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--- /dev/null
+The flowcharts given in Fig.~\ref{fig:sch} summarize our steganography scheme denoted as
+STABYLO for STeganography with cAnny, Bbs, binarY embedding at LOw cost.
+What follows successively details all the inner steps and flow inside
+the embedding stage (Fig.\ref{fig:sch:emb})
+and inside the extraction one(Fig.~\ref{fig:sch:ext}).
+
+
+\begin{figure*}[t]
+ \begin{center}
+ \subfloat[Data Embedding.]{
+ \begin{minipage}{0.49\textwidth}
+ \begin{center}
+ \includegraphics[width=5cm]{emb.pdf}
+ \end{center}
+ \end{minipage}
+ \label{fig:sch:emb}
+ }%\hfill
+ \subfloat[Data Extraction.]{
+ \begin{minipage}{0.49\textwidth}
+ \begin{center}
+ \includegraphics[width=5cm]{rec.pdf}
+ \end{center}
+ \end{minipage}
+ \label{fig:sch:ext}
+ }%\hfill
+ \end{center}
+ \caption{The STABYLO Scheme.}
+ \label{fig:sch}
+\end{figure*}
+
+
+\subsection{Steganalysis}
+
+LSB :
+"Adaptive steganalysis of Least Significant Bit replacement in grayscale natural images"
+
+Structural LSB Detectors:
+\verb+http://dde.binghamton.edu/download/structural_lsb_detectors/+
+
+Vainqueur du BOSS challenge
+
+
+ensemble:
+
+
+G. Gül and F. Kurugollu. A new methodology in steganalysis : Breaking highly
+undetactable steganograpy (HUGO). In Information Hiding, 13th International
+Workshop, volume 6958 of LNCS, pages 71–84, Prague, Czech Republic, May 18–
+20, 2011.
+
+
+\subsection{Data Embedding}
+
+
+
+
+\subsubsection{Edge Based Image Steganography}
+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. }
+
+
+Presentation des algos de detection de contour
+Caractéristiques
+
+Comparaison théoriques, références
+
+Algo de stegano basé juste sur cela : (pas de bbs, pas de stc, même message).
+
+Quels paramètres sont optimaux ? Combinaison ?
+
+
+
+
+
+\subsubsection{Security Considerations}
+
+Security aspect:
+BBS-based cryptographic version of the message
+
+\subsubsection{Minimizing Distortion with Syndrome-Treillis Codes}
+
+\input{stc}
+
+
+\subsection{Data Extraction}
\ No newline at end of file
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+ node [style = "", shape = diamond] cond [label="Size of x\n enough for m"];
+
+ start -> PI
+ start -> IC
+ start -> message
+ start -> key
+
+ key -> encrypt
+
+
+ message -> encrypt
+ IC -> ED
+ ED -> cond
+ encrypt -> cond
+ encrypt -> STC
+ cond -> STC [label="Yes"]
+ cond -> PI [label="No"]
+
+ PI -> ED
+ STC -> emb
+ emb -> end
+
+ }
\ No newline at end of file
--- /dev/null
+digraph start_up {
+ node [style = rounded, shape = rect] SI [label="Stego Image"]
+ node [style = "", shape = diamond] cond [label="Size of y\n enough"];
+ node [style = rounded, shape = rect] key
+ node [style = rounded, shape = rect] start end;
+ node [style = rounded, shape = rect] PI [label="Parameter\nInstantiation"]
+ node [style = rounded, shape = rect] STC [label="m = STCr(y,H)"]
+ node [style = rounded, shape = rect] ED [label="(y,H)= LSB(7 bits\nEdge Detection)"]
+ node [style = rounded, shape = rect] decrypt [label="message=decryption(m)"]
+
+
+
+ start -> SI
+ start -> PI
+ PI -> ED
+ start -> key
+
+ SI -> ED
+ ED -> cond
+ cond -> STC [label="Yes"]
+ cond -> PI [label="No"]
+
+ STC -> decrypt
+ key -> decrypt
+ decrypt -> end
+
+
+
+ }
\ No newline at end of file
--- /dev/null
+Let
+$x=(x_1,\ldots,x_n)$ be the $n$-bits cover vector of the image $X$,
+$m$ be the message to embed and
+$y=(y_1,\ldots,y_n)$ be the $n$-bits stego vector.
