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Uniformize notation

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Fabrice Mouhartem 2018-03-20 11:09:07 +01:00
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sec-pairings.tex
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@ -33,7 +33,7 @@ This hypothesis, from which the Diffie-Hellman key exchange relies its security
The \emph{Symmetric eXternal Diffie-Hellman} ($\SXDH$) assumption holds if the $\DDH$ assumption holds both in $\GG$ and $\Gh$. The \emph{Symmetric eXternal Diffie-Hellman} ($\SXDH$) assumption holds if the $\DDH$ assumption holds both in $\GG$ and $\Gh$.
\end{definition} \end{definition}
In Chapter~\ref{ch:sigmasig}, the security of the group signature scheme relies on the $\SXDH$ assumption, which is a well-studied assumption. 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. 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.
This gives a stronger security guarantee for the security of schemes proven under this kind of assumptions. This gives a stronger security guarantee for the security of schemes proven under this kind of assumptions.
@ -43,8 +43,8 @@ In the aforementioned chapter, we also rely on the following assumption, which g
\begin{definition}[$\SDL$] \begin{definition}[$\SDL$]
\label{de:SDL} \index{Pairings!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 $(\GG,\Gh,\GT^{})$ of prime order $p$, the \emph {Symmetric Discrete Logarithm} ($\SDL$) problem consists in, given
$(g,\hat{g},g^a,\hat{g}^a) \in \GG \times \hat{\GG}$ $(g,\hat{g},g^a,\hat{g}^a) \in \GG \times \Gh$
where $a \sample \ZZ_p^{}$, computing $a \in \ZZ_p^{}$. where $a \sample \ZZ_p^{}$, computing $a \in \ZZ_p^{}$.
\end{definition} \end{definition}