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\usepackage{amsmath}
\usepackage{bbm}
\usepackage{hyperref}
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\usepackage{stmaryrd}
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\usepackage{placeins}

\newcommand{\version}{\input{spec_version}}

\newcommand{\F}{\mathbbm{F}}
\newcommand{\G}{\mathbbm{G}}
\newcommand{\Z}{\mathbbm{Z}}
\newcommand{\N}{\mathbbm{N}}
\newcommand{\hash}{\mathbbm{H}}
\newcommand{\I}{\mathbbm{I}}
\newcommand{\B}{\mathbbm{B}}

\newcommand{\public}{\textsf{public}}
\newcommand{\shuffle}{\textsf{shuffle}}
\newcommand{\basesixfour}{\textsf{BASE64}}
\newcommand{\shatwo}{\textsf{SHA256}}

\newcommand{\jstring}{\texttt{string}}
\newcommand{\uuid}{\texttt{uuid}}
\newcommand{\tpk}{\texttt{trustee\_public\_key}}
\newcommand{\election}{\texttt{election}}
\newcommand{\ballot}{\texttt{ballot}}
\newcommand{\etally}{\texttt{encrypted\_tally}}
\newcommand{\pdecryption}{\texttt{partial\_decryption}}
\newcommand{\result}{\texttt{result}}

\newcommand{\cert}{\texttt{cert}}
\newcommand{\poly}{\texttt{polynomial}}
\newcommand{\vinput}{\texttt{vinput}}
\newcommand{\voutput}{\texttt{voutput}}

\newcommand{\vc}[1]{\textcolor{blue}{#1}}
\newcommand{\vcomment}[1]{\textcolor{violet}{#1}}

\title{Belenios specification}
\date{Version~\version}
\author{Stéphane Glondu}

\begin{document}
\maketitle
\tableofcontents

\section{Introduction}
{\it References.}
This document is a specification of the voting protocol implemented in
Belenios \version.
A high level description of Belenios and some statistics about its
usage can be found in~\cite{Belenios-Meadows2019}.
A security proof of the protocol for ballot privacy and verifiability
is presented in~\cite{Belenios-Easycrypt-CSF18}. The proof has been conducted with the
tool EasyCrypt. It focuses on the protocol aspects and assumes
security of the cryptographic primitives.
The cryptographic primitives have been introduced in various places
and their security proofs is spread across several references.
\begin{itemize}
\item
The threshold decryption scheme is
based on a ``folklore'' scheme:
Pedersen’s~\cite{Pedersen} Distributed Key Generation (DKG) that has several variations.
The variant considered in Belenios is proved
in~\cite{Pereira-proofDKG-eprint2024}.
%described in~\cite{wpes2013} and
%proved in~\cite{wpes2013,asiacrypt12}.
\item Ballots are composed of an ElGamal encryption of the votes and a
  zero-knowledge proof of well-formedness, as for the Helios
  protocol~\cite{Helios}. Compared to Helios, we support blank votes,
  which required to adapt the zero-knowledge proofs, as specified and
  proved in~\cite{note-Pierrick}. Additionally, ballots are signed to
  avoid ballot stuffing, as introduced in~\cite{CGGI-esorics14} and also
  described in~\cite{Belenios-Meadows2019}.
  Zero-knowledge proofs include the complete description of the group
  to avoid attacks described in~\cite{EVoteID20-Belenios}.
  \item During the tally phase, Belenios supports two modes. Ballots are either combined
    homomorphically or shuffled and randomized, using mixnets. The
    mixnet algorithms are taken from the CHVote specification~\cite{CHVote}.
\end{itemize}

% has been conducted with EasyCrypt and shows
% More discussion, theoretical explanations and
% bibliographical references can be found in an article
% available online.\footnote{\url{https://hal.inria.fr/hal-02066930/document}}

{\it Types of supported elections.}
Belenios supports two main types of questions.
In the \emph{homomorphic case}, voters can select between $k_1$ and
$k_2$ candidates out of $k$ candidates. This case is called
homomorphic because the result of the election for such questions is
the number of votes received for each candidate. No more information
is leaked.
In the \emph{non-homomorphic case}, voters can give a number to each
candidate. This can be used to rank candidates or grade them. Then the
(raw) result of the election is simply the list of votes, as emitted
by the voters, in
a random order, to preserve privacy.
Any counting method can then be  applied
(e.g. Condorcet, STV, or majority judgement) although Belenios does
not offer support for this.
The non-homomorphic case  therefore offers much more flexibility, at
the cost of extra steps during the tally (in order to securely shuffle
the ballots).
Belenios supports both types of questions and an election can even
mix homomorphic and non-homomorphic questions.
% and slightly less privacy
\medskip

{\it Group parameters.}
The cryptography involved in Belenios needs a cyclic group $\G$ where
discrete logarithms are hard to compute. We will denote by $g$ a
generator and $q$ its order. We use a multiplicative notation for the
group operation. In practice, $\G$ can be either a prime order multiplicative
subgroup of $\F^*_p$ (hence, all exponentiations are implicitly done
modulo $p$), or a prime order subgroup of the points of an elliptic curve. We
suppose the group parameters are agreed on
beforehand. Examples of supported groups are given in
section~\ref{default-group}.

\medskip

{\it Weights.}  In the homomorphic case (and only in the homomorphic
case), each voter has a weight: a
ballot is counted as many times as the weight of its owner. Usually,
the weight of all voters is 1 but sometimes, it may be useful to
assign different weights. We assume the sum of all weights is not too
big, so that it can be computed as the discrete logarithm of some
group element.

\section{Parties}

\newcommand{\pk}{\texttt{public\_key}}
\newcommand{\sk}{\texttt{private\_key}}
\newcommand{\proof}{\texttt{proof}}
\newcommand{\iproof}{\texttt{iproof}}
\newcommand{\ciphertext}{\texttt{ciphertext}}

\newcommand{\pklabel}{\textsf{public\_key}}
\newcommand{\pok}{\textsf{pok}}
\newcommand{\challenge}{\textsf{challenge}}
\newcommand{\response}{\textsf{response}}
\newcommand{\alphalabel}{\textsf{alpha}}
\newcommand{\betalabel}{\textsf{beta}}
\newcommand{\Hash}{\mathcal{H}}

\begin{itemize}
\item $\mathcal{A}$: server administrator
\item $\mathcal{C}$: credential authority
\item $\mathcal{T}_1,\dots,\mathcal{T}_m$: trustees
\item $\mathcal{V}_1,\dots,\mathcal{V}_n$: voters; each voter has a
    weight $w_i$ equal to 1 by default
%\item $\mathcal{M}_1,\dots,\mathcal{M}_p$: shufflers (if using non-homomorphic questions)
\item $\mathcal{S}$: voting server \\
  The voting server maintains the public data $D$ (see~\ref{public-data}) that
consists of a sequence of data and events, and is structured as a
hash chain. It contains in particular:
  \begin{itemize}
  \item the election data $E$
  \item the structure $PK$ that contains the verification keys of the
    trustees and other verification material
  \item the list $L$ of public credentials
  \item the list $B$ of accepted ballots
  \item the result of the election {\result} (once the election is tallied)
  \end{itemize}
  The voting server also produces a list $PB$ of pretty ballots, that
  is a list of hashes of ballots (the last ballot of each voter).
\end{itemize}

\section{Processes}
\label{processes}

\subsection{Election setup}
\label{election-setup}

\begin{enumerate}
\item $\mathcal{A}$ starts the preparation of an election, providing
  in particular the questions and the list of voters
\item $\mathcal{S}$ generates a fresh \hyperref[basic-types]{$\uuid$} $u$ and
  sends it to $\mathcal{C}$
\item $\mathcal{C}$ generates random \hyperref[credentials]{private
    credentials} $c_1,\dots,c_n$ and computes
  \[L=\textsf{sort}((\public(c_1),w_1,\mathcal{V}_1),\dots,(\public(c_n),w_n,\mathcal{V}_n))\]
\item for $j\in[1\dots n]$, $\mathcal{C}$ sends $c_j$ to $\mathcal{V}_j$
\item \label{item-forget} (optional) $\mathcal{C}$ forgets $c_1,\dots,c_n$
 \item $\mathcal{C}$ sends $L$ to $\mathcal{S}$
 \item $\mathcal{S}$ checks that the $\mathcal{V}_i$ and the $w_i$ in
   $L$ are correct, and that all public credentials are
   distinct;
 \item $\mathcal{S}$ defines the shape of the
   \hyperref[trustees]{$\texttt{trustees}$} structure that will be
   used in the election depending on $\mathcal{A}$'s instructions;
 \item $\mathcal{S}$ and $\mathcal{T}_1,\dotsc,\mathcal{T}_m$ run key
   establishment protocols (see \ref{process-filling-trustees}) as
     needed to fill in the $\texttt{trustees}$ structure;
\item $\mathcal{S}$ creates the \hyperref[elections]{$\election$} $E$
\item $\mathcal{C}$ checks that the list of public credentials in $L$
  is exactly the one that appears on the public data of the election.
\end{enumerate}
Step~\ref{item-forget} is optional. It offers a better protection
against ballot stuffing in case $\mathcal{C}$ unintentionally leaks
private credentials (but disables credential recovery, see
Section~\ref{sec:recovery}).

\subsubsection{Filling in the \texttt{trustees} structure}
\label{process-filling-trustees}

The \hyperref[trustees]{\texttt{trustees}} structure consists of
\texttt{"Single"} or \texttt{"Pedersen"} items. For each of these
items, one or several trustees run the corresponding protocol below to
produce a sub-key $y_\tau$. Once all protocols have been run,
$\mathcal{S}$ synthesizes the global election public key $y$ from the
sub-keys computed in each protocol by multiplying them:
\[
  y=\prod_\tau y_\tau
\]

\paragraph{\texttt{"Single"} protocol}

This protocol involves a single trustee $\mathcal{T}$, whose presence
will be required to compute the tally.
\begin{enumerate}
\item $\mathcal{T}$ generates a \hyperref[trustee-keys]{$\tpk$}
  $\gamma$ and sends it to $\mathcal{S}$
\item $\mathcal{S}$ checks $\gamma$
\end{enumerate}
Later, when the election is open:
\begin{enumerate}
\item $\mathcal{T}$ checks that $\gamma$ appears in the set of verification
  keys $PK$ of the election of {$\uuid$} $u$ (the id of the election
  should be publicly known)
\end{enumerate}
The sub-key for this protocol is the $\textsf{public\_key}$ field of
$\gamma$.

