short_elliptic_curves_and_pairings.tex

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\subsection{Pairings}
\label{sec:pairings}
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\noindent If $E$ is an elliptic curve defined over a prime field $\mathbb{F}_{p}$ of large characteristic $p$, 
we denote by $E(\mathbb{F}_{p})$ the abelian group containing all the points $(x, y) \in (\mathbb{F}_{p})^2$ 
on the curve along with the point at infinity. We will work with pairing friendly curves i.e., those with a secure~\cite{secure_pairings,pairings_for_cryptographers} 
efficiently computable, bilinear, non-degenerate mapping from a prime order subgroup of $E(\mathbb{F}_{p})$ and a subgroup of the curve over an extension field.
We will work with a \emph{pairing-friendly two-chain}, i.e., a pair of pairing friendly elliptic curves $\einn=E(\mathbb{F}_{p})$ (\emph{the inner curve}) and 
$\eout=E'(\mathbb{F}_{r})$ (\emph{the outer curve}), such that the pairing $\epinn$ on $\einn$ works on subgroups of order $r$. $\mathbb{F}_p$ is the \emph{base field} of 
$\einn=E(\mathbb{F}_{p})$ and $\mathbb{F}_r$ is its \emph{scalar field}. Note that another way of defining a pairing-friendly two-chain is by requiring a pairing friendly 
pair of curves such that the base field of the inner curve equals the scalar field of the outer curve. 
We write $\ginn{1}$, $\ginn{2}$, $\gtinn$, $\gout{1}$, $\gout{2}$, $\gtout$ for cyclic subgroups of 
$\einn$, $E(\mathbb{F}_{p^l}$),$\mathbb{F}_{p^k}$, $\eout$, $E'(\mathbb{F}_{r^{l'}}$), $\mathbb{F}_{r^{k'}}$ respectively for suitable $l,k,l',k'$ with the two pairings 
$\epinn:\ginn{1} \times \ginn{2} \rightarrow \gtinn$ and by $\epout:\gout{1} \times \gout{2} \rightarrow \gtout$. We write $\sginn{1}$, $\sginn{2}$, $\sgtinn$, $\sgout{1}$, 
$\sgout{2}$, $\sgtout$ respectively for randomly chosen generators of these groups. We use additive notation for group operations and write 
$[x]_{\indexoneinn} = x \cdot \sginn{1}$, $[x]_{\indextwoinn} = x \cdot \sginn{2}$. Concretely, our implementation uses BLS12-377~\cite{zexe} 
and BW6-761~\cite{BW6} for $\einn$ and $\eout$.
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