+The usual additive embbeding impact of replacing $x$ by $y$ in $X$
+is given by a distortion function
+$D_X(x,y)= \Sigma_{i=1}^n \rho_X(i,x,y)$ where the function $\rho_X$
+expressed the cost of replacing $x_i$ by $y_i$ in $X$.
+Let us consider that $x$ is fixed:
+this is for instance the LSBs of the image edge bits.
+The objective is thus to find $y$ that minimizes $D_X(x,y)$.
+
+Hamming embedding proposes a solution to this problem.
+Some steganographic
+schemes~\cite{DBLP:conf/ih/Westfeld01,DBLP:conf/ih/KimDR06,DBLP:conf/mmsec/FridrichPK07} are based on this binary embedding.
+Furthermore this code provides a vector $y$ s.t. $Hy$ is equal to
+$m$ for a given binary matrix $H$.
+
+Let us explain this embedding on a small illustrative example where
+$\rho_X(i,x,y)$ is identically equal to 1,
+$m$ and $x$ are respectively a 3 bits column
+vector and a 7 bits column vector.
+Let then $H$ be the binary Hamming matrix
+$$
+H = \left(
+\begin{array}{lllllll}
+ 0 & 0 & 0 & 1 & 1 & 1 & 1 \\
+ 0 & 1 & 1 & 0 & 0 & 1 & 1 \\
+ 1 & 0 & 1 & 0 & 1 & 0 & 1
+\end{array}
+\right).
+$$
+The objective is to modify $x$ to get $y$ s.t. $m = Hy$.
+In this algebra, the sum and the product respectively correspond to
+the exclusive \emph{or} and to the \emph{and} Boolean operators.
+If $Hx$ is already equal to $m$, nothing has to be changed and $x$ can be sent.
+Otherwise we consider the difference $\delta = d(m,Hx)$ which is expressed
+as a vector :
+$$
+\delta = \left( \begin{array}{l}
+\delta_1 \\
+\delta_2 \\
+\delta_3
+\end{array}
+\right)
+\textrm{ where $\delta_i$ is 0 if $m_i = Hx_i$ and 1 otherwise.}
+$$
+Let us thus consider the $j$th column of $H$ which is equal to $\delta$.
+We denote by $\overline{x}^j$ the vector we obtain by
+switching the $j$th component of $x$,
+that is, $\overline{x}^j = (x_1 , \ldots, \overline{x_j},\ldots, x_n )$.
+It is not hard to see that if $y$ is $\overline{x}^j$, then
+$m = Hy$.
+It is then possible to embed 3 bits in only 7 LSB of pixels by modifying
+1 bit at most.
+In the general case, when comunicating $n$ message bits in
+$2^n-1$ pixels needs $1-1/2^n$ average changes.
+
+
+
+Unfortunately,
+
+
+for any given $H$, finding $y$ that solves $Hy=m$ and that
+that minimizes $D_X(x,y)$ has exponential complexity with respect to $n$.
+The Syndrome-Trellis Codes (STC)
+presented by Filler et al. in~\cite{DBLP:conf/mediaforensics/FillerJF10}
+is a practical solution to this complexity. Thanks to this contribution,
+the solving algorithm has a linear complexity with resspect to $n$.
+
+First of all, Filler et al. compute the matrix $H$
+by placing a small sub-matrix $\hat{H}$ of size $h × w$ next
+to each other and shifted down by one row.
