a stego content $Y$ to
+produce the sequence $y$ of LSBs.
+If the STC approach has been selected in embedding, the STC reverse
+algorithm is directly executed to retrieve the encrypted message.
+This inverse function takes the $\hat{H}$ matrix as a parameter.
+Otherwise, \textit{i.e.}, if the \emph{sample} strategy is retained,
+the same random bit selection than in the embedding step
+is executed with the same seed, given as a key.
+Finally, the Blum-Goldwasser decryption function is executed and the original
+message is extracted.
+
+
+\subsection{Running example}\label{sub:xpl}
+In this example, the cover image is Lena,
+which is a $512\times512$ image with 256 grayscale levels.
+The message is the poem Ulalume (E. A. Poe), which is constituted by 104 lines, 667
+words, and 3,754 characters, \textit{i.e.}, 30,032 bits.
+Lena and the first verses are given in Fig.~\ref{fig:lena}.
+
+\begin{figure}
+\begin{center}
+\begin{minipage}{0.49\linewidth}
+\begin{center}
+\includegraphics[scale=0.20]{lena512}
+\end{center}
+\end{minipage}
+\begin{minipage}{0.49\linewidth}
+\begin{flushleft}
+\begin{scriptsize}
+The skies they were ashen and sober;\linebreak
+$\qquad$ The leaves they were crisped and sere—\linebreak
+$\qquad$ The leaves they were withering and sere;\linebreak
+It was night in the lonesome October\linebreak
+$\qquad$ Of my most immemorial year;\linebreak
+It was hard by the dim lake of Auber,\linebreak
+$\qquad$ In the misty mid region of Weir—\linebreak
+It was down by the dank tarn of Auber,\linebreak
+$\qquad$ In the ghoul-haunted woodland of Weir.
+\end{scriptsize}
+\end{flushleft}
+\end{minipage}
+\end{center}
+\caption{Cover and message examples} \label{fig:lena}
+\end{figure}
+
+The edge detection returns 18,641 and 18,455 pixels when $b$ is
+respectively 7 and 6 and $T=3$.
+These edges are represented in Figure~\ref{fig:edge}.
+When $b$ is 7, it remains one bit per pixel to build the cover vector.
+This configuration leads to a cover vector of size 18,641 if b is 7
+and 36,910 if $b$ is 6.
+
+\begin{figure}[t]
+ \begin{center}
+ \subfloat[$b$ is 7.]{
+ \begin{minipage}{0.49\linewidth}
+ \begin{center}
+ %\includegraphics[width=5cm]{emb.pdf}
+ \includegraphics[scale=0.20]{edge7}
+ \end{center}
+ \end{minipage}
+ %\label{fig:sch:emb}
+ }%\hfill
+ \subfloat[$b$ is 6.]{
+ \begin{minipage}{0.49\linewidth}
+ \begin{center}
+ %\includegraphics[width=5cm]{rec.pdf}
+ \includegraphics[scale=0.20]{edge6}
+ \end{center}
+ \end{minipage}
+ %\label{fig:sch:ext}
+ }%\hfill
+ \end{center}
+ \caption{Edge detection wrt $b$ with $T=3$}
+ \label{fig:edge}
+\end{figure}
+
+The STC algorithm is optimized when the rate between message length and
+cover vector length is lower than 1/2.
+So, only 9,320 bits are available for embedding
+in the configuration where $b$ is 7.
-%%RAPH: paragraphe en double :-)
+When $b$ is 6, we could have considered 18,455 bits for the message.
+However, first experiments have shown that modifying this number of bits is too
+easily detectable.
+So, we choose to modify the same amount of bits (9,320) and keep STC optimizing
+which bits to change among the 36,910 ones.
+In the two cases, about the third part of the poem is hidden into the cover.
+Results with {Adaptive} and {STC} strategies are presented in
+Fig.~\ref{fig:lenastego}.
+
+\begin{figure}[t]
+ \begin{center}
+ \subfloat[$b$ is 7.]{
+ \begin{minipage}{0.49\linewidth}
+ \begin{center}
+ %\includegraphics[width=5cm]{emb.pdf}
+ \includegraphics[scale=0.20]{lena7}
+ \end{center}
+ \end{minipage}
+ %\label{fig:sch:emb}
+ }%\hfill
+ \subfloat[$b$ is 6.]{
+ \begin{minipage}{0.49\linewidth}
+ \begin{center}
+ %\includegraphics[width=5cm]{rec.pdf}
+ \includegraphics[scale=0.20]{lena6}
+ \end{center}
+ \end{minipage}
+ %\label{fig:sch:ext}
+ }%\hfill
+ \end{center}
+ \caption{Stego images wrt $b$}
+ \label{fig:lenastego}
+\end{figure}
+
+
+Finally, differences between the original cover and the stego images
+are presented in Fig.~\ref{fig:lenadiff}. For each pair of pixel $X_{ij}$ and $Y_{ij}$ ($X$ and $Y$ being the cover and the stego content respectively),
+the pixel value $V_{ij}$ of the difference is defined with the following map
+$$
+V_{ij}= \left\{
+\begin{array}{rcl}
+0 & \textrm{if} & X_{ij} = Y_{ij} \\
+75 & \textrm{if} & \vert X_{ij} - Y_{ij} \vert = 1 \\
+150 & \textrm{if} & \vert X_{ij} - Y_{ij} \vert = 2 \\
+225 & \textrm{if} & \vert X_{ij} - Y_{ij} \vert = 3
+\end{array}
+\right.
+$$
+This function allows to emphasize differences between contents.
+Notice that since $b$ is 7 in Fig.~\ref{fig:diff7}, the embedding is binary
+and this image only contains 0 and 75 values.
+Similarly, if $b$ is 6 as in Fig.~\ref{fig:diff6}, the embedding is ternary
+and the image contains all the values in $\{0,75,150,225\}$.
+
+
+
+\begin{figure}[t]
+ \begin{center}
+ \subfloat[$b$ is 7.]{
+ \begin{minipage}{0.49\linewidth}
+ \begin{center}
+ %\includegraphics[width=5cm]{emb.pdf}
+ \includegraphics[scale=0.20]{diff7}
+ \end{center}
+ \end{minipage}
+ \label{fig:diff7}
+ }%\hfill
+ \subfloat[$b$ is 6.]{
+ \begin{minipage}{0.49\linewidth}
+ \begin{center}
+ %\includegraphics[width=5cm]{rec.pdf}
+ \includegraphics[scale=0.20]{diff6}
+ \end{center}
+ \end{minipage}
+ \label{fig:diff6}
+ }%\hfill
+ \end{center}
+ \caption{Differences with Lena's cover wrt $b$}
+ \label{fig:lenadiff}
+\end{figure}
-\subsection{Data Extraction}
-The message extraction summarized in Fig.~\ref{fig:sch:ext} follows data embedding
-since there exists a reverse function for all its steps.
-First of all, the same edge detection is applied (on the 7 first bits) to
-get the set of LSBs,
-which is sufficiently large with respect to the message size given as a key.
-Then the STC reverse algorithm is applied to retrieve the encrypted message.
-Finally, the Blum-Goldwasser decryption function is executed and the original
-message is extracted.
