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Symmetric Discrete Logarithm

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Fabrice Mouhartem 2018-03-20 11:07:21 +01:00
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sec-pairings.tex
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@ -7,7 +7,7 @@ Since then, many constructions have been proposed for cryptographic construction
Multiple constructions and parameter sets coexist for pairings. 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}. 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} %\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. 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} \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
$(g,\hat{g},g^a,\hat{g}^a) \in \GG \times \hat{\GG}$ $(g,\hat{g},g^a,\hat{g}^a) \in \GG \times \hat{\GG}$
where $a \sample \ZZ_p^{}$, computing $a \in \ZZ_p^{}$. where $a \sample \ZZ_p^{}$, computing $a \in \ZZ_p^{}$.
\end{definition} \end{definition}