preamble.tex

\newcommand{\err}{\mathsf{err}}

\newcommand{\cat}{|}
\newcommand{\dsum}{/}
\newcommand{\pdsum}[2]{\mathrm{diag}(#1,#2)}

\newcommand{\sblt}{\stackrel{s}{\bullet}}

\newcommand{\algSize}{normalsize} 
%\newcommand{\algSize}{footnotesize}
\newcommand{\sfleft}{{\mathsf{left}}}
\newcommand{\sfright}{{\mathsf{right}}}

\newcommand{\ps}{\Psi({\dist_k})}
\newcommand{\psws}{{\Psi(\overline{\dist}_k)}}
\newcommand{\sps}{{\Psi_{\mathsf{spl}}(\dist_k)}}
\newcommand{\Sps}{\Psi_{\mathsf{spl}}}
\newcommand{\spsws}{{\Psi_\mathsf{spl}(\overline{\dist}_k)}}
\newcommand{\Spsws}{\Psi_\mathsf{spl}}
\newcommand{\spsmas}{\Psi_{\sfsum}(\dist_k)}
\newcommand{\Spsmas}{\Psi_{\sfsum}}
\newcommand{\spswsmas}{\Psi_{\sfsum}(\overline{\dist}_k)}
\newcommand{\Spswsmas}{\Psi_{\sfsum}}
\newcommand{\spswscomm}{{\Psi_{\mathsf{com}}(\overline{\dist}_k)}}
\newcommand{\Spswscomm}{\Psi_{\mathsf{com}}}
\newcommand{\bbb}{\bar{b}}
\newcommand{\capprox}{\overset{c}{\approx}}

\newcommand{\latexDeMierdaEstupido}{]}

\newcommand{\Comm}{\mathsf{Comm}}
\newcommand{\Com}{\mathsf{Com}}
\newcommand{\vect}{\mathbf{vec}}

\newcommand{\lef}{{\mathtt{l}}}
\newcommand{\rig}{{\mathtt{r}}}
\newcommand{\stmnt}{\mathsf{stm}}
%Commitment keys

%Log Left Commitment Key
\newcommand{\llck}{\vecb{g}}
%Log Right Commitment Key
\newcommand{\lrck}{\vecb{h}}
%Left Commitment Key
\newcommand{\lck}{[{\llck}]_1}
%Right Commitment Key
\newcommand{\rck}{[{\lrck}]_2}
\newcommand{\rcks}{[{\lrck}]_1}

%Commitement keys matrices

%Log Left Commitment Keys
\newcommand{\Llck}{\matr{G}}
%Log Right Commitment Keys
\newcommand{\Lrck}{\matr{H}}
%Left Commitment Keys
\newcommand{\Lck}{[\Llck]_1}
%Right Commitment Keys
\newcommand{\Rck}{[{\Lrck}]_2}
\newcommand{\Rcks}{[{\Lrck}]_1}

%For quadratic info \lck\rck^\top
\newcommand{\Lqmatr}{\matr{C}}
\newcommand{\Qmatr}{{[\Lqmatr]_1}}
\newcommand{\Qspace}{\mathcal{C}}

% c_\Delta
\newcommand{\lccom}{\vecb{c}_\Delta}
\newcommand{\ccom}{\hvecb{c}_\Delta}

