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}).
Let us first focus on the data embedding. 

\begin{figure*}[t]
  \begin{center}
    \subfloat[Data Embedding.]{
      \begin{minipage}{0.49\textwidth}
        \begin{center}
          %\includegraphics[width=5cm]{emb.pdf}
          \includegraphics[scale=0.5]{emb.ps}
        \end{center}
      \end{minipage}
      \label{fig:sch:emb}
    }%\hfill
    \subfloat[Data Extraction.]{
      \begin{minipage}{0.49\textwidth}
        \begin{center}
          %\includegraphics[width=5cm]{rec.pdf}
          \includegraphics[scale=0.5]{rec.ps}
        \end{center}
      \end{minipage}
      \label{fig:sch:ext}
    }%\hfill
  \end{center}
  \caption{The STABYLO Scheme.}
  \label{fig:sch}
\end{figure*}








\subsection{Security Considerations}
Among 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}
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 
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
than $1/2$.
Equivalent formulations of such a property can
be found. They all lead to the fact that,
even if the encrypted message is extracted, 
it is impossible to retrieve the original one in 
polynomial time.   

Starting thus with a key $k$ and the message \textit{mess} to hide, 
this step computes a message $m$, which is the encrypted version  of \textit{mess}.


\subsection{Edge-Based Image Steganography}


The edge-based image
steganography schemes 
already presented \cite{Luo:2010:EAI:1824719.1824720,DBLP:journals/eswa/ChenCL10} differ 
in how carefully they select edge pixels, and  
how they modify them.

%Image Quality: Edge Image Steganography
%\JFC{Raphael, les fuzzy edge detection sont souvent utilisés. 
%  il faudrait comparer les approches en terme de nombre de bits retournés,
%  en terme de complexité. } \cite{KF11}
%\RC{Ben, à voir car on peut choisir le nombre de pixel avec Canny. Supposons que les fuzzy edge soient retourne un peu plus de points, on sera probablement plus détectable...  Finalement on devrait surement vendre notre truc en : on a choisi cet algo car il est performant en vitesse/qualité. Mais on peut aussi en utilisé d'autres :-)}

Many techniques have been proposed in the literature to  detect 
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, 
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.

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
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 $7$. 
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.
 
\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.   






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} 
\input{stc}



% Edge Based Image Steganography schemes 
% already studied~\cite{Luo:2010:EAI:1824719.1824720,DBLP:journals/eswa/ChenCL10,DBLP:conf/ih/PevnyFB10} differ 
% how they select edge pixels, and  
% how they modify these ones.

% 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.   

% 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.



%%RAPH: paragraphe en double :-)




\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.
