Add definition for SDL

sec-pairings.tex

@@ -29,7 +29,7 @@ described in \cref{de:DDH} and recalled here.
 
 This hypothesis, from which the Diffie-Hellman key exchange relies its security on, is then used to defined the $\SXDH$ assumption.
 
-\begin{definition}[{$\SXDH$~\cite[As.~1]{BGdMM05}}] \index{Pairings!SXDH}
+\begin{definition}[{$\SXDH$~\cite[As.~1]{BGdMM05}}] \index{Pairings!SXDH} \label{de:SXDH}
   The \emph{Symmetric eXternal Diffie-Hellman} ($\SXDH$) assumption holds if the $\DDH$ assumption holds both in $\GG$ and $\Gh$.
 \end{definition}
 
@@ -38,3 +38,15 @@ Moreover, this assumption is static, meaning that the size of the assumption is
 
 This gives a stronger security guarantee for the security of schemes proven under this kind of assumptions.
 For instance, Cheon gave an attack against $q$-Strong Diffie-Hellmann problem for large values of $q$~\cite{Che06} (which usually represents the number of adversarial queries).
+
+In the aforementioned chapter, we also rely on the following assumption, which generalizes the Discrete Logarithm problem to asymmetric groups.
+
+\begin{definition}[SDL]
+  \label{de:SDL} \index{Pairings!SDL}
+  In bilinear groups $(\GG,\hat{\GG},\GT^{})$  of  prime order $p$,  the \emph {Symmetric  Discrete Logarithm} (SDL) problem  consists in, given
+  $(g,\hat{g},g^a,\hat{g}^a) \in \GG \times \hat{\GG}$
+  where $a \sample \ZZ_p^{}$, computing $a \in \ZZ_p^{}$. 
+\end{definition}
+
+This assumption is still a static and non-interactive assumption.
+