\paragraph{\texttt{"Pedersen"} protocol}

This protocol involves $\mu$ trustees
$\mathcal{T}_1,\dots,\mathcal{T}_\mu$ such that only a subset of
$t+1$ of them will be needed to compute the tally.
The original Pedersen DKG scheme~\cite{Pedersen} assumes a secret and authenticated
channel between each pair of trustees. Since our trustees do not even
have a PKI, we first require that each trustee generates a fresh key
pair, that they then use to encrypt and sign messages for the other
trustees. All messages are sent and received (encrypted) through the voting server,
in order to simplify the communication infrastructure.
\begin{enumerate}
\item for $z\in[1\dots \mu]$,
  \begin{enumerate}
  \item $\mathcal{T}_z$ generates a \hyperref[certificates]{$\cert$} $\gamma_z$
    and sends it to $\mathcal{S}$
  \item $\mathcal{S}$ checks $\gamma_z$
  \end{enumerate}
\item $\mathcal{S}$ assembles $\Gamma=\gamma_1,\dotsc,\gamma_\mu$
\item for $z\in[1\dots \mu]$,
  \begin{enumerate}
  \item $\mathcal{S}$ sends $\Gamma$ to $\mathcal{T}_z$ and $\mathcal{T}_z$ checks it
  \item $\mathcal{T}_z$ generates a \hyperref[polynomials]{$\poly$} $P_z$ and
    sends it to $\mathcal{S}$. This corresponds to the actual first
    step of the Pedersen protocol. Each trustee generates a secret
    polynomial $f_z$. $P_z$ consists in particular of:
    \begin{itemize}
    \item the coefficients of $f_z$, as well as  $\Gamma$,
      self signed and encrypted, in order to store them for the next
      step
    \item the signed exponentiated coefficients of $f_z$,
      to be sent to all trustees
    \item the evaluation of the polynomial $s_{z,z'}=f_z(z')$, encrypted for
      trustee $\mathcal{T}_{z'}$ and signed by $\mathcal{T}_z$, for all $z'$
    \end{itemize}
  \item $\mathcal{S}$ checks $P_z$
  \end{enumerate}
\item for $z\in[1\dots \mu]$, $\mathcal{S}$ computes a
  \hyperref[vinputs]{$\vinput$} $\textsf{vi}_z$, that corresponds to
  the aggregation of the messages encrypted for $z$, collected from
  all the trustees
\item for $z\in[1\dots \mu]$,
  \begin{enumerate}
  \item $\mathcal{S}$ sends $\Gamma$ to $\mathcal{T}_z$ and $\mathcal{T}_z$ checks it
  \item $\mathcal{S}$ sends $\textsf{vi}_z$ to $\mathcal{T}_z$ and $\mathcal{T}_z$ checks it
  \item $\mathcal{T}_z$ computes a \hyperref[voutputs]{$\voutput$} $\textsf{vo}_z$ and
    sends it to $\mathcal{S}$. It corresponds to the resulting (signed) public
    key share of $\mathcal{T}_z$, with a proof of knowledge of the associated secret key.
  \item $\mathcal{S}$ checks $\textsf{vo}_z$
  \end{enumerate}
\item $\mathcal{S}$ extracts encrypted decryption keys $K_1,\dots,K_\mu$ and
  \hyperref[threshold-params]{threshold parameters}
\end{enumerate}
Later, when the election is open:
\begin{enumerate}
\item for $z\in[1\dots \mu]$, $\mathcal{T}_z$ checks that $\gamma_z$
  appears in the set of verification keys $PK$ of the election of {$\uuid$}
  $u$ (the id of the election should be publicly known).
\end{enumerate}
The sub-key for this protocol is computed from the polynomials of each
trustee as specified in section~\ref{polynomials}.

\subsection{Vote}

\begin{enumerate}
\item $\mathcal{V}$ gets public data of $E$
\item $\mathcal{V}$ uses the index in her secret credential $c$ to get
  her public credential $\hat c$ (from election public data), and checks that
  $\hat c=\public(c)$
\item $\mathcal{V}$ creates a \hyperref[ballots]{$\ballot$} $b$, she
  computes the hash $h$ of $b$, called tracking number, and submits $b$ to $\mathcal{S}$
\item $\mathcal{S}$ processes $b$:
  \begin{enumerate}
  \item let $C$ be the public credential used in $b$ (its
    $\textsf{credential}$ field)
  \item $\mathcal{S}$ checks that $(C,w_i,\mathcal{V})\in L$
  \item $\mathcal{S}$ checks all zero-knowledge proofs of $b$
  \item $\mathcal{S}$ adds $b$ to $D$
  \end{enumerate}
\item at any time (even after tally), $\mathcal{V}$ may check that $h$
  (if it is her last ballot)
  appears in the list of pretty ballots $PB$ and the weight of her
  ballot as it appears in $PB$ is equal to her weight
\end{enumerate}


\subsection{Credential recovery}\label{sec:recovery}

If $\mathcal C$ has forgotten the private credentials of the voter
(optional step~\ref{item-forget} of the setup) then credentials cannot
be recovered.

If $\mathcal C$ has the list of private credentials (associated to the
voters), credentials can be recovered:
\begin{enumerate}
\item $\mathcal{V}_i$ contacts $\mathcal{C}$
\item $\mathcal{C}$ looks up $\mathcal{V}_i$'s private credential $c_i$
\item $\mathcal{C}$ sends $c_i$ to $\mathcal{V}_i$
\end{enumerate}

\subsection{Tally}

\begin{enumerate}
\item $\mathcal{A}$ stops $\mathcal{S}$,  $\mathcal{S}$ computes the initial
  \hyperref[tally]{$\etally$} $\Pi_0$, and publishes it in $D$
\item $\mathcal{S}$ extracts the non-homomorphic ciphertexts from the
  encrypted tally (see
  section~\ref{shuffles}):
  \[\tilde\Pi_0=\textsf{nh\_ciphertexts}(\Pi_0)\]
\item if the election contains a non-homomorphic part, that is, if
  $\tilde\Pi_0\neq []$,
 then for $z\in[1\dots m]$:
  \begin{enumerate}
  \item $\mathcal{S}$ sends $\tilde\Pi_{z-1}$ to $\mathcal{T}_z$
  \item $\mathcal{T}_z$ verifies consistency of $D$
  \item $\mathcal{T}_z$ runs the shuffle algorithm, producing a
    \hyperref[shuffles]{$\texttt{shuffle}$} $\sigma_z$ and sends it to
    $\mathcal{S}$
  \item $\mathcal{S}$ verifies $\sigma_z$, publishes it in $D$, and extracts
    $\tilde\Pi_z=\textsf{ciphertexts}(\sigma_z)$
  \end{enumerate}
\item $\mathcal{S}$ merges shuffled non-homomorphic ciphertexts with
  homomorphic ciphertexts, i.e. builds $\Pi$ such
  that:\[\tilde\Pi_m=\textsf{nh\_ciphertexts}(\Pi)\]
\item for $z\in[1\dots m]$ (or, if in threshold mode, a subset of it
  of size at least $t+1$),
  \begin{enumerate}
  \item $\mathcal{S}$ sends $\Pi$ (and $K_z$ if in threshold mode) to
    $\mathcal{T}_z$
  \item $\mathcal{T}_z$ verifies consistency of $D$
  \item $\mathcal{T}_z$ generates a \hyperref[tally]{$\pdecryption$} $\delta_z$
    and sends it to $\mathcal{S}$
  \item $\mathcal{S}$ verifies $\delta_z$ and adds it to $D$
  \end{enumerate}
\item $\mathcal{S}$ combines all the partial decryptions, computes and publishes
  the election \hyperref[election-result]{\result}
\item $\mathcal{T}_z$ checks that $\delta_z$ and $\sigma_z$ (if any) appear in
  {\result}
\end{enumerate}

\subsection{Audit}

Belenios can be publicly audited: anyone having access to the (public)
election data can check that the ballots are well formed and that the
result corresponds to the ballots. Ideally, the list of ballots should
also be monitored during the voting phase, to guarantee that no ballot
disappears.

\subsubsection{During the voting phase}
\label{sec:audit-voting}
At any time, an auditor can retrieve the public board and check its consistency. She should
always record at least the last audited board. Then:
\begin{enumerate}
\item she gets the public data $D$ and retrieves the list $L$ of
  public credentials;
  \begin{itemize}
  \item she records $D$;
  \item for each ballot $b$ in $D$, she checks that the proofs of $b$ are valid and that
  the signature of $b$ is valid and corresponds to one of the keys in
  $L$; she also checks that the weights correspond;
\item she computes $\hat{B}=\textsf{last}(B)$, the list of ballots
  obtained from $B$ by removing all ballots that have the same
  credential except the last one (only the last vote is kept for each
  voter, see section~\ref{tallied-ballots}). She checks that the list
  of hashed ballots in $\hat{B}$ corresponds to the pretty ballots
  $PB$;
  \item she checks that $D$ is correctly chained, that is, each event correctly
    refers to the hash of its parent's event;
    \item she checks that her view is consistent with the fingerprints
      displayed on the election main page.
  \end{itemize}
\item she retrieves the previously recorded election data $D'$ (if it
  exists) and gets the hash $h$ of the last event in $D'$.
She checks that $h$ appears as the hash of a parent of an event in
$D$.
This ensures that nothing has been removed from $D'$.
\end{enumerate}

There is no tool support on the web interface for these checks,
instead the command line tools \texttt{election verify} and
\texttt{election verify-diff} can be used.

\subsubsection{During and after the tally}
The auditor retrieves the public data $D$ and in particular the list
$B$ of ballots, the list $\Sigma$ of \hyperref[shuffles]{shuffles} (if
any), the list $\Delta$ of \hyperref[partial-decryptions]{partial
  decryptions} and the \hyperref[election-result]{\result} $r$. Then:

  \begin{enumerate}
   \item she checks consistency of $B$, that is, performs all
    the checks described at step 1 of section~\ref{sec:audit-voting};
  \item she checks that $B$ corresponds to the board
      monitored so far, thus performs all
    the checks described at step 2 of section~\ref{sec:audit-voting};
    \item she computes $\hat{B}=\textsf{last}(B)$, that is, she keeps
      only the last ballots (see section~\ref{tallied-ballots});
      \item she checks that the   \hyperref[tally]{$\etally$}
        corresponds to $\hat{B}$;
 \item as soon as they are available, she checks that the proofs in $\Sigma$ and $\Delta$, and the result $r$, are valid
   w.r.t. $\hat{B}$;
      \item she checks that her view is consistent with the fingerprints
      displayed on the election main page.
\end{enumerate}
To ease verification of the trustees and the credential authorities,
it is possible to display the hash of their public data (e.g. the
public keys and the partial decryptions of the trustees, the hash of
the list of the public credentials) in some human-readable form. In
that case, the audit should also check that this human-readable data is
consistent with the election data.

There is no tool support on the web interface for these checks,
instead the command line tool \texttt{election verify} can be used.

\section{Messages}
\label{messages}

\subsection{Conventions}

Structured data is encoded in JSON (RFC 4627). When serialized, data
must be in compact form, and order of fields must be respected. We use
the notation $\textsf{field}(o)$ to access the field \textsf{field} of
$o$.

\subsection{Basic types}
\label{basic-types}

\begin{itemize}
\item $\jstring$: JSON string
\item $\uuid$: election identifier (a string of
  Base58 characters\footnote{Base58 characters are:
    \texttt{123456789ABCDEFGHJKLMNPQRSTUVWXYZabcdefghijkmnopqrstuvwxyz}}
  of size at least 14), encoded as a JSON string
\item $\I$: small integer, encoded as a JSON number
\item $\B$: boolean, encoded as a JSON boolean
\item $\N$, $\Z_q$: big integer, written in base 10 and encoded as a
  JSON string
\item $\G$: a JSON string, whose interpretation depends on the group
\item $\hash$: hash (typically SHA256), written in hexadecimal and
  encoded as a JSON string
\end{itemize}

\subsection{Common structures}
\label{common}

\begin{gather*}
  \proof=\left\{
    \begin{array}{rcl}
      \challenge&:&\Z_q\\
      \response&:&\Z_q
    \end{array}
  \right\}
  \qquad
  \ciphertext=\left\{
    \begin{array}{rcl}
      \alphalabel&:&\G\\
      \betalabel&:&\G
    \end{array}
  \right\}
\end{gather*}

\subsection{Public data}
\label{public-data}

During an election, all public data is published in a dynamic
file. This file can be used to perform verifications and
monitoring. It is actually an old-style tar archive and evolves in an
append-only fashion. It stops evolving when the election is tallied.

