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Consistency of symbols

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Fabrice Mouhartem 2018-02-06 18:40:48 +01:00
parent a6d8100824
commit 16e717f08b
2 changed files with 5 additions and 5 deletions
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8
sec-lattices.tex
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@ -18,16 +18,16 @@ Worst-case lattice problems have been extensively studied in the last past years
\label{fig:lattice-basis}
\end{figure}
A (full-rank) lattice~$\Lambda$ is defined as the set of all integer linear combinations of some linearly independent basis vectors~$(\mathbf{b}_i)_{i\leq n}$ belonging to some~$\RR^n$.
A (full-rank) lattice~$\Lambda$ is defined as the set of all integer linear combinations of some linearly independent basis vectors~$(\mathbf{b}_i)_{i\leq n}$ belonging to some~$\RR^n_{}$.
We can notice that this basis is not unique, as illustrated in Figure~\ref{fig:lattice-basis}.
In the following, we work with $q$-ary lattices, for some prime $q$.
\begin{definition} \label{de:qary-lattices} \index{Lattices}
Let~$m \geq n \geq 1$, a prime~$q \geq 2$, $\mathbf{A} \in \ZZ_q^{n \times m}$ and $\mathbf{u} \in \ZZ_q^n$, define
\begin{align*}
\Lambda_q(\mathbf{A}) & \triangleq \{ \mathbf{e} \in \ZZ^m \mid \exists \mathbf{s} \in \ZZ_q^n ~\text{ s.t. }~\mathbf{A}^T \cdot \mathbf{s} = \mathbf{e} \bmod q \} \text{ as well as}\\
\Lambda_q^{\perp} (\mathbf{A}) & \triangleq \{\mathbf{e} \in \ZZ^m \mid \mathbf{A} \cdot \mathbf{e} = \mathbf{0}^n \bmod q \} \text{, and}\\
\Lambda_q^{\mathbf{u}} (\mathbf{A}) & \triangleq \{\mathbf{e} \in \ZZ^m \mid \mathbf{A} \cdot \mathbf{e} = \mathbf{u} \bmod q \}.
\Lambda_q^{}(\mathbf{A}) & \triangleq \{ \mathbf{e} \in \ZZ^m_{} \mid \exists \mathbf{s} \in \ZZ_q^n ~\text{ s.t. }~\mathbf{A}^T_{} \cdot \mathbf{s} = \mathbf{e} \bmod q \} \text{ as well as}\\
\Lambda_q^{\perp} (\mathbf{A}) & \triangleq \{\mathbf{e} \in \ZZ^m_{} \mid \mathbf{A} \cdot \mathbf{e} = \mathbf{0}^n \bmod q \} \text{, and}\\
\Lambda_q^{\mathbf{u}} (\mathbf{A}) & \triangleq \{\mathbf{e} \in \ZZ^m_{} \mid \mathbf{A} \cdot \mathbf{e} = \mathbf{u} \bmod q \}.
\end{align*}
For any lattice point $\mathbf{t} \in \Lambda_q^{\mathbf{u}} (\mathbf{A})$, it holds that $\Lambda_q^{\mathbf{u}}(\mathbf{A})=\Lambda_q^{\perp}(\mathbf{A}) + \mathbf{t}$. Meaning that $\Lambda_q^{\mathbf{u}} (\mathbf{A}) $

2
sec-pairings.tex
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@ -38,7 +38,7 @@ defined in Definition~\ref{de:DDH} and recalled here.
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!Symmetric external Diffie-Hellman (SXDH)}
\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}