Symmetric Discrete Logarithm

sec-pairings.tex

@@ -7,7 +7,7 @@ Since then, many constructions have been proposed for cryptographic construction
 Multiple constructions and parameter sets coexist for pairings.
 Real-world implementation are based on elliptic curves~\cite{BN06, KSS08}, but recent advances in cryptanalysis makes it hard to evaluate the security level of pairing-based cryptography~\cite{KB16,MSS17,BD18}.
 
-In the following, we rely on the black-box definition of cryptographic pairings as bilinear maps, and on the assumed hardness of a classical assumption over pairings, namely $\SXDH$.
+In the following, we rely on the black-box definition of cryptographic pairings as bilinear maps, and on the assumed hardness of classical assumptions over pairings, namely $\SXDH$ and $\SDL$.
 
 
 %\subsection{Bilinear maps}
@@ -41,9 +41,9 @@ 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{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
+  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}