Add sigmasig

sec-pairings.tex

@@ -11,14 +11,14 @@ In the following, we rely on the black-box definition of cryptographic pairings
 
 
 %\subsection{Bilinear maps}
-\begin{definition}[Pairings~\cite{BSS05}] \label{de:pairings} \index{Pairings}
+\begin{restatable}[Pairings~\cite{BSS05}]{definition}{defPairings} \label{de:pairings} \index{Pairings}
   A pairing is a map $e: \GG \times \Gh \to \GT$ over cyclic groups of order $p$ that verifies the following properties for any $g \in \GG, \hat{g} \in \Gh$:
   \begin{enumerate}[\quad (i)]
     \item bilinearity: for any $a, b \in \Zp$, we have $e(g^a, \hat{g}^b) = e(g^b, \hat{g}^a) = e(g, \hat{g})^{ab}$.
     \item non-degeneracy: $e(g,\hat{g}) = 1_{\GT} \iff g = 1_{\GG}$ or $\hat{g} = 1_{\Gh}$.
     \item the map is computable in polynomial time in the size of the input.
   \end{enumerate}
-\end{definition}
+\end{restatable}
 
 For cryptographic purpose, pairings are usually defined over elliptic curves, hence $\GT$ is a multiplicative subgroup of the multiplicative group of a finite field.
 
@@ -29,9 +29,9 @@ 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} \label{de:SXDH}
+\begin{restatable}[{$\SXDH$~\cite[As.~1]{BGdMM05}}]{definition}{defSXDH} \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}
+\end{restatable}
 
 In \cref{ch:sigmasig}, the security of the group signature scheme relies on the $\SXDH$ assumption, which is a well-studied assumption.
 Moreover, this assumption is static, meaning that the size of the assumption is independent of any parameters, and is non-interactive, in the sense that it does not involve any oracle.
@@ -41,12 +41,12 @@ For instance, Cheon gave an attack against $q$-Strong Diffie-Hellmann problem fo
 
 In the aforementioned chapter, we also rely on the following assumption, which generalizes the Discrete Logarithm problem to asymmetric groups.
 
-\begin{definition}[$\SDL$]
+\begin{restatable}[$\SDL$]{definition}{defSDL}
   \label{de:SDL} \index{Pairings!SDL}
   In bilinear groups $\bigl(\GG,\Gh,\GT^{}\bigr)$  of  prime order $p$,  the \emph {Symmetric  Discrete Logarithm} ($\SDL$) problem  consists in, given
   $\bigl(g,\hat{g},g^a_{},\hat{g}^a_{}\bigr) \in \bigl(\GG \times \Gh\bigr)^2_{}$
   where $a \sample \ZZ_p^{}$, computing $a \in \ZZ_p^{}$.
-\end{definition}
+\end{restatable}
 
 This assumption is still a static and non-interactive assumption.