The archive starts with a \texttt{BELENIOS} file containing a JSON
structure that acts as a header for the whole archive. It contains a
version number and the timestamp of the beginning of the election.  It
is then followed by JSON files of two kinds: data (whose names end in
\texttt{.data.json}) and events (whose names end in
\texttt{.event.json}). The file names start with the SHA256 hash of
their contents, encoded in hexadecimal.

\begin{gather*}
  \texttt{header}=\left\{
    \begin{array}{rcl}
      \textsf{version}&:&\I\\
      \textsf{timestamp}&:&\I
    \end{array}
  \right\}
  \qquad
  \texttt{event}=\left\{
    \begin{array}{rcl}
      ?\textsf{parent}&:&\hash\\
      \textsf{height}&:&\I\\
      \textsf{type}&:&\texttt{event\_type}\\
      ?\textsf{payload}&:&\hash
    \end{array}
  \right\}\\
  \texttt{event\_type}=
  \begin{array}{l}
    \texttt{"Setup"}\\
    \mid\texttt{"Ballot"}
    \mid\texttt{"EndBallots"}\\
    \mid\texttt{"EncryptedTally"}\\
    \mid\texttt{"Shuffle"}
    \mid\texttt{"EndShuffles"}\\
    \mid\texttt{"PartialDecryption"}
    \mid\texttt{"Result"}
  \end{array}
\end{gather*}

An \texttt{event} structure is appended at each important step of the
election (e.g. when somebody votes). This structure refers to its
predecessor (except for the first one) through its \textsf{parent}
field, and can refer to a payload by its hash. Its \textsf{height}
field is an integer, set to 0 in the first event, and incremented in
each new event. Typically, the payload will precede the \texttt{event}
structure in the archive, and can itself refer to other payloads that
precede. The type of the payload depends on the \textsf{type} field of
the \texttt{event} structure.

The typical sequence of data and events occurring in an archive is
described in the following table:

\begin{gather*}
\begin{array}{|c|c|c|}
  \hline
  \text{Data}&\text{Event}&\text{Defined in section}\\
  \hline
  \texttt{election}&&\ref{elections}\\
  \texttt{trustees}&&\ref{trustees}\\
  \texttt{public\_credentials}&&\ref{credentials}\\
  \texttt{setup\_data}&&\ref{elections}\\
  &\texttt{Setup}& \\
  \hline
  \texttt{ballot}&&\ref{ballots}\\
  &\texttt{Ballot}&\\
  \vdots&\vdots&\\
  \hline
  &\texttt{EndBallots}&\\
  \hline
  \texttt{encrypted\_tally}&&\ref{tally}\\
  \texttt{sized\_encrypted\_tally}&&\ref{tally}\\
  &\texttt{EncryptedTally}&\\
  \hline
  \texttt{shuffle}&&\ref{shuffles}\\
  \texttt{owned\_shuffle}&&\ref{shuffles}\\
  &\texttt{Shuffle}&\\
  \vdots&\vdots&\\
  \hline
  &\texttt{EndShuffles}&\\
  \hline
  \texttt{partial\_decryption}&&\ref{partial-decryptions}\\
  \texttt{owned\_partial\_decryption}&&\ref{partial-decryptions}\\
  &\texttt{PartialDecryption}&\\
  \vdots&\vdots&\\
  \hline
  \texttt{result}&&\ref{election-result}\\
  &\texttt{Result}&\\
  \hline
\end{array}
\end{gather*}

\subsection{Verification keys}
\label{trustee-keys}

\begin{gather*}
  \pk=\G\qquad\sk=\Z_q\\
  \tpk=\left\{
    \begin{array}{rcl}
      \pok&:&\proof\\
      \pklabel&:&\pk\\
      \textsf{?signature}&:&\proof
    \end{array}
  \right\}
\end{gather*}

A private key is a number $x$ modulo $q$, chosen at random in the
basic decryption mode, and computed after several interactions in the
threshold mode.
The corresponding
$\pklabel$ is $X=g^x$. A $\tpk$ is a bundle of this public key with a
\hyperref[common]{$\proof$} of knowledge computed as follows:
\begin{enumerate}
\item pick a random $w\in\Z_q$
\item compute $A=g^w$
\item $\challenge=\Hash_\pok(X,A)\mod q$
\item $\response=w-x\times\challenge\mod q$
\end{enumerate}
where $\Hash_\pok$ is computed as follows:
\[\Hash_\pok(X,A) = \shatwo(\verb=pok|=G\verb=|=X\verb=|=A) \]
where $\pok$ and the vertical bars are verbatim,
and $G$ is the string specifying the group in the
$\election$ structure. The result is interpreted as a 256-bit
big-endian number. The proof is verified as follows:
\begin{enumerate}
\item compute $A={g^\response}\times{X^\challenge}$
\item check that $\challenge=\Hash_\pok(X,A)\mod q$
\end{enumerate}

The \textsf{signature} field is present only in threshold mode, and is
the signature of $\pklabel$ by its owner, as described in
section~\ref{pki}.

\subsection{Messages specific to threshold decryption support}
\label{threshold}

\subsubsection{Public key infrastructure}
\label{pki}

Establishing a public key so that threshold decryption is supported
requires private communications between trustees. To achieve this,
Belenios uses a custom public key infrastructure. During the key
establishment protocol, each trustee starts by generating a secret
seed (at random), then derives from it encryption and decryption keys,
as well as signing and verification keys. These four keys are then
used to exchange messages between trustees by using $\mathcal{S}$ as a proxy.

The secret seed $s$ is a 22-character string, where characters are
taken from the set:
\[\texttt{123456789ABCDEFGHJKLMNPQRSTUVWXYZabcdefghijkmnopqrstuvwxyz}\]

\paragraph{Deriving keys}

The (private) signing key $\textsf{sk}$ is derived by computing the
SHA256 of $s$ prefixed by the string \verb/sk|/. The corresponding
(public) verification key is $g^{\textsf{sk}}$. The (private)
decryption key $\textsf{dk}$ is derived by computing the SHA256 of $s$
prefixed by the string \verb/dk|/. The corresponding (public)
encryption key is $g^{\textsf{dk}}$.

\paragraph{Signing}
\label{pki-signing}

Signing takes a signing key $\textsf{sk}$ and a \textsf{message} $M$
(as a $\jstring$), computes a \textsf{signature} and produces a
$\texttt{signed\_msg}$. For the signature, we use a (Schnorr-like)
non-interactive zero-knowledge proof.

\begin{gather*}
  \texttt{signed\_msg}=\left\{
    \begin{array}{rcl}
      \textsf{message}&:&\jstring\\
      \textsf{signature}&:&\texttt{proof}
    \end{array}
  \right\}
\end{gather*}
To compute the \textsf{signature},
\begin{enumerate}
\item pick a random $w\in\Z_q$
\item compute the commitment $A=g^w$
\item compute the \textsf{challenge} as
  $\textsf{SHA256}(\texttt{sigmsg|}M\texttt{|}A)$, where
  the result is interpreted as a 256-bit big-endian
  number
\item compute the \textsf{response} as
  $w-\textsf{sk}\times\textsf{challenge}\mod q$
\end{enumerate}
To verify a \textsf{signature} using a verification key \textsf{vk},
\begin{enumerate}
\item compute the commitment $A=g^{\textsf{response}}\times\textsf{vk}^{\textsf{challenge}}$
\item check that $\textsf{challenge}=\textsf{SHA256}(\texttt{sigmsg|}M\texttt{|}A)$
\end{enumerate}

\paragraph{Encrypting}

Encrypting takes an encryption key $\textsf{ek}$ and a message $M$ (as
a $\jstring$), computes an \texttt{encrypted\_msg} and serializes it
as a $\jstring$. We use an El Gamal-like system.

\begin{gather*}
  \texttt{encrypted\_msg}=\left\{
    \begin{array}{rcl}
      \textsf{?algorithm}&:&\jstring\\
      \textsf{alpha}&:&\G\\
      \textsf{beta}&:&\G\\
      \textsf{data}&:&\jstring
    \end{array}
  \right\}
\end{gather*}
To compute the \texttt{encrypted\_msg}:
\begin{enumerate}
\item pick random $r,s\in\Z_q$
\item compute $\textsf{alpha}=g^r$
\item compute $\textsf{beta}=\textsf{ek}^r\times g^s$
\item compute $\textsf{data}$ as the hexadecimal encoding of the
  (symmetric) encryption of $M$ using $\textsf{algorithm}$
  (\texttt{AES-CCM} by default, can also be \texttt{AES-GCM}) with
  $\textsf{SHA256}(\texttt{key|}g^s)$ as the key and
  $\textsf{SHA256}(\texttt{iv|}g^r)$ as the initialization vector
\end{enumerate}
To decrypt an \texttt{encrypted\_msg} using a decryption key \textsf{dk}:
\begin{enumerate}
\item compute the symmetric key as $\textsf{SHA256}(\texttt{key|}\textsf{beta}/(\textsf{alpha}^{\textsf{dk}}))$
\item compute the initialization vector as $\textsf{SHA256}(\texttt{iv|}\textsf{alpha})$
\item decrypt $\textsf{data}$
\end{enumerate}

\subsubsection{Certificates}
\label{certificates}

A certificate is a \texttt{signed\_msg} encapsulating a serialized
\texttt{cert\_keys} structure, itself filled with the public keys
generated as described in section~\ref{pki}. Each certificate comes
with a \texttt{context} containing the string description of the
group, the number of trustees participating in the Pedersen protocol,
the threshold and the index (starting at 1) of the owner of the
certificate.
\begin{gather*}
  \texttt{context}=\left\{
    \begin{array}{rcl}
      \textsf{group}&:&\jstring\\
      \textsf{size}&:&\I\\
      \textsf{threshold}&:&\I\\
      \textsf{index}&:&\I
    \end{array}
  \right\}
  \qquad
  \texttt{cert\_keys}=\left\{
    \begin{array}{rcl}
      \textsf{context}&:&\texttt{context}\\
      \textsf{verification}&:&\G\\
      \textsf{encryption}&:&\G
    \end{array}
  \right\}\\
  \texttt{cert}=\texttt{signed\_msg}
\end{gather*}
The message is signed with the signing key associated to
\textsf{verification}.

\subsubsection{Channels}
\label{channels}

A \textsf{message} is sent securely from \textsf{sk} (a signing key)
to \textsf{recipient} (an encryption key) by encapsulating it in a
\texttt{channel\_msg}, serializing it as a $\jstring$, signing it with
\textsf{sk} and serializing the resulting \texttt{signed\_msg} as a
$\jstring$, and finally encrypting it with \textsf{recipient}. The
resulting $\jstring$ will be denoted by
$\textsf{send}(\textsf{sk},\textsf{recipient},\textsf{message})$, and
can be transmitted using a third-party (typically the election server).
\begin{gather*}
  \texttt{channel\_msg}=\left\{
    \begin{array}{rcl}
      \textsf{recipient}&:&\G\\
      \textsf{message}&:&\jstring
    \end{array}
  \right\}
\end{gather*}
When decoding such a message, \textsf{recipient} must be checked.

