From 16e717f08bb1796eef19a0e3009a5deafaaf95e6 Mon Sep 17 00:00:00 2001 From: Fabrice Mouhartem Date: Tue, 6 Feb 2018 18:40:48 +0100 Subject: [PATCH] Consistency of symbols --- sec-lattices.tex | 8 ++++---- sec-pairings.tex | 2 +- 2 files changed, 5 insertions(+), 5 deletions(-) diff --git a/sec-lattices.tex b/sec-lattices.tex index d3f6549..ad5488b 100644 --- a/sec-lattices.tex +++ b/sec-lattices.tex @@ -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}) $ diff --git a/sec-pairings.tex b/sec-pairings.tex index 2d26d3f..8704722 100644 --- a/sec-pairings.tex +++ b/sec-pairings.tex @@ -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}