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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 classical constant-size assumptions over pairings, namely $\SXDH$ and $\SDL$.


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

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

The most standard assumptions over pairings are derived from the equivalent of the Diffie-Hellman assumptions from cyclic groups,
described in \cref{de:DDH} and recalled here.

\defDDH*

This hypothesis, from which the Diffie-Hellman key exchange relies its security on, is then used to defined the $\SXDH$ assumption.

\begin{restatable}[{$\SXDH$~\cite[As.~1]{BGdMM05}}]{definition}{defSXDH} \index{Pairings!SXDH} \label{de:SXDH}
  The \emph{Symmetric eXternal Diffie-Hellman} ($\SXDH$) assumption holds if the $\DDH$ assumption holds both in $\GG$ and $\Gh$.
\end{restatable}

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.

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).

In the aforementioned chapter, we also rely on the following assumption, which generalizes the Discrete Logarithm problem to asymmetric groups.

\begin{restatable}[$\SDL$]{definition}{defSDL}
  \label{de:SDL} \index{Pairings!SDL}
  In bilinear groups $\bigl(\GG,\Gh,\GT^{}\bigr)$  of  prime order $p$,  the \emph {Symmetric  Discrete Logarithm} ($\SDL$) problem  consists in, given
  $\bigl(g,\hat{g},g^a_{},\hat{g}^a_{}\bigr) \in \bigl(\GG \times \Gh\bigr)^2_{}$
  where $a \sample \ZZ_p^{}$, computing $a \in \ZZ_p^{}$.
\end{restatable}

This assumption is still a static and non-interactive assumption.