\subsubsection{Polynomials}
\label{polynomials}

Let $\Gamma=\gamma_1,\dotsc,\gamma_m$ be the certificates of all
trustees. We will denote by $\textsf{vk}_z$ (resp. $\textsf{ek}_z$)
the \textsf{verification} key (resp. the \textsf{encryption} key) of
$\gamma_z$. Each trustee must compute a \texttt{polynomial} structure
in step 3 of the key establishment protocol.
\begin{gather*}
  \texttt{polynomial}=\left\{
    \begin{array}{rcl}
      \textsf{polynomial}&:&\jstring\\
      \textsf{secrets}&:&\jstring^\ast\\
      \textsf{coefexps}&:&\texttt{coefexps}\\
      \textsf{signature}&:&\proof
    \end{array}
  \right\}
\end{gather*}
Suppose $\mathcal{T}_i$ is the trustee who is computing. Therefore, $\mathcal{T}_i$ knows
the signing key $\textsf{sk}_i$ corresponding to $\textsf{vk}_i$ and the
decryption key $\textsf{dk}_i$ corresponding to $\textsf{ek}_i$. $\mathcal{T}_i$
first checks that keys indeed match. Then $\mathcal{T}_i$ picks a random
polynomial
\[
  f_i(x)=a_{i0}+a_{i1}x+\dotsb+a_{it}x^t\in\Z_q[x]
\]
and computes $A_{ik}=g^{a_{ik}}$ for $k=0,\dotsc,t$ and
$s_{ij}=f_i(j)\mod q$ for $j=1,\dotsc,m$. $\mathcal{T}_i$ then fills the
\texttt{polynomial} structure as follows:
\begin{itemize}
\item the \textsf{polynomial} field is
  $\textsf{send}(\textsf{sk}_i,\textsf{ek}_i,M)$ where $M$ is a
  serialized \texttt{raw\_polynomial} structure
  \begin{gather*}
    \texttt{raw\_polynomial}=\left\{
      \begin{array}{rcl}
        \textsf{polynomial}&:&\Z_q^\ast
      \end{array}
    \right\}
  \end{gather*}
  filled with $a_{i0},\dotsc,a_{it}$
\item the \textsf{secrets} field is
  $\textsf{send}(\textsf{sk}_i,\textsf{ek}_1,M_{i1}),\dotsc,\textsf{send}(\textsf{sk}_i,\textsf{ek}_m,M_{im})$
  where $M_{ij}$ is a serialized \texttt{secret} structure
  \begin{gather*}
    \texttt{secret}=\left\{
      \begin{array}{rcl}
        \textsf{secret}&:&\Z_q
      \end{array}
    \right\}
  \end{gather*}
  filled with $s_{ij}$
\item the \textsf{coefexps} field is a signed message containing a
  serialized \texttt{raw\_coefexps} structure
  \begin{gather*}
    \texttt{coefexps}=\texttt{signed\_msg}
    \qquad
    \texttt{raw\_coefexps}=\left\{
      \begin{array}{rcl}
        \textsf{coefexps}&:&\G^\ast
      \end{array}
    \right\}
  \end{gather*}
  filled with $A_{i0},\dotsc,A_{it}$
\item the \textsf{signature} field is computed as follows:
  \begin{enumerate}
  \item let $M$ be the string \verb=certs_sig|= followed by the
    compact JSON serialization of the following structure:
    \begin{gather*}
      \texttt{certs}=\left\{
        \begin{array}{rcl}
          \textsf{certs}&:&\texttt{cert}^\ast\\
          \textsf{coefexps}&:&\G^\ast
        \end{array}
      \right\}
    \end{gather*}
    where \textsf{certs} is set to $\Gamma=\gamma_1,\dotsc,\gamma_m$
    and \textsf{coefexps} is set to $A_{i0},\dotsc,A_{it}$
  \item the field is set to the \textsf{signature} of $M$ with
    $\textsf{sk}_i$ as described in~\ref{pki-signing}
  \end{enumerate}
\end{itemize}

The sub-key for this protocol run will be:
\[
y=\prod_{z\in[1\dots m]}g^{f_z(0)}=\prod_{z\in[1\dots m]}A_{z0}
\]

\subsubsection{Vinputs}
\label{vinputs}

Once we receive all the \texttt{polynomial} structures
$P_1,\dotsc,P_m$, we compute (during step 4) input data (called
\texttt{vinput}) for a verification step performed later by the
trustees. Step 4 can be seen as a routing step.
\begin{gather*}
  \texttt{vinput}=\left\{
    \begin{array}{rcl}
      \textsf{polynomial}&:&\jstring\\
      \textsf{secrets}&:&\jstring^\ast\\
      \textsf{coefexps}&:&\texttt{coefexps}^\ast\\
      \textsf{signatures}&:&\proof^\ast
    \end{array}
  \right\}
\end{gather*}
Suppose we are computing the \texttt{vinput} structure $\textsf{vi}_j$
for trustee $\mathcal{T}_j$. We fill it as follows:
\begin{itemize}
\item the \textsf{polynomial} field is the same as the one of $P_j$
\item the \textsf{secret} field is
  $\textsf{secret}(P_1)_j,\dotsc,\textsf{secret}(P_m)_j$
\item the \textsf{coefexps} field is
  $\textsf{coefexps}(P_1),\dotsc,\textsf{coefexps}(P_m)$
\item the \textsf{signatures} field is
  $\textsf{signature}(P_1),\dotsc,\textsf{signature}(P_m)$
\end{itemize}
Note that the \textsf{coefexps} field is the same for all trustees.

In step~5, $\mathcal{T}_j$ checks consistency of $\textsf{vi}_j$ by unpacking it
and checking that, for $i=1,\dotsc,m$,
\[
g^{s_{ij}}=\prod_{k=0}^t(A_{ik})^{j^k}
\]

\subsubsection{Voutputs}
\label{voutputs}

In step 5 of the key establishment protocol, a trustee $\mathcal{T}_j$ receives
$\Gamma$ and $\textsf{vi}_j$, and produces a \texttt{voutput}
$\textsf{vo}_j$.
\begin{gather*}
  \texttt{voutput}=\left\{
    \begin{array}{rcl}
      \textsf{private\_key}&:&\jstring\\
      \textsf{public\_key}&:&\texttt{trustee\_public\_key}
    \end{array}
  \right\}
\end{gather*}
Trustee $\mathcal{T}_j$ fills $\textsf{vo}_j$ as follows:
\begin{itemize}
\item \textsf{private\_key} is set to
  $\textsf{send}(\textsf{sk}_j,\textsf{ek}_j,S_j)$, where $S_j$ is $\mathcal{T}_j$'s
  (private) decryption key:
  \[
    S_j=\sum_{i=1}^m s_{ij}\mod q
  \]
\item \textsf{public\_key} is set to a
  \hyperref[trustee-keys]{\texttt{trustee\_public\_key}} structure
  built using $S_j$ as private key, which computes the corresponding
  public key and a proof of knowledge of $S_j$.
\end{itemize}
The administrator checks $\textsf{vo}_j$ as follows:
\begin{itemize}
\item check that:
  \[
    \textsf{public\_key}(\textsf{public\_key}(\textsf{vo}_j))=\prod_{i=1}^m \prod_{k=0}^t (A_{ik})^{j^k}
  \]
\item check $\textsf{pok}(\textsf{public\_key}(\textsf{vo}_j))$
\end{itemize}

\subsubsection{Threshold parameters}
\label{threshold-params}

The \texttt{threshold\_parameters} structure embeds data that is
published during the election.
\begin{gather*}
  \texttt{threshold\_parameters}=\left\{
    \begin{array}{rcl}
      \textsf{threshold}&:&\I\\
      \textsf{certs}&:&\texttt{cert}^\ast\\
      \textsf{coefexps}&:&\texttt{coefexps}^\ast\\
      \textsf{signatures}&:&\proof^\ast\\
      \textsf{verification\_keys}&:&\texttt{trustee\_public\_key}^\ast
    \end{array}
  \right\}
\end{gather*}
The administrator fills it as follows:
\begin{itemize}
\item \textsf{threshold} is set to $t+1$
\item \textsf{certs} is set to $\Gamma=\gamma_1,\dotsc,\gamma_m$
\item \textsf{coefexps} is set to the same value as the
  \textsf{coefexps} field of \texttt{vinput}s
\item \textsf{signatures} is set to the same value as the
  \textsf{signatures} field of \texttt{vinput}s
\item \textsf{verification\_keys} is set to
  $\textsf{public\_key}(\textsf{vo}_1),\dotsc,\textsf{public\_key}(\textsf{vo}_m)$
\end{itemize}

\subsection{Trustees}
\label{trustees}

\begin{gather*}
  \texttt{trustees}=\texttt{trustee\_kind}^\ast\\
  \texttt{trustee\_kind}=
  [\texttt{"Single"},\texttt{trustee\_public\_key}]\mid
  [\texttt{"Pedersen"},\texttt{threshold\_parameters}]
\end{gather*}

A $\texttt{trustees}$ structure is associated to each election. Such a
structure is a list of either a single verification key as described in
section~\ref{trustee-keys}, or threshold parameters as described in
section~\ref{threshold}.  Each item describes how a partial decryption
is computed: either a specific (mandatory) verification key is used to
compute a share, or a subset of a set of (optional) verification keys
are used to compute a share.

The generality of this definition allows to mix mandatory and optional
trustees during decryption. For example, in an election with 3
mandatory trustees, the $\texttt{trustees}$ structure will look like:
\[
  [[\texttt{"Single"},\dotsc],[\texttt{"Single"},\dotsc],[\texttt{"Single"},\dotsc]]
\]
and in an election where only one trustee is mandatory, and a subset
of another set of trustees (with a threshold) is needed to decrypt the
result, will have a $\texttt{trustees}$ structure that looks like:
\[
  [[\texttt{"Single"},\dotsc],[\texttt{"Pedersen"},\dotsc]]
\]
As explained in section~\ref{process-filling-trustees}, the sub-keys
of each item (\texttt{"Single"} or \texttt{"Pedersen"}) are then
combined to form the global election key.

The server itself must always have a mandatory key, which must be
different in each election. Other (third-party) keys may be imported
from one election to another.