\newcommand{\pke}{\mathsf{PKE}}
\newcommand{\kem}{\mathsf{KEM}}
\newcommand{\prf}{\mathsf{PRF}}
\newcommand{\ev}{\mathsf{F}}
\newcommand{\KEM}{\mathsf{KEM}}
\newcommand{\gen}{\mathsf{Gen}}
\newcommand{\enc}{\mathsf{Enc}}
\newcommand{\Enc}{\mathsf{Enc}}
\newcommand{\dec}{\mathsf{Dec}}
\newcommand{\Dec}{\mathsf{Dec}}
\newcommand{\pk}{\mathit{pk}}
\newcommand{\sk}{\mathit{sk}}
\newcommand{\cdh}{\ensuremath{\mathsf{CDH}}}
\newcommand{\ddh}{\ensuremath{\mathsf{DDH}}}
\newcommand{\sxdh}{\ensuremath{\mathsf{SXDH}}}
\newcommand{\mddh}{\ensuremath{\mathsf{MDDH}}}
\newcommand{\mcdh}{\ensuremath{\mathsf{MCDH}}}
\newcommand{\fmdh}{\ensuremath{\mathsf{KerMDH}}}
\newcommand{\bddh}{\ensuremath{\mathsf{BDDH}}}
\newcommand{\mat}[1]{\ensuremath{#1\mbox{-}\mathsf{Mat}}}
\newcommand{\pddh}[1]{\ensuremath{#1\mbox{-}\mathsf{PDDH}}}
\newcommand{\mlddh}[1]{\ensuremath{#1\mbox{-}\mathsf{MLDDH}}}
\newcommand{\eddh}[1]{\ensuremath{#1\mbox{-}\mathsf{EDDH}}}
\newcommand{\casc}[1]{\ensuremath{#1\mbox{-}\mathsf{Casc}}}
\newcommand{\scasc}[1]{\ensuremath{#1\mbox{-}\mathsf{SCasc}}}
\newcommand{\lin}[1]{\ensuremath{#1\mbox{-}\mathsf{Lin}}}
\newcommand{\rlin}[1]{\ensuremath{#1\mbox{-}\mathsf{RLin}}}
\newcommand{\re}{\mathsf{RE}_\G}
\newcommand{\kcirc}[1]{\ensuremath{#1\mbox{-}\mathsf{Circ}}}
\newcommand{\escQE}{\gamma}
\newcommand{\EscQE}{\Gamma}


% M \in Z^{\la \times \lb}, N\in Z^{\la \times \lc}, \Lambda\in Z^{\ld\times \lb}
% x \in \Z_q^\la, b\in \Z_q^\lb, w\in\Z_q^\lc, \alpha\in \Z_q^\ld
\newcommand{\la}{{\ell_1}}
\newcommand{\lb}{{m}}
\newcommand{\lc}{{\ell_2}}
\newcommand{\ld}{{\ell_3}}


\newcommand{\LangMN}{{\Lang_{[\matr{M}]_1,[\matr{N}]_1,\matr{\Lambda},\grkb{\alpha}}}}



\newcommand{\skermdh}{\ensuremath{\mathsf{SKerMDH}}}
\newcommand{\kermdh}{\ensuremath{\mathsf{KerMDH}}}

\newcommand{\KG}{\mathsf{KeyGen}}
\newcommand{\GS}{{\mathsf{GS}}}

%HPS definitions
\newcommand{\univo}{universal$_1$\xspace}
\newcommand{\univt}{universal$_2$\xspace}
\newcommand{\distance}[2]{\Delta\left[#1 \,,\, #2\right]}
\newcommand{\entropic}{entropic\xspace}
\newcommand{\ciphertext}{{c}}
\def\params{\mathit{params}}
\newcommand{\structure}{\mathcal{S}}
\newcommand{\ciphersp}{\mathcal{C}}
\newcommand{\conssp}{\mathcal{V}}
\newcommand{\primeorder}{p}
\newcommand{\keysp}{\mathcal{K}}
\def\hps{\varfont{hps}}
\newcommand{\advBhps}{\calB}
\newcommand{\advBhpso}{\calB_{1}}
\newcommand{\advBhpst}{\calB_{2}}
\newcommand{\PK}{\mathcal{PK}}
\newcommand{\SK}{\mathcal{SK}}
\newcommand{\hash}{\mu}
\newcommand{\bigiota}{\mathcal{I}}
\newcommand{\var}[1]{{\mathsf{#1}}}
\newcommand{\vvar}[1]{{\mathbf{\var{#1}}}}
\newcommand{\varb}{\mathsf{b}}
\newcommand{\varx}{\mathsf{x}}
\newcommand{\vvarx}{\textbf{\textsf{x}}}
\newcommand{\vary}{\mathsf{y}}
\newcommand{\vvary}{\textbf{\textsf{y}}}
\newcommand{\varz}{\mathsf{z}}
\newcommand{\varX}{\mathsf{X}}
\newcommand{\varY}{\mathsf{Y}}
\newcommand{\varZ}{\mathsf{Z}}
\newcommand{\varvecx}{\vec{\varx}}
\newcommand{\varvecy}{\vec{\vary}}
\newcommand{\varvecz}{\vec{\varx}}
\newcommand{\hcx}{(\hat{x}_1,\ldots,\hat{x}_m)}
\newcommand{\matrB}{\matr{B}} 
\newcommand{\vecL}{(\hat{l}_1,\ldots,\hat{l}_n)}



