%% Common header
\usepackage{amsmath,amsfonts,amssymb,url}
\usepackage{mdwlist}
\usepackage{graphicx,graphpap}
\usepackage{soul}

\usepackage{color}
\newcommand{\todo}[1]{\color{red}{\bf To do: #1} \color{black}}
\newcommand{\myul}[2][red]{\setulcolor{#1}\ul{#2}\setulcolor{red}}

\newcommand{\squish}{
      \setlength{\topsep}{0pt}
      \setlength{\itemsep}{0ex}
      \vspace{-1ex}
      \setlength{\parskip}{0pt}}
\newcommand{\squishend}{\vskip -1ex\relax}

\newenvironment{gamequote}
               {\list{}{\rightmargin0pt\relax}\item\relax}
               {\endlist}
               
\newcommand{\Adv}{\mathcal{A}}
\newcommand{\game}{\mathrm{Game~}}
\newcommand{\negl}{\mathit{negl}}

\newcommand{\hash}{\ensuremath{F}}
\newcommand{\hashone}{\ensuremath{H_1}}
\newcommand{\hashtwo}{\ensuremath{H_2}}

\newcommand{\bigO}{\ensuremath{\mathcal{O}}}
\newcommand{\Z}{\ensuremath{\mathbb{Z}}}
\newcommand{\F}{\ensuremath{\mathbb{F}}}
\newcommand{\R}{\ensuremath{\mathbb{R}}}
\newcommand{\C}{\ensuremath{\mathbb{C}}}
\newcommand{\Q}{\ensuremath{\mathbb{Q}}}
\newcommand{\G}{\ensuremath{\mathbb{G}}}
\newcommand{\GT}{\ensuremath{\mathbb{G}_T}}
\newcommand{\map}{e}
\newcommand{\BMsetup}{\mathsf{BSetup}}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{cor}[theorem]{Corollary}
\newtheorem{fact}[theorem]{Fact}
%\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}

\newcommand{\bit}{\{0,1\}}
\newcommand{\half}{{\frac{1}{2}}}
\newcommand{\Zp}{\ensuremath{{\Z_p}}}
\newcommand{\Zps}{\ensuremath{{\Z_p^*}}}
\newcommand{\SSS}{\ensuremath{\Sigma}}
\newcommand{\Attrmax}{\ensuremath{Attr_{\mathrm{max}}}}
\newcommand{\lmax}{\ensuremath{\ell_{\mathrm{max}}}}
\newcommand{\nmax}{\ensuremath{n_{\mathrm{max}}}}
\newcommand{\kmax}{\ensuremath{k_{\mathrm{max}}}}
\newcommand{\MM}{\ensuremath{\mathcal{M}}}
\newcommand{\weil}{\ensuremath{\hat{e}}}
\newcommand{\tuple}[1]{\langle #1 \rangle}
\newcommand{\e}{\epsilon}
%\newcommand{\st}{\;\;\mbox{s.t.}\;\;}
\newcommand{\rgets}{\stackrel{\scriptscriptstyle{R}}{\leftarrow}}

\newcommand{\Setup}{\mathsf{Setup}}
\newcommand{\Encrypt}{\mathsf{Encrypt}}
\newcommand{\Decrypt}{\mathsf{Decrypt}}
\newcommand{\KeyGen}{\mathsf{KeyGen}}

%\newcommand{\Pr}{\ensuremath{\mathrm{Pr}}}
\newcommand{\CT}{\ensuremath{\mathrm{CT}}}
\newcommand{\TK}{\ensuremath{\mathrm{TK}}}
\newcommand{\SK}{\ensuremath{\mathrm{SK}}}
\newcommand{\MK}{\ensuremath{\mathrm{MK}}}
\newcommand{\PK}{\ensuremath{\mathrm{PK}}}
\newcommand{\PP}{\ensuremath{\mathrm{PP}}}
\newcommand{\MSK}{\ensuremath{\mathrm{MSK}}}
\newcommand{\GP}{\ensuremath{\mathrm{GP}}}
\newcommand{\ASK}{\ensuremath{\mathrm{ASK}}}
\newcommand{\GID}{\ensuremath{\mathrm{GID}}}

\newcommand{\Phve}{P^{\scriptscriptstyle{\mathrm{HVE}}}}
\newcommand{\AlgA}{\ensuremath{{\cal A}}}
\newcommand{\AlgB}{\ensuremath{{\cal B}}}
\newcommand{\Pdist}[1]{{\cal P}_{#1}}
\newcommand{\Rdist}[1]{{\cal R}_{#1}}
\newcommand{\EXP}[1]{\mathsf{EXP}_{\scriptscriptstyle{\text{#1}}}}
\newcommand{\deq}{\mathrel{:=}}
\newcommand{\GG}{\ensuremath{\mathcal{G}}}
\newcommand{\PhiHVE}{\ensuremath{\Phi_{\scriptscriptstyle \mathrm{HVE}}}}
\newcommand{\PhiEq}{\ensuremath{\Phi_{\mathrm{eq}}}}
\newcommand{\true}{\ensuremath{\textsf{true}}}

\newcommand{\ID}{\ensuremath{\mathcal{I}}}
\newcommand{\EE}{\ensuremath{\mathcal{E}}}
\newcommand{\Etr}{\ensuremath{\mathcal{E}_{\scriptscriptstyle{\text{TR}}}}}

\newcommand{\abort}{\ensuremath{\mathsf{abort}}}
\newcommand{\valid}{\ensuremath{\mathsf{valid}}}
\newcommand{\invalid}{\ensuremath{\mathsf{invalid}}}

\newcommand{\eDBDH}{\ensuremath{\e_{\scriptscriptstyle{DBDH}}}}
\newcommand{\eDTPDH}{\ensuremath{\e_{\scriptscriptstyle{D3DH}}}}
\newcommand{\eSD}{\ensuremath{\e_{\scriptscriptstyle{SD}}}}
\newcommand{\eBSD}{\ensuremath{\e_{\scriptscriptstyle{BSD}}}}
\newcommand{\eDSTPDH}{\ensuremath{\e_{\scriptscriptstyle{DS3DH}}}}

\newcommand{\inputenc}{I_{enc}}
\newcommand{\inputkey}{I_{key}}

\newtheorem{defn}{Definition}

\def\x{{\vec{x}}}
\def\y{{\vec{y}}}

\newenvironment{tightenum}
{\begin{enumerate}
\setlength{\itemsep}{1pt}
\setlength{\parskip}{0pt}
\setlength{\parsep}{0pt}}
{\end{enumerate}}

\newcommand{\qed}{\hfill\Box}
\newenvironment{proof}{
\vspace*{-\parskip}\noindent\textit{Proof.}}{$\qed$

\medskip
}

\newenvironment{proofsketch}{
\vspace*{-\parskip}\noindent\textit{Proof sketch.}}{$\qed$

\medskip
}

\newcommand{\ourparagraph}[1]{\smallskip\noindent {\bf #1}  }
\newcommand{\betterparagraph}[1]{\smallskip\noindent {\em #1.}  }

\newcommand{\etal}{{\em et al.}}
\newcommand{\var}{\mathrm}