\subsection{Credentials}
\label{credentials}

\newcommand{\secret}{\textsf{secret}}

A secret \emph{credential} $c$ is a string of the form
$\texttt{XXXXX-XXXXXX-XXXXX-XXXXXX}$, where the 22 $\texttt{X}$
characters are taken from the Base58 alphabet:
\[\texttt{123456789ABCDEFGHJKLMNPQRSTUVWXYZabcdefghijkmnopqrstuvwxyz}\]
From $c$, a secret exponent $x=\secret(c)$ is derived as follows:
\begin{itemize}
\item for $i=0,1$, let
  $x_i=\textsf{SHA256}(\texttt{derive\_credential|}\uuid\texttt{|}i\texttt{|}c)$,
  where $\uuid$ is replaced with the $\uuid$ of the election, $i$ is
  written in base 10, and the result is written in hexadecimal;
\item $x$ is the concatenation of $x_0$ and $x_1$, interpreted as a
  big-endian (512-bit) hexadecimal number, taken modulo $q$.
\end{itemize}
From this secret exponent, a public key $\public(c)=g^x$ is
computed.

\[\texttt{public\_credentials}=\jstring^\ast\]
Public credentials, published as part of public data, are a list of strings, each one
being:
\begin{itemize}
\item either a public credential by itself (if weights are not being
  used),
\item or a public credential, followed by a comma and the weight (if
  weights are being used).
\end{itemize}

\subsection{Questions}
\label{questions}

\newcommand{\question}{\texttt{question}}

\newcommand{\blank}{\textsf{blank}}
\newcommand{\minlabel}{\textsf{min}}
\newcommand{\maxlabel}{\textsf{max}}
\newcommand{\answers}{\textsf{answers}}

\begin{gather*}
  \texttt{question\_h}=\left\{
    \begin{array}{rcl}
      \answers&:&\jstring^\ast\\
      ?\blank&:&\B\\
      \minlabel&:&\I\\
      \maxlabel&:&\I\\
      \textsf{question}&:&\jstring
    \end{array}
  \right\}
  \qquad
  \texttt{question\_nh}=\left\{
    \begin{array}{rcl}
      \answers&:&\jstring^\ast\\
      \textsf{question}&:&\jstring
    \end{array}
  \right\}
\end{gather*}

\begin{gather*}
  \texttt{question\_l}=\left\{
    \begin{array}{rcl}
      \answers&:&\jstring^{\ast\ast}\\
      \textsf{question}&:&\jstring
    \end{array}
  \right\}
\end{gather*}

\begin{gather*}
  \texttt{question\_gen}=\left\{
    \begin{array}{rcl}
      \textsf{type}&:&\jstring\\
      \textsf{value}&:&\texttt{json}\\
      ?\textsf{extra}&:&\texttt{json}
    \end{array}
  \right\}
\end{gather*}

\begin{gather*}
  \question=\texttt{question\_h}\mid\texttt{question\_gen}
\end{gather*}

There are three types of questions: homomorphic (H) ones,
non-homomorphic (NH) ones and lists (L) ones. A key difference is the
outcome of the election: with H or L questions, only the pointwise sum
of all the answers (see~\ref{answers}) will be revealed at the end of
the election whereas with a NH question, each individual answer will
be revealed.

\subsubsection{Homomorphic questions}

Homomorphic questions are represented directly (first
alternative). They are the first type of question that was implemented
in Belenios. They are suitable for many elections, like the ones where
the voter is invited to select one choice among several (as in a
referendum).

The $\blank$ field of $\texttt{question\_h}$ is optional. When present
and true, the voter can vote blank for this question. In a blank vote,
all answers are set to $0$ regardless of the values of $\minlabel$ and
$\maxlabel$ ($\minlabel$ doesn't need to be $0$).

\subsubsection{Non-homomorphic questions}

Non-homomorphic questions are represented nested in a
\texttt{question\_gen} structure (second alternative), where the
\textsf{type} property is set to \verb=NonHomomorphic=, and the
\textsf{value} property is set to a \texttt{question\_nh}
structure. They are needed when homomorphic questions are not
suitable, for example when answers represent preferences or are too
big. An \textsf{extra} field may be present, to give a hint on the
intended counting method (Majority Judgment, Condorcet, STV, \dots).

\subsubsection{Lists questions}

Lists questions are represented nested in a \texttt{question\_gen}
structure (second alternative), where the \textsf{type} property is
set to \verb=Lists=, and the \textsf{value} property is set to a
\texttt{question\_l} structure. Such questions are suitable to elect
candidates grouped in lists. Here, $\answers$ is a vector of vectors
of strings with the shape:
\[ [[L_1,C_{1,1},\dots],\dots,[L_n,C_{n,1},\dots]] \] where $L_i$ is
the name of list $i$ and $C_{i,j}$ is the name of candidate $j$ of
list $i$. The voter is asked to select one list $L_i$ among
$L_1,\dots,L_n$ and, in this list, at least one candidate among
$C_{i,1},\dots,C_{i,n_i}$.

\subsection{Elections}
\label{elections}

\newcommand{\answer}{\texttt{answer}}
\newcommand{\signature}{\texttt{signature}}
\newcommand{\iproofs}{\textsf{individual\_proofs}}
\newcommand{\oproof}{\textsf{overall\_proof}}
\newcommand{\bproof}{\textsf{blank\_proof}}
\newcommand{\choices}{\textsf{choices}}
\newcommand{\iprove}{\textsf{iprove}}

\begin{gather*}
  \election=\left\{
    \begin{array}{rcl}
      \textsf{version}&:&\I\\
      \textsf{description}&:&\jstring\\
      \textsf{name}&:&\jstring\\
      \textsf{group}&:&\jstring\\
      \textsf{public\_key}&:&\G\\
      \textsf{questions}&:&\texttt{question}^\ast\\
      \textsf{uuid}&:&\texttt{uuid}\\
      \textsf{?administrator}&:&\jstring\\
      \textsf{?credential\_authority}&:&\jstring
    \end{array}
  \right\}
\end{gather*}

The $\election$ structure includes all public data related to an
election and is sent to each voter, serialized as a string which must
be always the same throughout the election. The $\textsf{version}$ is
set to 1 in this version of the specification. It is incremented in
case of backward-incompatible changes. The group is specified by the
$\textsf{group}$ member, a short string unambiguously describing the
group (see section~\ref{default-group}).

The election public key, which is denoted by $y$ throughout this
document, is computed during the setup phase, and stored in the
$\textsf{public\_key}$ member.

During an election, the following data need to be public in order to
verify the setup phase and to validate ballots:
\begin{itemize}
\item the $\election$ structure described above;
\item the $\texttt{trustees}$ structure described in
  section~\ref{trustees};
\item the \texttt{public\_credentials} structure described in
  section~\ref{credentials}.
\end{itemize}
These three structures are referred to by the following structure, used
as payload of the \texttt{Setup} event in public data.
\begin{gather*}
  \texttt{setup\_data}=\left\{
    \begin{array}{rcl}
      \textsf{election}&:&\hash\\
      \textsf{trustees}&:&\hash\\
      \textsf{credentials}&:&\hash
    \end{array}
  \right\}
\end{gather*}

\label{varphi}
Additionally, we will denote throughout this document by $\varphi$ the
Base64 encoding of the \textsf{election} field of
\texttt{setup\_data}, without padding.

\subsection{Encrypted answers}
\label{answers}

\begin{gather*}
  \texttt{answer\_h}=\left\{
    \begin{array}{rcl}
      \choices&:&\ciphertext^\ast\\
      \iproofs&:&\iproof^\ast\\
      \oproof&:&\iproof\\
      ?\bproof&:&\proof^2
    \end{array}
  \right\}
\end{gather*}

\begin{gather*}
  \texttt{answer\_nh}=\left\{
    \begin{array}{rcl}
      \choices&:&\ciphertext\\
      \textsf{proof}&:&\proof
    \end{array}
  \right\}
\end{gather*}

\begin{gather*}
  \texttt{answer\_l}=\left\{
    \begin{array}{rcl}
      \choices&:&\ciphertext^{\ast\ast}\\
      \iproofs&:&\iproof^{\ast\ast}\\
      \oproof&:&\proof\\
      \textsf{list\_proofs}&:&\proof^{\ast\ast}\\
      \textsf{nonzero\_proof}&:&\texttt{nonzero\_proof}
    \end{array}
  \right\}
\end{gather*}

\begin{gather*}
  \answer=\texttt{answer\_h}\mid\texttt{answer\_nh}\mid\texttt{answer\_l}
\end{gather*}

The structure of an answer to a \hyperref[elections]{$\question$}
depends on the type of the question. In all cases, a
\hyperref[credentials]{credential} $c$ is needed. Let $s$ be the
number $\secret(c)$, and $S_0$ be the string $\varphi$ followed by a
vertical bar and the serialization of $g^s$.

\subsubsection{Homomorphic answers}

An answer to a homomorphic question is the vector
$\choices$ of encrypted values given to each answer. When $\blank$ is
false (or absent), a blank vote is not allowed and this vector has the
same length as $\answers$; otherwise, a blank vote is allowed and this
vector has an additionnal leading value corresponding to whether the
vote is blank or not.  Each value comes with a proof (in \iproofs,
same length as \choices) that it is $0$ or $1$. The whole answer also
comes with additional proofs that values respect constraints.

More concretely, each value $m\in[0\dots1]$ is encrypted (in an El
Gamal-like fashion) into a $\ciphertext$ as follows:
\begin{enumerate}
\item pick a random $r\in\Z_q$
\item $\alphalabel=g^r$
\item $\betalabel=y^rg^m$
\end{enumerate}
where $y$ is the election public key. The resulting vector is then
used to compute $S$ as follows:
\begin{enumerate}
\item let $a$ be the vector $\choices$, where each ciphertext $c$ is
  replaced by the serialization of its $\alphalabel$ field, a comma,
  and the serialization of its $\betalabel$ field;
\item let $b$ be the concatenation of all strings in $a$, separated
  by commas;
\item let $S$ be the string $S_0$ followed by a vertical bar and $b$.
\end{enumerate}

The individual proof that $m\in[0\dots1]$ is computed by running
$\iprove(S_0,r,m,0,1)$ (see section~\ref{iproof}).

When a blank vote is not allowed, $\oproof$ proves that
$M\in[\minlabel\dots\maxlabel]$ and is computed by running
$\iprove(S,R,M-\minlabel,\minlabel,\dots,\maxlabel)$ where $R$ is the
sum of the $r$ used in ciphertexts, and $M$ the sum of the $m$. There
is no $\bproof$.

When a blank vote is allowed, and there are $n$ choices, the answer is
modeled as a vector $(m_0,m_1,\dotsc,m_n)$, when $m_0$ is whether this
is a blank vote or not, and $m_i$ (for $i>0$) is whether choice $i$
has been selected. Each $m_i$ is encrypted and proven equal to $0$ or
$1$ as above. Let $m_\Sigma=m_1+\dotsb+m_n$. The additional proofs
are as follows:
\begin{itemize}
\item $\bproof$ proves that $m_0=0\lor m_\Sigma=0$;
\item $\oproof$ proves that $m_0=1\lor m_\Sigma\in[\minlabel\dots\maxlabel]$.
\end{itemize}
They are computed as described in section~\ref{bproof}.