% Las instancias de SPLHS se van a llamr \Phi
\newcommand{\SPLHSinst}{\Phi}
\newcommand{\SG}{\mathsf{SignGen}}
\newcommand{\SN}{\mathsf{Sign}}
\newcommand{\SD}{\mathsf{SignDerive}}
\newcommand{\SV}{\mathsf{Verify}}
\newcommand{\SP}{\ensuremath{\mathsf{SP}}}
\newcommand{\poly}{\mathsf{poly}}

%Comicmens a los eltos de la listas
\newcommand{\lcom}{\vecb{f}}
\newcommand{\Lcom}{\matr{F}}

%Definition
\newcommand{\MP}{\mathsf{MP}}
\newcommand{\GScom}{\mathsf{GS.Com_{\hvecb{U}}}}
\newcommand{\MPcomg}{\mathsf{MP.Com}_{\hmatr{G}}}
\newcommand{\MPcomh}{\mathsf{MP.Com}_{\hmatr{H}}}




% QA-NIZK for linear spaces
\newcommand{\ZKLinInst}{\mathsf{ZKLin}}
\newcommand{\ZKLinK}{\mathsf{ZKLin.K}_0}
\newcommand{\ZKLinKK}{\mathsf{ZKLin.K}_1}
\newcommand{\ZKLinCRS}{\mathsf{ZKLin.crs}}

\newcommand{\QANIZKsum}{{\spswsmas}}
\newcommand{\QANIZKcomms}{{\spswscomm}}
\newcommand{\QANIZKsym}{{\Psi_{\mathsf{sym}}}}

\newcommand{\nb}{{\overline{n}}}

\newcommand{\bb}{\overline{b}}
\newcommand{\bub}{{b(\overline{b}-1)}}
\newcommand{\tm}{\tilde{m}}

\newcommand{\rank}{\mathbf{rank}}
\newcommand{\HPSscheme}{\mathsf{HPS}}
%\newcommand{\THPSscheme}{\schemefont{HPS_t}}
% \newcommand{\THPSscheme}{\mathsf{HPS^{td}}}
% \newcommand{\HHPSscheme}{\mathsf{HPS}_2}
\newcommand{\HPSsys}{\mathsf{Param}}
\newcommand{\HPSpub}{\mathsf{Pub}}
\newcommand{\HPSpriv}{\mathsf{Priv}}
\newcommand{\HPSdec}{\mathsf{Decide}}
\newcommand{\eval}{\Lambda}
%\newcommand{\hpscu}{{\notionfont{cu}_2}}
%\newcommand{\ExpHPScu}[2]{\Exp^{\hpscu}_{#1,#2}}
%\newcommand{\AdvHPScu}[2]{\Adv^{\hpscu}_{#1,#2}}
%\newcommand{\hpscub}{{\notionfont{\hpscu\mbox{-}b}}}
%\newcommand{\hpscuz}{{\notionfont{\hpscu\mbox{-}0}}}
%\newcommand{\hpscuo}{{\notionfont{\hpscu\mbox{-}1}}}
%\newcommand{\ExpHPScub}[2]{\Exp^{\hpscub}_{#1,#2}}
%\newcommand{\ExpHPScuo}[2]{\Exp^{\hpscuo}_{#1,#2}}
%\newcommand{\ExpHPScuz}[2]{\Exp^{\hpscuz}_{#1,#2}}
% \newcommand{\pcol}{\delta}
\newcommand{\prcol}{\delta}
\newcommand{\trapdoor}{\omega}
\newcommand{\witness}{r}
\newcommand{\algD}{\mathsf{D}}
\newcommand{\algK}{\mathsf{K}}
\newcommand{\algG}{\mathsf{G}}
\newcommand{\algP}{\mathsf{P}}
\newcommand{\algV}{\mathsf{V}}
\newcommand{\algS}{\mathsf{S}}
\newcommand{\algF}{\mathsf{F}}
\newcommand{\algVrfy}{\mathsf{Vrfy}}

