% Games \newcommand{\game}{\textbf{Game} } \newcommand{\oracle}{\textbf{Oracle} } \newcommand{\inp}{\cdot} \newcommand{\field}[1]{\mathbb{Z}_{#1}} \newcommand{\secparamter}{\lambda} \newcommand{\randomsample}{\leftarrow} \newcommand{\randomassign}{\leftarrow} \newcommand{\assign}{:=} \newcommand{\encoded}[1]{\underline{#1}} \newcommand{\m}{m} \newcommand{\signature}{\sigma} \newcommand{\pubkey}{pk} \newcommand{\privkey}{sk} \newcommand{\groupelement}[1]{#1} \newcommand{\agmgroupelement}[2]{[\groupelement{#1}]_{\overset{\rightharpoonup}{#2}}} \newcommand{\group}[1]{\mathbb{#1}} \newcommand{\oraclequeries}{q_o} \newcommand{\hashqueries}{q_h} \newcommand{\test}{\overset{?}{=}} \newcommand{\ch}{\textbf{ch}} \newcommand{\messagespace}{\mathcal{M}} % Special Dlog \newcommand{\sdlog}{\text{DLog'} } \newcommand{\somdl}{\text{OMDL'} } % SIM algotithm \newcommand{\simalg}{\textit{Sim} } % Intermediate Game \newcommand{\igame}{\text{Game Z} } \newcommand{\ioracle}{\textit{Chall} } % EdDSA procedures \newcommand{\keygen}{\textit{KeyGen} } \newcommand{\sign}{\textit{Sign} } \newcommand{\verify}{\textit{Verify} } % Security Notions \newcommand{\cma}{\text{SUF-CMA} } \newcommand{\adversary}[1]{\mathcal{#1}} \newcommand{\advantage}[2]{\text{Adv}_{#1}^{#2}} \newcommand{\prone}[1]{Pr[#1 \Rightarrow 1]} % Oracle \newcommand{\Osign}{\textit{Sign} } \newcommand{\Odl}{\textif{DL} } % Structrues \newcommand{\curve}{E}