\subsubsection{Non-homomorphic answers}
\label{nh-answers}

The plaintext answer to a non-homomorphic question is a vector
$[v_1,\dots,v_n]$ of small integers, one for each possible
choice. When an election contains such a question, $\G$ must support
the \textsf{to\_ints} and \textsf{of\_ints} operations (see
section~\ref{default-group}). The answer is encrypted as follows:
\begin{itemize}
\item $\xi=\textsf{of\_ints}([v_1,\dots,v_n])$
\item $\textsf{choices}$ is set to an El Gamal encryption of $\xi$ as
  follows:
  \begin{enumerate}
  \item pick a random $r\in\Z_q$
  \item $\alphalabel=g^r$
  \item $\betalabel=y^r\xi$
  \end{enumerate}
  where $y$ is the election public key;
\item $\textsf{proof}$ is computed as follows:
  \begin{enumerate}
  \item pick a random $w\in\Z_q$
  \item compute $A=g^w$
  \item $\challenge=\Hash_{\textsf{raweg}}(S,y,\alphalabel,\betalabel,A)$
  \item $\response=w-r\times\challenge$
  \end{enumerate}
  where $\Hash_{\textsf{raweg}}$ is computed as follows:
  \[
    \Hash_{\textsf{raweg}}(S,y,\alpha,\beta,A)=\shatwo(\verb=raweg|=S\verb=|=y\verb=,=\alpha\verb=,=\beta\verb=|=A)\mod q
  \]
  where \verb=raweg=, vertical bars and commas are verbatim. The
  result is interpreted as a 256-bit big-endian number.
\end{itemize}
The proof is verified as follows:
\begin{enumerate}
\item compute $A=g^\response\times\alphalabel^\challenge$
\item check that $\challenge=\Hash_{\textsf{raweg}}(S,y,\alphalabel,\betalabel,A)$
\end{enumerate}

\subsubsection{Lists answers}
\label{l-answers}

The plaintext answer to a lists question is a vector of vectors of
bits \[m=[[L_1,C_{1,1},\dots],\dots,[L_n,C_{n,1},\dots]]\] where:
\begin{itemize}
\item $L_i$ represents if list $i$ is selected;
\item $C_{i,j}$ represents if candidate $j$ of list $i$ is selected.
\end{itemize}
The corresponding encrypted answer consists of:
\begin{itemize}
\item $\choices$: the pointwise encryptions of bits of $m$,
  like in homomorphic answers;
\item $\iproofs$: proofs that $m$ contains only bits ($0$ or $1$),
  like in homomorphic answers;
\item $\oproof$: a proof that $\sum_i L_i=1$ (\emph{exactly one list
    is selected});
\item $\textsf{list\_proofs}$: a vector of proofs
  $[\pi_1,\dots,\pi_n]$ where each $\pi_i$ proves that
  $L_i=1\lor\sum_jC_{i,j}=0$ (\emph{each list is either selected, or
    all its candidates are not selected});
\item $\textsf{nonzero\_proof}$: a proof that
  $\sum_i\sum_jC_{i,j}\neq 0$ (\emph{at least one candidate is
    selected}).
\end{itemize}

\paragraph{Encryptions} Each bit $m_{i,j}$ in $m$ is encrypted (in an
El Gamal-like fashion) into a $\ciphertext$ $c_{i,j}$ as follows:
\begin{enumerate}
\item pick a random $r_{i,j}\in\Z_q$
\item $\alphalabel=g^{r_{i,j}}$
\item $\betalabel=y^{r_{i,j}}g^{m_{i,j}}$
\end{enumerate}
\paragraph{Individual proofs} Each individual proof that
$m_{i,j}\in\{0,1\}$ is computed by running:
\[\iprove(S_0,r_{i,j},m,0,1)\]
Like in homomorphic answers, other proofs in lists answers use a
string $S$ built as follows:
\begin{itemize}
\item let $a$ be $\choices$, where each ciphertext $c$ is replaced by
  the serialization of its $\alphalabel$ field, a comma, and the
  serialization of its $\betalabel$ field;
\item let $b$ be the concatenation of all strings in $a$, separated by
  commas;
\item let $S$ be the string $S_0$ followed by a vertical bar and $b$.
\end{itemize}

\paragraph{Overall proof} The overall proof that
$\sum_i L_i=1(=\sum_i m_{i,0})$ is computed by running:
\[\iprove\left(S,\left(\sum_i r_{i,0}\right),\left(\sum_i
      m_{i,0}\right),1,1\right)\]

\paragraph{List proofs} See section~\ref{list-proofs}.

\paragraph{Non-zero proof} To prove that $\sum_i\sum_jC_{i,j}\neq 0$:
\begin{enumerate}
\item compute $r=\sum_i\sum_{j\neq 0}r_{i,j}$,
  $\alpha=\prod_i\prod_{j\neq 0}\alphalabel(c_{i,j})$ and
  $\beta=\prod_i\prod_{j\neq 0}\betalabel(c_{i,j})$;
  \begin{itemize}
  \item $(\alpha,\beta)$ is the encryption of $\sum_i\sum_jC_{i,j}$
    with random $r$;
  \end{itemize}
\item prove that $(\alpha,\beta)$ encrypts a non-zero value with
  random $r$ (see section~\ref{nonzero-proof}).
\end{enumerate}

\subsection{Proofs of interval membership}
\label{iproof}

\begin{gather*}
  \iproof=\proof^\ast
\end{gather*}

Given a pair $(\alpha,\beta)$ of group elements, one can prove that it
has the form $(g^r,y^rg^{M_i})$ with $M_i\in[M_0,\dots,M_k]$ by
creating a sequence of $\proof$s $\pi_0,\dots,\pi_k$ with the
following procedure, parameterised by a string $S$:
\begin{enumerate}
\item for $j\neq i$:
  \begin{enumerate}
  \item create $\pi_j$ with a random $\challenge$ and a random
    $\response$
  \item compute
    \[A_j={g^\response}\times{\alpha^\challenge}\quad\text{and}\quad
    B_j={y^\response}\times{(\beta/g^{M_j})^\challenge}\]
  \end{enumerate}
\item $\pi_i$ is created as follows:
  \begin{enumerate}
  \item pick a random $w\in\Z_q$
  \item compute $A_i=g^w$ and $B_i=y^w$
  \item $\challenge(\pi_i)=\Hash_\iprove(S,\alpha,\beta,A_0,B_0,\dots,A_k,B_k)-\sum_{j\neq
      i}\challenge(\pi_j)\mod q$
  \item $\response(\pi_i)=w-r\times\challenge(\pi_i)\mod q$
  \end{enumerate}
\end{enumerate}
In the above, $\Hash_\iprove$ is computed as follows:
\[\Hash_\iprove(S,\alpha,\beta,A_0,B_0,\dots,A_k,B_k)=\shatwo(\verb=prove|=S\verb=|=\alpha\verb=,=\beta\verb=|=A_0\verb=,=B_0\verb=,=\dots\verb=,=A_k\verb=,=B_k)\mod q\]
where \verb=prove=, vertical bars and commas are verbatim.
The result is interpreted as a 256-bit
big-endian number. We will denote the whole procedure by
$\iprove(S,r,i,M_0,\dots,M_k)$.

The proof is verified as follows:
\begin{enumerate}
\item for $j\in[0\dots k]$, compute
  \[A_j={g^{\response(\pi_j)}}\times{\alpha^{\challenge(\pi_j)}}\quad\text{and}\quad
  B_j={y^{\response(\pi_j)}}\times{(\beta/g^{M_j})^{\challenge(\pi_j)}}\]
\item check that
  \[\Hash_\iprove(S,\alpha,\beta,A_0,B_0,\dots,A_k,B_k)=\sum_{j\in[0\dots
    k]}\challenge(\pi_j)\mod q\]
\end{enumerate}

\subsection{Proofs of possibly-blank votes}
\label{bproof}

In this section, we suppose:
\[
(\alpha_0,\beta_0)=(g^{r_0},y^{r_0}g^{m_0})
\quad\text{and}\quad
(\alpha_\Sigma,\beta_\Sigma)=(g^{r_\Sigma},y^{r_\Sigma}g^{m_\Sigma})
\]
Note that $\alpha_\Sigma$, $\beta_\Sigma$ and $r_\Sigma$ can be easily
computed from the encryptions of $m_1,\dotsc,m_n$ and their associated
secrets.

Additionally, let $M_1,\dotsc,M_k$ be the sequence
$\minlabel,\dots,\maxlabel$ ($k=\maxlabel-\minlabel+1$).

\subsubsection{Non-blank votes ($m_0=0$)}
\label{non-blank-votes}

\paragraph{Computing \bproof}
In $m_0=0\lor m_\Sigma=0$, the first case is true. The proof $\bproof$
of the whole statement is the couple of proofs $(\pi_0,\pi_\Sigma)$
built as follows:
\begin{enumerate}
\item pick random $\challenge(\pi_\Sigma)$ and $\response(\pi_\Sigma)$
  in $\Z_q$
\item compute
  $A_\Sigma=g^{\response(\pi_\Sigma)}\times\alpha_\Sigma^{\challenge(\pi_\Sigma)}$
  and
  $B_\Sigma=y^{\response(\pi_\Sigma)}\times\beta_\Sigma^{\challenge(\pi_\Sigma)}$
\item pick a random $w$ in $\Z_q$
\item compute $A_0=g^w$ and $B_0=y^w$
\item compute \[\challenge(\pi_0)=\Hash_{\mathsf{bproof0}}(S,A_0,B_0,A_\Sigma,B_\Sigma)-\challenge(\pi_\Sigma)\mod q\]
\item compute $\response(\pi_0)=w-r_0\times\challenge(\pi_0)\mod q$
\end{enumerate}
In the above, $\Hash_{\mathsf{bproof0}}$ is computed as follows:
\[\Hash_{\mathsf{bproof0}}(\dotsc)=
\shatwo(\verb=bproof0|=S\verb=|=A_0\verb=,=B_0\verb=,=A_\Sigma\verb=,=B_\Sigma)\mod q\]
where \verb=bproof0=, vertical bars and commas are verbatim.
The result is interpreted as a 256-bit big-endian number.

\paragraph{Computing \oproof}
In $m_0=1\lor m_\Sigma\in[M_1\dots M_k]$, the second case
is true. Let $i$ be such that $m_\Sigma=M_i$. The proof of the whole
statement is a $(k+1)$-tuple $(\pi_0,\pi_1,\dotsc,\pi_k)$ built as
follows:
\begin{enumerate}
\item pick random $\challenge(\pi_0)$ and $\response(\pi_0)$
  in $\Z_q$
\item compute
  $A_0=g^{\response(\pi_0)}\times\alpha_0^{\challenge(\pi_0)}$
  and
  $B_0=y^{\response(\pi_0)}\times(\beta_0/g)^{\challenge(\pi_0)}$
\item for $j>0$ and $j\neq i$:
  \begin{enumerate}
  \item create $\pi_j$ with a random $\challenge$ and a random
    $\response$ in $\Z_q$
  \item compute
    $A_j={g^\response}\times{\alpha_\Sigma^\challenge}$ and
    $B_j={y^\response}\times{(\beta_\Sigma/g^{M_j})^\challenge}$
  \end{enumerate}
\item pick a random $w\in\Z_q$
\item compute $A_i=g^w$ and $B_i=y^w$
\item compute
  \[\challenge(\pi_i)=\Hash_{\textsf{bproof1}}(S,A_0,B_0,\dots,A_k,B_k)-\sum_{j\neq i}\challenge(\pi_j)\mod q\]
\item compute $\response(\pi_i)=w-r_\Sigma\times\challenge(\pi_i)\mod q$
\end{enumerate}
In the above, $\Hash_{\mathsf{bproof1}}$ is computed as follows:
\[\Hash_{\mathsf{bproof1}}(\dotsc)=
\shatwo(\verb=bproof1|=S\verb=|=A_0\verb=,=B_0\verb=,=\dotsc\verb=,=A_k\verb=,=B_k)\mod q\]
where \verb=bproof1=, vertical bars and commas are verbatim.
The result is interpreted as a 256-bit big-endian number.