+Thanks to this special form of $H$, one can represent
+every solution of $m=Hy$ as a path through a trellis.
+
+Next, the process of finding $y$ consists of a forward and a backward part:
+\begin{enumerate}
+\item Forward construction of the trellis that depends on $\hat{H}$, on $x$, on $m$, and on $\rho$;
+\item Backward determinization of $y$ which minimizes $D$ starting with
+the complete path with minimal weight
+\end{enumerate}
+Let us now give some details about these two parts.
+
+
--- /dev/null
+from random import *
+import numpy as np
+from math import *
+import gc
+
+infinity = 100000000000
+
+
+
+# forward part
+
+
+def dec(ch,n):
+ l = len(ch)
+ acc = 0
+ for i in xrange(l):
+ if ch[i]==1:
+ acc = acc + 2**(n-i-1)
+ return acc
+
+
+def bin(elem,n):
+ """Convertit un nombre en binaire"""
+ q = -1
+ res = [0 for i in xrange(n)]
+ i = 1
+ while q != 0:
+ q = elem // 2
+ r = elem % 2
+ res[n-i] = r
+ elem = q
+ i+=1
+ return res
+
+
+
+def xorb(a,b):
+ return 1 if a != b else 0
+
+def xor(e1,e2,h):
+ e1b,e2b = bin(e1,h),bin(e2,h)
+ d = dec([xorb(e1b[j],e2b[j]) for j in xrange(h)],h)
+ return d
+
+def lit(d,(indx,indy)):
+ if (indx,indy) in d :
+ return d[(indx,indy)]
+ else :
+ return 0
+
+
+
+
+def forward(H_hat,x,message,lnm,rho):
+ (h,w) = int(log(max(H_hat),2))+1, len(H_hat)
+ path = dict()
+ nbblock = lnm
+ wght = [infinity for _ in xrange(int(2**h))]
+ wght[0]=0
+ newwght = [0 for _ in xrange(int(2**h))]
+# rho = 1
+# rho= [1 for _ in xrange(len(x))]
+ indx,indm = 0,0
+ i=0
+ while i < nbblock: # pour chaque bit du message
+ for j in xrange(w): # pour chaque colonne de H_hat
+ #print indx, "en entrant",wght
+ k = 0
+ while k < int(2**h): # pour chaque ligne de H
+ w0 = wght[k] + x[indx]*rho[indx]
+ w1 = wght[xor(k,H_hat[j],h)] + (1-x[indx])*rho[indx]
+ if w1 < w0 :
+ path[(indx,k)] = 1
+ else :
+ if (indx,k) in path:
+ del path[(indx,k)]
+ newwght[k] = min(w0,w1)
+ k +=1
+ indx +=1
+ wght = [t for t in newwght]
+ #print " apres calcul",wght
+
+ for j in xrange(int(2**(h-1))): # pour chaque colonne de H
+ wght[j] = wght[2*j + message[indm]]
+ wght = wght[:int(pow(2,h-1))] + [infinity for _ in xrange(int(pow(2,h)-pow(2,h-1)))]
+ indm +=1
+ i +=1
+ start = np.argmin(wght)
+ return (start,path)
+
+
+def backward(start,H_hat,x,message,lnm,path):
+ (h,w) = int(log(max(H_hat),2))+1, len(H_hat)
+ indx,indm = len(x)-1,lnm-1
+ state = 2*start + message[indm]
+ indm -=1
+ # l'initialisation de state n'est pas optimale...
