X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/hdrcouchot.git/blobdiff_plain/01003216d61845263ad195f3ecf7334817d60407..523864c862215a63c5133568a9771f5b8f60c89e:/annexePreuveMarquageCorrectioncompletude.tex?ds=inline

diff --git a/annexePreuveMarquageCorrectioncompletude.tex b/annexePreuveMarquageCorrectioncompletude.tex
index dd6ad42..76e6e5c 100644
--- a/annexePreuveMarquageCorrectioncompletude.tex
+++ b/annexePreuveMarquageCorrectioncompletude.tex
@@ -1,64 +1,34 @@
-\begin{theorem}
-La condition de l'algorithme de marquage est nécressaire et suffisante
-pour permettre l'extraction du message du média marqué.
-\end{theorem}
+
+\marquagecorrectioncompl*
 
 \begin{proof}
-For sufficiency, let $d_i$ be the last iteration (date) the element $i \in \Im(S_p)$
-of $x$ has been modified:% is defined by 
+Pour la suffisance, soit $d_i$ la dernière itération où l'élément $i \in \Im(S_p)$
+de la configuration $x$ a été modifié:% is defined by 
 $$
 d_i = \max\{j | S^j_p = i \}.
 $$ 
-Let $D=\{d_i|i \in \Im(S_p) \}$.
-The set $\Im(S_c)_{|D}$ is thus 
-the restriction of  the image of $S_c$ to $D$.
+Soit $D=\{d_i|i \in \Im(S_p) \}$.
+L'ensemble $\Im(S_c)_{|D}$ est donc la restriction de l'image de $S_c$ à $D$.
 
 
-The host that results from this iteration scheme is thus
-$(x^l_0,\ldots,x^l_{\mathsf{N}-1})$ where 
-$x^l_i$ is either $x^{d_i}_i$ if $i$ belongs to $\Im(S_p)$ or $x^0_i$ otherwise.
-Moreover, for each $i \in \Im(S_p)$, the element $x^{d_i}_i$ is equal to 
+Le vecteur qui résutle de ces itérations est donc
+$(x^l_0,\ldots,x^l_{\mathsf{N}-1})$ où
+$x^l_i$ est soit  $x^{d_i}_i$ si $i$ appartient à $\Im(S_p)$ ou $x^0_i$ sinon.
+De plus, pour chaque $i \in \Im(S_p)$, l'élément $x^{d_i}_i$ est égal à 
 $m^{d_i-1}_{S^{d_i}_c}$. 
-Thanks to constraint \ref{itm2:Sc}, all the indexes 
-$j \in \llbracket 0 ;\mathsf{P} -1 \rrbracket$ belong to 
+Sous hypothèse que la contrainte imposée soit réalisée,  tous les indices
+$j \in \llbracket 0 ;\mathsf{P} -1 \rrbracket$ appartiennent à
 $\Im(S_c)_{|D}$.
-Let then $j \in \llbracket 0 ;\mathsf{P} -1 \rrbracket$ s.t.
+On a alors $j \in \llbracket 0 ;\mathsf{P} -1 \rrbracket$ tel que 
 $S^{d_i}_c=j$. 
-Thus we have all the elements $m^._j$ of the vector $m$.
-Let us focus now on some  $m^{d_i-1}_j$.
-Thus the value of $m^0_j$ can be  immediately
-deduced by counting in $S_c$ how many
-times the component $j$ has been switched 
-before $d_i-1$.  
+On retrouve ainsi tous les éléments $m^._j$ du vecteur $m$.
+A partir de $m^{d_i-1}_j$, 
+la valeur de  $m^0_j$ peut être déduite en comptant dans  $S_c$ combien de fois 
+l'élément $j$ a été invoqué avant $d_i-1$.  
 
-Let us focus now on necessity. 
-If $\Im(S_c)_{|D} \subsetneq 
+Réciproquement, si $\Im(S_c)_{|D} \subsetneq 
 \llbracket 0 ;\mathsf{P} -1 \rrbracket$,
-there exist some $j \in \llbracket 0 ;\mathsf{P} -1 \rrbracket$ that 
-do not belong to $\Im(S_c)_{|\Im(S_p)}$. 
-Thus $m_j$ is  not present in $x^l$ and the message cannot be extracted.
+i lexiste un $j \in \llbracket 0 ;\mathsf{P} -1 \rrbracket$ qui n'appartient pas à $\Im(S_c)_{|\Im(S_p)}$. 
+Ainsi, $m_j$ n'est pas présent dans $x^l$ et le message ne peut pas extrait.
 \end{proof}
 
-When the constraint \ref{itm2:Sc} is satisfied, we obtain a scheme
-that always finds the original message provided the watermarked media 
-has not been modified. 
-In that context, correctness and completeness are established. 
-
-
-Thanks to constraint~\ref{itm2:Sc}, the cardinality $k$ of
-$\Im(S_p)$ is larger than $\mathsf{P}$. 
-Otherwise the cardinality of $D$  would be smaller than $\mathsf{P}$ 
-and similar to the cardinality of $\Im(S_c)_{|D}$,
-which is contradictory.
-
-One bit of index $j$ of the original message $m^0$
-is thus embedded at least twice in $x^l$. 
-By counting the number of times this bit has been switched in $S_m$, the value of 
-$m_j$ can be deduced in many places. 
-Without attack, all these values are equal and the message is immediately 
-obtained.
- After an attack,  the value of $m_j$ is obtained as mean value of all 
-its occurrences. 
-The scheme is thus complete.
-Notice that if the cover is not attacked, the returned message is always equal to the original 
-due to the definition of the mean function.