\subsubsection{Blank votes ($m_0=1$)}

\paragraph{Computing \bproof}
In $m_0=0\lor m_\Sigma=0$, the second case is true. The proof
$\bproof$ of the whole statement is the couple of proofs
$(\pi_0,\pi_\Sigma)$ built as in section~\ref{non-blank-votes}, but
exchanging subscripts $0$ and $\Sigma$ everywhere except in the call
to $\Hash_{\textsf{bproof0}}$.

\paragraph{Computing \oproof}
In $m_0=1\lor m_\Sigma\in[M_1\dots M_k]$, the first case is
true. The proof of the whole statement is a $(k+1)$-tuple
$(\pi_0,\pi_1,\dotsc,\pi_k)$ built as follows:
\begin{enumerate}
\item for $j>0$:
  \begin{enumerate}
  \item create $\pi_j$ with a random $\challenge$ and a random
    $\response$ in $\Z_q$
  \item compute
    $A_j={g^\response}\times{\alpha_\Sigma^\challenge}$ and
    $B_j={y^\response}\times{(\beta_\Sigma/g^{M_j})^\challenge}$
  \end{enumerate}
\item pick a random $w\in\Z_q$
\item compute $A_0=g^w$ and $B_0=y^w$
\item compute
  \[\challenge(\pi_0)=\Hash_{\textsf{bproof1}}(S,A_0,B_0,\dots,A_k,B_k)-\sum_{j>0}\challenge(\pi_j)\mod q\]
\item compute $\response(\pi_0)=w-r_0\times\challenge(\pi_0)\mod q$
\end{enumerate}

\subsubsection{Verifying proofs}

\paragraph{Verifying \bproof}
A proof of $m_0=0\lor m_\Sigma=0$ is a couple of proofs
$(\pi_0,\pi_\Sigma)$ such that the following procedure passes:
\begin{enumerate}
\item compute
  $A_0=g^{\response(\pi_0)}\times\alpha_0^{\challenge(\pi_0)}$
  and
  $B_0=y^{\response(\pi_0)}\times\beta_0^{\challenge(\pi_0)}$
\item compute
  $A_\Sigma=g^{\response(\pi_\Sigma)}\times\alpha_\Sigma^{\challenge(\pi_\Sigma)}$
  and
  $B_\Sigma=y^{\response(\pi_\Sigma)}\times\beta_\Sigma^{\challenge(\pi_\Sigma)}$
\item check that
  \[\Hash_{\mathsf{bproof0}}(S,A_0,B_0,A_\Sigma,B_\Sigma)=\challenge(\pi_0)+\challenge(\pi_\Sigma)\mod q\]
\end{enumerate}

\paragraph{Verifying \oproof}
A proof of $m_0=1\lor m_\Sigma\in[M_1\dots M_k]$ is a $(k+1)$-tuple
$(\pi_0,\pi_1,\dotsc,\pi_k)$ such that the following procedure passes:
\begin{enumerate}
\item compute
  $A_0=g^{\response(\pi_0)}\times\alpha_0^{\challenge(\pi_0)}$
  and
  $B_0=y^{\response(\pi_0)}\times(\beta_0/g)^{\challenge(\pi_0)}$
\item for $j>0$, compute
  \[A_j=g^{\response(\pi_j)}\times\alpha_\Sigma^{\challenge(\pi_j)}
  \quad\text{and}\quad
  B_j=y^{\response(\pi_j)}\times(\beta_\Sigma/g^{M_j})^{\challenge(\pi_j)}\]
\item check that
  \[\Hash_{\textsf{bproof1}}(S,A_0,B_0,\dots,A_k,B_k)=\sum_{j=0}^k\challenge(\pi_j)\mod q\]
\end{enumerate}

\subsection{Proofs of list constraints}
\label{list-proofs}

\paragraph{Proving}
For each $i$, the proof of the list constraint that
$L_i=1\lor\sum_jC_{i,j}=0$ is a pair $\pi_i=(\pi_{i,0},\pi_{i,1})$
computed as follows:
\begin{itemize}
\item let $r_0=r_{i,0}$, $\alpha_0=\alphalabel(c_{i,0})$ and
  $\beta_0=\betalabel(c_{i,0})$:
  \begin{itemize}
  \item $(\alpha_0,\beta_0)$ is the encryption of $L_i$ with random $r_0$;
  \end{itemize}
\item compute $r=\sum_{j\neq 0}r_{i,j}$,
  $\alpha=\prod_{j\neq 0}\alphalabel(c_{i,j})$ and
  $\beta=\prod_{j\neq 0}\betalabel(c_{i,j})$:
  \begin{itemize}
  \item $(\alpha,\beta)$ is the encryption of $\sum_jC_{i,j}$ with random $r$;
  \end{itemize}
\item if $L_i=1$:
  \begin{enumerate}
  \item pick random $\challenge_1$ and $\response_1$ in $\Z_q$;
  \item compute $A_1=g^{\response_1}\times\alpha^{\challenge_1}$ and
    $B_1=y^{\response_1}\times\beta^{\challenge_1}$;
  \item pick a random $w\in\Z_q$;
  \item compute $A_0=g^w$ and $B_0=y^w$;
  \item compute $h=\Hash_{\textsf{lproof}}(S,A_0,B_0,A_1,B_1)$;
  \item compute $\challenge_0=h-\challenge_1$ and
    $\response_0=w-r_0\times\challenge_0$;
  \end{enumerate}
\item if $\sum_jC_{i,j}=0$:
  \begin{enumerate}
  \item pick random $\challenge_0$ and $\response_0$ in $\Z_q$;
  \item compute $A_0=g^{\response_0}\times\alpha_0^{\challenge_0}$ and
    $B_0=y^{\response_0}\times(\beta_0/g)^{\challenge_0}$;
  \item pick a random $w\in\Z_q$;
  \item compute $A_1=g^w$ and $B_1=y^w$;
  \item compute $h=\Hash_{\textsf{lproof}}(S,A_0,B_0,A_1,B_1)$;
  \item compute $\challenge_1=h-\challenge_0$ and
    $\response_1=w-r\times\challenge_1$;
  \end{enumerate}
\item for $k\in\{0,1\}$, build $\pi_{i,k}$ from $\challenge_k$ and
  $\response_k$.
\end{itemize}
In the above, $\Hash_{\textsf{lproof}}$ is computed as follows:
\[
  \Hash_{\textsf{lproof}}(\dotsc)=
  \shatwo(\verb=lproof|=S\verb=|=A_0\verb=,=B_0\verb=,=A_1\verb=,=B_1)\mod
  q
\]

\paragraph{Verifying} A proof composed of $\challenge_k$ and
$\response_k$ can be verified as follows:
\begin{enumerate}
\item compute $\alpha_0$, $\beta_0$, $\alpha$ and $\beta$ as above;
\item compute $A_0=g^{\response_0}\times\alpha_0^{\challenge_0}$ and
  $B_0=y^{\response_0}\times(\beta_0/g)^{\challenge_0}$;
\item compute $A_1=g^{\response_1}\times\alpha^{\challenge_1}$ and
  $B_1=y^{\response_1}\times\beta^{\challenge_1}$;
\item check that
  $\Hash_{\textsf{lproof}}(S,A_0,B_0,A_1,B_1) =
  \challenge_0+\challenge_1$.
\end{enumerate}

\subsection{Proof of encryption of non-zero}
\label{nonzero-proof}

\begin{gather*}
  \texttt{nonzero\_proof}=\left\{
    \begin{array}{rcl}
      \textsf{commitment}&:&\G\\
      \challenge&:&\Z_q\\
      \response&:&\Z_q\times\Z_q
    \end{array}
  \right\}
\end{gather*}

\paragraph{Proving}
To prove that $(\alpha,\beta)$ is the encryption of $m\neq 0$ with
random $r$:
\begin{enumerate}
\item pick a random $s\in\Z_q$ such that $s\neq 0$;
\item compute $A_0=\beta^s\times y^{-s\times r}$:
  \begin{itemize}
  \item notice that $m\neq 0$ is equivalent to $A_0\neq 1$;
  \item we also have $1=\alpha^s\times g^{-s\times r}$;
  \item hence, we resort to proving that $A_0$ (in base $(\beta,y)$)
    and $1$ (in base $(\alpha,g)$) have the same representation;
  \end{itemize}
\item pick random $w_1$ and $w_2$ in $\Z_q$;
\item compute $A_1=\alpha^{w_1}\times g^{w_2}$ and
  $A_2=\beta^{w_1}\times y^{w_2}$;
\item compute $c=\Hash_{\textsf{nonzero}}(S,A_0,A_1,A_2)$;
\item compute $t_1=w_1-s\times c$ and $t_2=w_2+s\times r\times c$;
\item set $\textsf{commitment}$ to $A_0$, $\challenge$ to $c$ and $\response$
  to $(t_1,t_2)$.
\end{enumerate}
In the above, $\Hash_{\textsf{nonzero}}$ is computed as follows:
\[
  \Hash_{\textsf{nonzero}}(\dotsc)=
  \shatwo(\verb=nonzero|=S\verb=|=A_0\verb=,=A_1\verb=,=A_2)\mod
  q
\]

\paragraph{Verifying} Following the same notations as above, a proof
composed of $A_0$, $c$, $t_1$ and $t_2$ can be verified as follows:
\begin{enumerate}
\item check that $A_0\neq 1$;
\item compute $A_1=\alpha^{t_1}\times g^{t_2}$ and
  $A_2=\beta^{t_1}\times y^{t_2}\times A_0^c$;
\item check that $\Hash_{\textsf{nonzero}}(S,A_0,A_1,A_2)=c$.
\end{enumerate}

\subsection{Signatures}
\label{signatures}

\begin{gather*}
  \signature=\left\{
    \begin{array}{rcl}
      \textsf{hash}&:&\jstring\\
      \textsf{proof}&:&\proof
    \end{array}
  \right\}
\end{gather*}

\newcommand{\siglabel}{\textsf{signature}}

Each ballot contains a (Schnorr-like) digital signature to avoid
ballot stuffing. The signature needs a
\hyperref[credentials]{credential} $c$ and uses the $\textsf{hash}$ of
the surrounding ballot (without the $\siglabel$ field). It is computed
as follows:
\begin{enumerate}
\item compute $s=\secret(c)$
\item pick a random $w\in\Z_q$
\item compute $A=g^w$
\item compute $\textsf{proof}$ as follows:
  \begin{enumerate}
  \item $\challenge=\Hash_\siglabel(\textsf{hash},A)\mod q$
  \item $\response=w-s\times\challenge\mod q$
  \end{enumerate}
\end{enumerate}
In the above, $\Hash_\siglabel$ is computed as follows:
\[
\Hash_\siglabel(H,A)=\shatwo(\verb=sig|=H\verb=|=A)
\]
where \verb=sig=, vertical bars and commas are verbatim.
The result is interpreted as a 256-bit big-endian number.