\newcommand{\R}{\mathcal{R}}
\newcommand{\M}{\mathcal{M}}
\newcommand{\dist}{\mathcal{D}}
\newcommand{\distw}{\mathcal{W}}
\newcommand{\distk}{\mathcal{K}}
\newcommand{\distlin}{\mathcal{L}}
\newcommand{\distrlin}{\mathcal{RL}}
\newcommand{\distc}{\mathcal{C}}
\newcommand{\distsc}{\mathcal{SC}}
\newcommand{\distcirc}{\mathcal{CI}}
\newcommand{\distu}{\mathcal{U}}
\newcommand{\distp}{\mathcal{P}}
\newcommand{\distink}{\dist_k^{m,i}}
\newcommand{\distjnk}{\dist_1^{m,i}}

\newcommand{\distinmk}{\dist_k^{mn,i}}
\newcommand{\distzeronmk}{\dist_k^{mn,0}}
\newcommand{\distzeronk}{\dist_k^{m,0}}
\newcommand{\distlininone}{\distlin_1^{m,i}}
\newcommand{\distlinisnone}{\distlin_1^{m,i^*}}
\newcommand{\distlinizeroone}{\distlin_1^{m,0}}
\newcommand{\distlinjsnzero}{\distlin_1^{n,j^*}}
\newcommand{\block}[1]{
  \underbrace{\begin{matrix}1 & \cdots & 1\end{matrix}}_{#1}
}
%\newcommand{\gets}{\leftarrow}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\G}{\mathsf{Gen}}
\newcommand{\advD}{\mathsf{D}}
\newcommand{\advA}{\mathsf{A}}
\newcommand{\advB}{\mathsf{B}}
\newcommand{\adv}{\mathbf{Adv}}
\newcommand{\group}{{gk}}
\newcommand{\gk}{\group}
\newcommand{\pgroup}{\mathcal{PG}}
\newcommand{\mgroup}[1]{\mathcal{MG}_{#1}}
\newcommand{\ggen}{\mathsf{Gen}}
\newcommand{\pggen}{\mathsf{PGen}}
\newcommand{\mggen}[1]{\mathsf{MGen}_{#1}}
\newcommand{\heading}[1]{\smallskip\noindent{\sc{#1}}}
\newcommand{\vecb}[1]{{\boldsymbol{#1}}}
\newcommand{\vecbt}[1]{\vec{#1}^{\ \top}}
\newcommand{\uvecb}[1]{{\vect({\vecb{#1}})}}
\newcommand{\tvecb}[1]{{\tilde{\vecb{#1}}}}
\newcommand{\tgrkb}[1]{\tilde{\grkb{#1}}}
\newcommand{\ovecb}[1]{{\overline{\vecb{#1}}}}
\newcommand{\Pt}{\mathcal{P}}
\newcommand{\Opt}{\mathcal{O}}
\newcommand{\pt}[1]{\mathcal{#1}}
\newcommand{\stbl}{\ \tilde \bullet \ }
\newcommand{\bilgroup}{\mathcal{PG}}
\newcommand{\matr}[1]{\mathbf{{#1}}}
\newcommand{\vecw}{\vecb{w}}
\newcommand{\vecr}{\vecb{r}}
\newcommand{\vecz}{\vecb{z}}
\newcommand{\vecy}{\vecb{y}}
\newcommand{\vecx}{\vecb{x}}
\newcommand{\veca}{\vecb{a}}
\newcommand{\matrA}{\matr{A}}
\newcommand{\hmatrA}{\hmatr{A}}
\newcommand{\cmatrA}{\cmatr{A}}
\newcommand{\smallpmatrix}[1]{\left(\begin{smallmatrix}#1\end{smallmatrix}\right)}
