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                    % \section{Pairing-Based Cryptography} %
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Pairing-based cryptography was introduced by Antoine Joux~\cite{Jou00} to generalize Diffie-Hellman key exchange to three users in one round.
Since then, many constructions have been proposed for cryptographic constructions, such as identity-based encryption~\cite{BF01,Wat05} or group signature~\cite{ACJT00,BBS04}.
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 $\SXDH$.


%\subsection{Bilinear maps}
\begin{definition}[Pairings~\cite{BSS05}] \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}

For cryptographic purpose, pairings are usually defined over elliptic curves, hence $\GT$ is a multiplicative subgroup of the multiplicative group of a finite field.

Most standard assumptions over pairings are derived from the equivalent of the Diffie-Hellman assumptions from cyclic groups,
defined in Definition~\ref{de:DDH} and recalled here.

\begin{definition}[$\DDH$ (recall)]  \index{Discrete Logarithm!Decisional Diffie-Hellman}
  Let $\GG$ be a cyclic group of order $p$. The \emph{decisional Diffie-Hellman} ($\DDH$) problem is the following.
  Given $(g, g^a, g^b, g^c) \in \GG^4$, the goal is to decide if $c = ab$ or if $c$ is sampled uniformly in $\GG$.

  The DDH assumption is the intractability of the problem for any $\ppt$ algorithm.

%  Let us now define the $\DDH$ language as
%  $L_\DDH = \bigl\{ (g, g^a, g^b, g^{c}) \in \GG^4 \mid c = a \cdot b \bigr\}.$
%  Thus the $\DDH$ problem is equivalently the question of whether $L_\DDH \in \mathsf{PP}$ or not.
\end{definition}

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}
  The \emph{Symmetric eXternal Diffie-Hellman} ($\SXDH$) assumption holds if the $\DDH$ assumption holds both in $\GG$ and $\Gh$.
\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.
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.
For instance, Cheon gave an attack against $q$-Strong Diffie-Hellmann problem for large values of $q$~\cite{Che06} (which usually represents the number of adversarial queries).
