-\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 résulte 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.
+il existe 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.