+ nbblock = lnm
+ y=np.zeros(len(x))
+ i=0
+ while i < nbblock:
+ l = range(w)
+ l.reverse()
+ for j in l: # pour chaque colonne de H_hat
+ y[indx] = lit(path,(indx,state))
+ state = xor(state,y[indx]*H_hat[j],h)
+ indx -=1
+ state = 2*state + message[indm]
+ indm -=1
+ i +=1
+ return [int(t) for t in y]
+
+
+
+
+
+
+def trouve_H_hat(n,m,h):
+ assert h ==7
+ alpha = float(n)/m
+ assert alpha >= 2
+ index = min(int(alpha),9)
+ mat = {
+ 2 : [71,109],
+ 3 : [95, 101, 121],
+ 4 : [81, 95, 107, 121],
+ 5 : [75, 95, 97, 105, 117],
+ 6 : [73, 83, 95, 103, 109, 123],
+ 7 : [69, 77, 93, 107, 111, 115, 121],
+ 8 : [69, 79, 81, 89, 93, 99, 107, 119],
+ 9 : [69, 79, 81, 89, 93, 99, 107, 119, 125]
+ }[index]
+ return(mat, index*m)
+
+
+def stc(x,rho,message):
+ lnm = len(message)
+ (mat,taille_suff) = trouve_H_hat(len(x),len(message),7)
+ x_b = x[:taille_suff]
+ (start,path) = forward(mat,x_b,message,lnm,rho)
+ return (x_b,backward(start,mat,x_b,message,lnm,path),mat)
+
+
+
+
+
+def nbdif(x,y):
+ r,it = 0,0
+ l = len(y)
+ while it < l :
+ if x[it] != y[it] :
+ r +=1
+ it += 1
+ return float(r)/l
+
+
+
+
+
+
+def prod(H_hat,lnm,y):
+ (h,w) = int(log(max(H_hat),2))+1, len(H_hat)
+ i=0
+ H =[]
+ V=[0 for _ in range(len(y))]
+ sol=[]
+ while i < lnm: # pour chaque ligne
+ V=[0 for _ in range(len(y))]
+ k = max([(i-h+1)*w,0])
+ dec = max([i-h+1,0])
+ for j in xrange(min([i+1,h])): #nbre de blocks presents sur la ligne i
+ for l in xrange(w): # pour chaque collone de H_hat
+ V[k] = bin(H_hat[l],h)[h-i-1+j+dec]
+ k+=1
+
+ sol.append(np.dot(np.array(V),np.array(y)))
+ i+=1
+ #H += [V]
+ #H = np.array(H)
+ #y = np.array(y)
+ #print "dot",np.dot(H,y),H.shape
+ #print "sol",sol
+ return sol#list(np.dot(H,y))
+
+def equiv(x,y):
+ lx = len(x)
+ assert lx == len(y)
+ i=0
+ while i < lx :
+ if x[i] % 2 != y[i]%2 :
+ return False
+ i += 1
+ return True
+
+"""
+################
+
+x = [randint(0,1) for _ in xrange(65000)]
+rho = [randint(1,9) for _ in xrange(65000)]
+message = [randint(0,1) for _ in xrange(26000)]
+
+
+
+#(x_b,y,H_hat) = stc(x,rho,message)
+# x_b est la sous partie de x qui va etre modifiee
+# y est le vecteur des bits modifies
+# H_hat est la sous-matrice retenue qui est embarquee dans H
+
+#print nbdif(x_b,y)
+
+#print message
+#print x
+#print rho
+#print y
+#message2 = [x%2 for x in prod(H_hat,len(message),y)]
+#print message2
+print max([message[l]-message2[l] for l in xrange(len(message))])
+print equiv(message,message2)
+
+"""
+count = 0
+
+while count < 100 and eval :
+ lx = randint(500,1000)
+ x = [randint(0,1) for _ in xrange(lx)]
+ rho = [randint(1,9) for _ in xrange(lx)]
+
+ lm = randint(lx/10,lx/2)
+ message = [randint(0,1) for _ in xrange(lm)]
+ (x_b,y,H_hat) = stc(x,rho,message)
+ eval = equiv(message, prod(H_hat,len(message),y))
+ if not (eval):
+ print x
+ print message
+
+ count +=1
+ if count % 10 == 0 :
+ print count
+
+print count
+
--- /dev/null
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