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 \medskip  \hfill  \begin{minipage}[t]{.45\textwidth}   \textbf{$\LWE_{n,q,\chi}$ problem:}\\Given $m \geq 1$, $\mathbf{A} \sample \mathcal U(\Zq^{n \times m})$,\\   $\textcolor{orange!70!black}{\mathbf{e}} \sample \chi^m$ \\[.5em]   $\left(   \begin{tikzpicture}   \tikzstyle{matA}=[fill=blue!10]   \tikzstyle{vecS}=[color=red!70!black]   \tikzstyle{vecE}=[color=orange!70!black]   \path[use as bounding box] (-2.1, .5) rectangle (2.4, 1.8);   \draw[matA, xshift=-2cm, yshift=.5cm] (0,0) rectangle node {$\mathbf{A}$} (1.5,1);   \node at (-.2, .75) {$,$};   \draw[matA] (0,0) rectangle node {$\mathbf{A}^T_{}$} (1,1.5);   \draw[|-|, vecS] (1.2, 1.5) -- node[right] {$\mathbf{s}$} ++(0, -1);   \node at (1.8, .75) {$+$};   \draw[|-|, vecE] (2.1, 1.5) -- node[right] {$\mathbf{e}$} ++ (0, -1.5);   \end{tikzpicture}   \right)$\\[.5em]   $\in \Zq^{n \times m} \times \Zq^{m}$,   find $\textcolor{red!70!black}{\mathbf{s}} \in \Zq^n.$  \end{minipage} \hfill  \begin{minipage}[t]{.4\textwidth}   \textbf{$\SIS_{n,m,q,\beta}$ problem:}\\Given\\[.5em]   $\tikz[baseline=.3cm]{ \draw[fill=blue!10] (0,0) rectangle node {$\mathbf{A}$} (1.5,1); } \in \Zq^{n \times m},$   find $\textcolor{red!70!black}{\mathbf{x}} \in \ZZ^m_{}$ such that\\[.5em]   $\begin{tikzpicture}[baseline=.25cm]   \tikzstyle{matA}=[fill=blue!10]   \tikzstyle{vecX}=[color=red!70!black]   \draw[matA] (0,0) rectangle node {$\mathbf{A}$} (1.5,1);   \draw[|-|, vecX] (1.7, 1) -- node[right] {$\mathbf{x}$} ++ (0, -1.5);   \node at (2.4, .25) {$=$};   \draw[|-|] (2.8, 1) -- node[right] {$\mathbf{0}^n$} ++ (0, -1);   \end{tikzpicture},$ and\\$0< \|\textcolor{red!70!black}{\mathbf{x}}\| \leq \beta$.  \end{minipage}  \hfill  \medskip