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+ Restatable definition

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Fabrice Mouhartem
2018-02-28 18:02:06 +01:00
parent 106498a095
commit 79cc6c5806
2 changed files with 5 additions and 13 deletions
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chap-proofs.tex
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@ -128,12 +128,13 @@ To illustrate this, let us consider the two following assumptions:
The \textit{discrete logarithm assumption} is the intractability of this problem.
\end{definition}
\begin{definition}[Decisional Diffie-Hellman] \label{de:DDH} \index{Discrete Logarithm!Decisional Diffie-Hellman}
\begin{restatable}[Decisional Diffie-Hellman]{definition}{defDDH}
\index{Discrete Logarithm!Decisional Diffie-Hellman} \label{de:DDH}
Let $\GG$ be a cyclic group of order $p$. The \emph{decisional Diffie-Hellman} ($\DDH$) problem is the following.
Given the tuple $(g, g_1^{}, g_2^{}, g_3^{}) = (g, g^a_{}, g^b{}, g^c_{}) \in \GG^4_{}$, the goal is to decide whether $c = ab$ or $c$ is sampled uniformly in $\GG$.
The \textit{\DDH assumption} is the intractability of the problem for any $\ppt$ algorithm.
\end{definition}
\end{restatable}
The discrete logarithm assumption is implied by the decisional Diffie-Hellman assumption for instance.
Indeed, if one is able to solve the discrete logarithm problem, then it suffices to compute the discrete logarithm of $g_1$, let us say $\alpha$, and then check whether $g_2^\alpha = g_3^{}$ or not.