\newcommand{\pmatri}[1]{\left(\begin{matrix}#1\end{matrix}\right)}
\newcommand{\bmatri}[1]{\left[\begin{matrix}#1\end{matrix}\right]}
\newcommand{\matri}[1]{{\begin{matrix}#1\end{matrix}}}
\newcommand{\smatri}[1]{{\begin{smallmatrix}#1\end{smallmatrix}}}
\newcommand{\sfsplit}{\mathsf{spl}}
\newcommand{\bulletsp}{ \bullet }
\newcommand{\newf}{\widehat{f}}
\newcommand{\newF}{\widehat{F}}
\newcommand{\com}{\mathsf{com}}
\newcommand{\eq}{\mathsf{eq}}
\newcommand{\eqd}{\equiv}
\newcommand{\negl}{\mathsf{negl}}
\newcommand{\A}{\mathcal{A}}
\newcommand{\GG}{\mathbb{G}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\Gr}{\ensuremath{\mathbb{G}_1}}
\newcommand{\Hr}{\ensuremath{\mathbb{G}_2}}
\newcommand{\T}{\ensuremath{\mathbb{T}}}
\newcommand{\SSDP}{\ensuremath{\mathsf{SSDP}}}
\newcommand{\PermP}{\ensuremath{\mathsf{PermP}}}
\newcommand{\PP}{\ensuremath{\mathsf{PP^*}}}
\newcommand{\bmatr}[1]{\left[\matr{#1}\right]}
\newcommand{\hmatr}[1]{{\hat{\matr{#1}}}}
\newcommand{\cmatr}[1]{\check{\matr{#1}}}
\newcommand{\bvecb}[1]{\left[\vecb{#1}\right]}
\newcommand{\hvecb}[1]{{\hat{\vecb{#1}}}}
\newcommand{\cvecb}[1]{\check{\vecb{#1}}}
\newcommand{\bits}{\{0,1\}}
\newcommand{\rmIm}{\mathbf{Im}}
\newcommand{\sfGame}{\mathsf{Game}}
\newcommand{\sfReal}{\mathsf{Real}}
\newcommand{\grkb}[1]{{\boldsymbol #1}}
\newcommand{\ugrkb}[1]{{\underline{\grkb{#1}}}}
\newcommand{\hgrkb}[1]{\hat{\grkb{#1}}}
\newcommand{\cgrkb}[1]{\check{\grkb{#1}}}
\newcommand{\SDP}{\ensuremath{\mathsf{SDP}}}
\newcommand{\Span}{\mathbf{Span}}
\newcommand{\Group}{G}
\newcommand{\Forger}{\mathsf{F}}
\newcommand{\advSound}{\mathsf{P}^*}
\newcommand{\Lang}{\mathcal{L}}
\newcommand{\crs}{\mathsf{crs}}
\newcommand{\sfproof}{\mathsf{proof}}
\newcommand{\sfbits}{\mathsf{bits}}
\newcommand{\sfbitsn}{{\mathsf{bits},n}}
\newcommand{\sflin}{\mathsf{lin}}
\newcommand{\sfcom}{\mathsf{com}}
\newcommand{\sfbin}{\mathsf{bin}}
\newcommand{\sfset}{\mathsf{set}}
\newcommand{\sfsum}{\mathsf{sum}}
\newcommand{\rp}{{\mathsf{range}\mbox{-}\mathsf{proof}}}
\newcommand{\ovG}{\overline{\matr{G}}}
\newcommand{\ovc}{\overline{\vecb{c}}}
\newcommand{\ovb}{\overline{\vecb{b}}}

