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diff --git a/ourapproach.tex b/ourapproach.tex
new file mode 100644 (file)
index 0000000..5fc4f7e
--- /dev/null
+++ b/ourapproach.tex
@@ -0,0 +1,87 @@
+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

diff --git a/rec.pdf b/rec.pdf
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diff --git a/schema.dot b/schema.dot
new file mode 100644 (file)
index 0000000..165432b
--- /dev/null
+++ b/schema.dot
@@ -0,0 +1,36 @@
+digraph start_up { 
+       node [style = rounded, shape = rect] IC [label="Image\nCover X "]
+       node [style = rounded, shape = rect] message [label="message"]
+       node [style = rounded, shape = rect] key 
+
+       node [style = rounded, shape = rect] PI [label="Parameter\nInstantiation"]               
+       node [style = rounded, shape = rect] encrypt [label="m=encryption(message)"]             
+       node [style = rounded, shape = rect] STC [label="y= STC(x,m)"]           
+       node [style = rounded, shape = rect] emb [label="stego=embedding(y,X)"]                  
+
+       node [style = rounded, shape = rect] start end;
+       node [style = rounded, shape = rect] ED [label="x= LSB(7 bits\nEdge Detection)"]  
+       
+       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

diff --git a/schema2.dot b/schema2.dot
new file mode 100644 (file)
index 0000000..4f47d83
--- /dev/null
+++ b/schema2.dot
@@ -0,0 +1,29 @@
+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

diff --git a/stc.tex b/stc.tex
new file mode 100644 (file)
index 0000000..6570c73
--- /dev/null
+++ b/stc.tex
@@ -0,0 +1,85 @@
+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.
+
+

diff --git a/stc/exp/python/stc_r.py b/stc/exp/python/stc_r.py
new file mode 100644 (file)
index 0000000..854c780
--- /dev/null
+++ b/stc/exp/python/stc_r.py
@@ -0,0 +1,241 @@
+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
+

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new file mode 100644 (file)
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