53 lines
3.8 KiB
TeX
53 lines
3.8 KiB
TeX
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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.
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Since then, many constructions have been proposed for cryptographic constructions, such as identity-based encryption~\cite{BF01,Wat05} or group signature~\cite{BBS04}.
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Multiple constructions and parameter sets coexist for pairings.
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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}.
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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 $\SDL$.
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The notations $1_{\GG}^{}$, $1_{\Gh}^{}$ and $1_{\GT}^{}$ denote the unit element in $\GG$, $\Gh$ and $\GT$ respectively.
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\begin{restatable}[Pairings~\cite{BSS05}]{definition}{defPairings} \label{de:pairings} \index{Pairings}
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A pairing is a map $e: \GG \times \Gh \to \GT$ over cyclic groups of order $p$ that verifies the following properties for any $g \in \GG, \hat{g} \in \Gh$:
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\begin{enumerate}[\quad (i)]
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\item bilinearity: for any $a, b \in \Zp$, we have $e(g^a, \hat{g}^b) = e(g^b, \hat{g}^a) = e(g, \hat{g})^{ab}$.
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\item non-degeneracy: $e(g,\hat{g}) = 1_{\GT} \iff g = 1_{\GG}$ or $\hat{g} = 1_{\Gh}$.
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\item the map is computable in polynomial time in the size of the input.
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\end{enumerate}
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\end{restatable}
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For cryptographic purpose, pairings are usually defined over elliptic curves, hence $\GT$ is a multiplicative subgroup of the multiplicative group of a finite field.
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The most standard assumptions over pairings are derived from the equivalent of the Diffie-Hellman assumptions from cyclic groups,
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described in \cref{de:DDH} and recalled here.
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\defDDH*
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This hypothesis, from which the Diffie-Hellman key exchange relies its security on, is then used to defined the $\SXDH$ assumption.
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\begin{restatable}[{$\SXDH$~\cite[As.~1]{BGdMM05}}]{definition}{defSXDH} \index{Pairings!SXDH} \label{de:SXDH}
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The \emph{Symmetric eXternal Diffie-Hellman} ($\SXDH$) assumption holds if the $\DDH$ assumption holds both in $\GG$ and $\Gh$.
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\end{restatable}
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The advantages of the best $\ppt$ adversary against $\DDH$ in group $\GG$ and $\Gh$ are written $\advantage{\DDH}{\GG}$ and $\advantage{\DDH}{\Gh}$ respectively. Both of those quantities are assumed negligible under the $\SXDH$ assumption.
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In \cref{ch:sigmasig}, the security of the group signature scheme relies on the $\SXDH$ assumption, which is a well-studied assumption.
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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.
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This gives a stronger security guarantee for the security of schemes proven under this kind of assumptions.
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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).
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In the aforementioned chapter, we also rely on the following assumption, which generalizes the Discrete Logarithm problem to asymmetric groups.
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\begin{restatable}[$\SDL$]{definition}{defSDL}
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\label{de:SDL} \index{Pairings!SDL}
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In bilinear groups $\bigl(\GG,\Gh,\GT^{}\bigr)$ of prime order $p$, the \emph {Symmetric Discrete Logarithm} ($\SDL$) problem consists in, given
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$\bigl(g,\hat{g},g^a_{},\hat{g}^a_{}\bigr) \in \bigl(\GG \times \Gh\bigr)^2_{}$
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where $a \sample \ZZ_p^{}$, computing $a \in \ZZ_p^{}$.
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\end{restatable}
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This assumption is also static and non-interactive.
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