\newcommand{\dmatrix}[1]{\begin{pamtrix}#1 & \cdots & \vecb{0}\\\vdots & \ddots & \vdots\\\vecb{0}& \ldots & #1\end{pmatrix}}
\newcommand{\sdmatrix}[1]{\smallpmatrix{#1 & \cdots & \vecb{0}\\\vdots & \ddots & \vdots\\\vecb{0}& \ldots & #1}}


%weas q hay que hacer pa q no webee el latex
\newsavebox{\smlmat}% Box to store smallmatrix content
\newsavebox{\smat}
\savebox{\smlmat}{$\left(\begin{smallmatrix}
\matr{G}_1 & \ldots & \vecb{0}   & \vecb{g}_{n+1} & \ldots & \vecb{0}\\
\vdots     & \ddots & \vdots     & \vdots         & \ddots & \vdots\\
\vecb{0}   & \ldots & \matr{G}_1 & \vecb{0}       & \ldots & \vecb{g}_{n+1}
\end{smallmatrix}\right)$}

\savebox{\smat}{$\left(\begin{matrix}
s_1 & \ldots & s_n\\
0   & \ldots & 0
\end{matrix}\right)$}






\newcommand{\sG}{|\GG_1|}
\newcommand{\sH}{|\GG_2|}
\newcommand{\s}{(\sG+\sH)}

\newcommand{\vu}{\hat{\vecb{u}}}
%\newcommand{\vv}{\check{\vecb{v}}}
\newcommand{\vc}{\hat{\vecb{c}}}
\newcommand{\vd}{\check{\vecb{d}}}
\newcommand{\zip}{\mathbf{zip}}

\newcommand{\indexSet}[2]{\mathcal{I}_{#1,#2}}
\newcommand{\SignaturesSet}{\mathcal{S}}
\newcommand{\KGen}{\mathsf{KGen}}
\newcommand{\Sign}{\mathsf{Sign}}
\newcommand{\Ver}{\mathsf{Ver}}

\newcommand{\bit}{\mathsf{bit}}
\newcommand{\sfts}{\mathsf{ts}}

%El-Gamal keys
\newcommand{\egpk}{\hat{x}}
\newcommand{\egsk}{x}
\newcommand{\egvpk}{\hvecb{k}}
\newcommand{\egvsk}{\vecb{k}}

%The set of permutation matrices
\newcommand{\matrPerms}{\mathcal{S}}
%The set of permutations
\newcommand{\Perms}{S}

\newcommand{\duda}[1]{{\iffalse\color{red}#1\fi}}

\newcommand{\cambio}[2]{{\iffalse\color{blue}Ahora: \fi#1}{\iffalse\color{red}(Antes: #2)\fi}}

\newenvironment{code}
   {\begin{tabbing}
   \hspace{4mm} \= \hspace{4mm} \= \hspace{4mm} \= \hspace{4mm} \= \hspace{4mm} \= \kill \\
   }
   {\end{tabbing}}

%\newcommand{\authnote}[2]{\medskip \noindent {\bf #1 says:} #2}
\newcommand{\authnote}[2]{\medskip \noindent {\bf #1 says:} {\textcolor{blue}{#2}}}

%\newtheorem{corollary}{Corollary}

\newtheorem{fact}{Fact}
\newtheorem{observation}{Observation}

\newcommand{\Am}{A}
\newcommand{\vX}{\ensuremath{\hat{\vecb{x}}}}
\newcommand{\VX}{\ensuremath{\hat{\vecb{v}}}}
\newcommand{\WX}{\ensuremath{\hat{\vecb{w}}}}
\newcommand{\VY}{\ensuremath{\check{\vecb{v}}}}
\newcommand{\WY}{\ensuremath{\check{\vecb{w}}}}
\newcommand{\vy}{\ensuremath{\vecb{y}}}
\newcommand{\vY}{\ensuremath{\check{\vecb{y}}}}
\newcommand{\vx}{\ensuremath{\vecb{x}}}


\newcommand{\U}{\ensuremath{\vecb{u}}}
\newcommand{\V}{\ensuremath{\vecb{v}}}
\newcommand{\vr}{\ensuremath{\vecb{r}}}
\newcommand{\vs}{\ensuremath{\vecb{s}}}
\newcommand{\vt}{\ensuremath{\vecb{t}}}


\newcommand{\ux}{\ensuremath{\hat{\vecb{u}}}}
\newcommand{\uy}{\ensuremath{\check{\vecb{u}}}}

\newcommand{\minitbl}[2]{\begin{tabular}{l}{#1}\\{#2}\end{tabular}}

\newcommand{\ef}{\iffalse}
%
%\makeatletter
%\newcommand*{\inlineequation}[2][]{%
%  \begingroup
%    % Put \refstepcounter at the beginning, because
%    % package `hyperref' sets the anchor here.
%    \refstepcounter{equation}%
%    \ifx\\#1\\%
%    \else
%      \label{#1}%
%    \fi
%    % prevent line breaks inside equation
%    \relpenalty=10000 %
%    \binoppenalty=10000 %
%    \ensuremath{%
%      % \displaystyle % larger fractions, ...
%      #2%
%    }%
%    ~\@eqnnum
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%}
%\makeatother
%
%\makeatletter
%\renewcommand*{\@opargbegintheorem}[3]{\trivlist
%  \item[\hskip \labelsep{\bfseries #1\ #2}] \textbf{(#3)}\ \itshape}
%\makeatother