Signatures are verified as follows ($\textsf{credential}$ and
$\textsf{hash}$ can be obtained from the surrounding ballot):
\begin{enumerate}
\item compute $A=g^\response\times \textsf{credential}^\challenge$
\item check that $\challenge=\Hash_\siglabel(\textsf{hash},A)\mod q$
\end{enumerate}

\subsection{Ballots}
\label{ballots}

\newcommand{\json}{\textsf{JSON}}

\begin{gather*}
  \ballot=\left\{
    \begin{array}{rcl}
      \textsf{election\_uuid}&:&\uuid\\
      \textsf{election\_hash}&:&\jstring\\
      \textsf{credential}&:&\G\\
      \answers&:&\hyperref[answers]{\answer}^\ast\\
      \siglabel&:&\hyperref[signatures]{\signature}
    \end{array}
  \right\}
\end{gather*}
A ballot references in its $\textsf{credential}$ member the public
credential $S=g^{\secret(c,s)}$ ($c$ being the secret credential) of the voter. The hash (or
\emph{fingerprint}) of the election is $\varphi$ (see
section~\ref{varphi}).

To compute the $\textsf{hash}$ used in signatures, the ballot without
the $\siglabel$ field is first serialized as a JSON compact string,
where object fields are ordered as specified in this document. The
hash is the compact Base64 encoding of the SHA256 of this string.
The same hashing function is used on the serialization of the whole
$\ballot$ structure to produce a so-called \emph{smart ballot
  tracker}.

The weight of a ballot $b$, denoted by $\textsf{weight}(b)$, is the
weight associated to $\textsf{credential}(b)$ in the list of public
credentials $L$.

Ballots are appended to public data as payloads of \texttt{Ballot}
events. The end of ballots is marked by an \texttt{EndBallots} event
without payload.

\paragraph{Tallied ballots}
\label{tallied-ballots}
The list of tallied ballots $\hat B$ can be computed from the list of
all accepted ballots $B$ with the $\textsf{last}$ function
($\hat B=\textsf{last}(B)$), defined as follows:
\begin{itemize}
\item initialize $\hat B$ with the empty list
\item for $b\in B$:
  \begin{itemize}
  \item if there is a ballot in $\hat B$ with the same credential as
    $b$, remove it
  \item add $b$ to $\hat B$
  \end{itemize}
\end{itemize}

\subsection{Encrypted tally}
\label{tally}

\begin{gather*}
  \texttt{ciphertexts\_h}=\ciphertext^\ast
  \qquad
  \texttt{ciphertexts\_nh}=\ciphertext^\ast
  \\
  \etally=(\texttt{ciphertexts\_h}\mid\texttt{ciphertexts\_nh})^\ast\\
  \texttt{sized\_encrypted\_tally}=\left\{
    \begin{array}{rcl}
      \textsf{num\_tallied}&:&\I\\
      \textsf{total\_weight}&:&\I\\
      \textsf{encrypted\_tally}&:&\hash
    \end{array}
  \right\}
\end{gather*}
A so-called \emph{encrypted tally} is constructed out of the accepted
ballots $\hat B=\textsf{last}(B)=b_1,\dots,b_n$ (see section~\ref{tallied-ballots}).
It is an array $[C_1,\dots,C_m]$ where $m$ is the number of
questions. Each element $C_i$ is itself an array of ciphertexts that
is built differently depending on the type of the question:
\begin{itemize}
\item for homomorphic questions, each element of $C_i$
  ($\texttt{ciphertexts\_h}$) is the pointwise product of the $i$-th
  ciphertext of all the ballots, raised to the power of its weight:
  \[
    C_{i,j}=\prod_{k}\choices(\answers(b_k)_i)_j^{\textsf{weight}(b_k)}
  \]
  where the product of two ciphertexts $(\alpha_1,\beta_1)$ and
  $(\alpha_2,\beta_2)$ is $(\alpha_1\alpha_2,\beta_1\beta_2)$;
\item for non-homomorphic questions, $C_i$ is directly made from the
  list of ciphertexts corresponding to the question:
  \[
    C_{i,k}=\choices(\answers(b_k)_i)
  \]
  In this case, it is an error if $\textsf{weight}(b_k)\neq 1$.
\end{itemize}
In the end, in both cases, the encrypted tally is isomorphic to an
array of arrays of ciphertexts:
\[
\etally\approx\ciphertext^\ast{}^\ast
\]

The \texttt{sized\_encrypted\_tally} structure contains the number of
ballots taken into account, their total weight, and a reference to the
$\etally$ structure. It is used as the payload of the
\texttt{EncryptedTally} event.

\subsection{Shuffles}
\label{shuffles}

If the election has non-homomorphic questions, let us say $n$ out of
$m$ ($1\leq n\leq m$), non-homomorphic ciphertexts must be
shuffled. They are first extracted from the encrypted tally~$a$: if
$i_1,\dots,i_n$ are the indices of the non-homomorphic questions,
\[
  b=\textsf{nh\_ciphertexts}(a)=[a_{i_1},\dots,a_{i_n}]
\]
where $a$ is the $\etally$ structure defined
in~\ref{tally}. Conversely, once ciphertexts are shuffled as $b'$ (see
later), they must be merged into the encrypted tally as $a'$ such that
$b'=\textsf{nh\_ciphertexts}(a')$.

Shuffles are done in the same way as the CHVote system\footnote{See
  version 1.3.2 of the CHVote System Specification at~\cite{CHVote}}.
%  \url{https://eprint.iacr.org/2017/325}}.
For each non-homomorphic
question, its ciphertexts are re-encrypted and randomly
permuted, and a zero-knowledge proof of the permutation is
computed. All these shuffles are then assembled into a
$\texttt{shuffle}$ structure:
\begin{gather*}
  \texttt{shuffle}=\left\{
    \begin{array}{rcl}
      \textsf{ciphertexts}&:&\ciphertext^\ast{}^\ast\\
      \textsf{proofs}&:&\texttt{shuffle\_proof}^\ast
    \end{array}
  \right\}
\end{gather*}
which uses the following auxiliary types:
\begin{gather*}
  \begin{array}{rcl}
    \texttt{shuffle\_commitment\_rand}&=&\G\times\G\times\G\times(\G\times\G)\times\G^\ast\\
    \texttt{shuffle\_response}&=&\Z_q\times\Z_q\times\Z_q\times\Z_q\times\Z_q^\ast\times\Z_q^\ast\\
    \texttt{shuffle\_commitment\_perm}&=&\G^\ast\\
    \texttt{shuffle\_chained\_challenges}&=&\G^\ast\\
    \texttt{shuffle\_proof}&=&\texttt{shuffle\_commitment\_rand}\\
                                      &\times&\texttt{shuffle\_response}\\
                                      &\times&\texttt{shuffle\_commitment\_perm}\\
                                      &\times&\texttt{shuffle\_chained\_challenges}
  \end{array}
\end{gather*}
For each non-homomorphic question $i$:
\begin{enumerate}
\item let $\textbf{e}=b_i=[e_1,\dots,e_N]$ be the array of ciphertexts
  corresponding to question $i$ ($N$ being the number of ballots);
\item let
  $(\textbf{e}',\textbf{r}',\psi)=\textsf{GenShuffle}(\textbf{e},y)$
  ($y$ being the public key of the election);
\item let
  $\pi=\textsf{GenShuffleProof}(\textbf{e},\textbf{e}',\textbf{r}',\psi,y)$;
  \item set $\textsf{ciphertexts}_i$ to $\textbf{e}'$ and
    $\textsf{proofs}_i$ to $\pi$.
\end{enumerate}
The functions $\textsf{GenShuffle}$ and $\textsf{GenShuffleProof}$ are
the same as in CHVote and are given in
section~\ref{shuffle-algorithms}. Typically, several shuffles will be
computed sequentially by different persons.

\begin{gather*}
  \texttt{owned\_shuffle}=\left\{
    \begin{array}{rcl}
      \textsf{owner}&:&\I\\
      \textsf{payload}&:&\hash
    \end{array}
  \right\}
\end{gather*}
The \texttt{owned\_shuffle} structure links a shuffle with the trustee
that did it and is used as payload of \texttt{Shuffle} events.

\subsection{Partial decryptions}
\label{partial-decryptions}

\newcommand{\dfactors}{\textsf{decryption\_factors}}
\newcommand{\dproofs}{\textsf{decryption\_proofs}}
\newcommand{\decrypt}{\textsf{decrypt}}

\begin{gather*}
  \pdecryption=\left\{
    \begin{array}{rcl}
      \dfactors&:&\G^\ast{}^\ast\\
      \dproofs&:&\proof^\ast{}^\ast
    \end{array}
  \right\}
\end{gather*}
From the encrypted tally $a'$ (where answers to non-homomorphic questions
have been shuffled), each trustee computes a partial decryption using
the \hyperref[trustee-keys]{private key} $x$ (and the corresponding
public key $X=g^x$) he generated during election setup. It consists of
so-called \emph{decryption factors}:
\[
\dfactors_{i,j}=\alphalabel(a'_{i,j})^x
\]
and proofs that they were correctly computed. Each $\dproofs_{i,j}$ is
computed as follows:
\begin{enumerate}
\item pick a random $w\in\Z_q$
\item compute $A=g^w$ and $B=\alphalabel(a'_{i,j})^w$
\item $\challenge=\Hash_\decrypt(X,A,B)$
\item $\response=w-x\times\challenge\mod q$
\end{enumerate}
In the above, $\Hash_\decrypt$ is computed as follows:
\[
\Hash_\decrypt(X,A,B)=\shatwo(\verb=decrypt|=\varphi\verb=|=X\verb=|=A\verb=,=B)\mod q
\]
where \verb=decrypt=, vertical bars and commas are verbatim.
The result is interpreted as a 256-bit big-endian number.

These proofs are verified using the $\tpk$ structure $k$ that the
trustee sent to the administrator during the election setup:
\begin{enumerate}
\item compute
\[
\begin{array}{rcl}
A&=&{g^\response}\times{\pklabel(k)^\challenge}\\
B&=&{\alphalabel(a'_{i,j})^\response}\times{\dfactors_{i,j}^\challenge}
\end{array}
\]
\item check that $\Hash_\decrypt(\pklabel(k),A,B)=\challenge$
\end{enumerate}

\begin{gather*}
  \texttt{owned\_partial\_decryption}=\left\{
    \begin{array}{rcl}
      \textsf{owner}&:&\I\\
      \textsf{payload}&:&\hash
    \end{array}
  \right\}
\end{gather*}
The \texttt{owned\_partial\_decryption} structure links a partial
decryption with the trustee that did it and is used as payload of
\texttt{PartialDecryption} events.

\subsection{Election result}
\label{election-result}

\newcommand{\ntallied}{\textsf{num\_tallied}}
\newcommand{\etallylabel}{\textsf{encrypted\_tally}}
\newcommand{\pdlabel}{\textsf{partial\_decryptions}}
\newcommand{\resultlabel}{\textsf{result}}

\begin{gather*}
  \result=\left\{
    \begin{array}{rcl}
      \resultlabel&:&(\I^\ast\mid\I^\ast{}^\ast)^\ast
    \end{array}
  \right\}
\end{gather*}

The decryption factors are combined for each ciphertext to build
synthetic ones $F_{i,j}$. The way this combination is done depends on
the $\texttt{trustees}$ structure, the list $PK$. For each item of
index $\tau$ in $PK$, a sub-factor $F_{i,j,\tau}$ is computed:
\begin{itemize}
\item for a \texttt{"Single"} item corresponding to trustee
  $\mathcal{T}_z$:
  \[
    F_{i,j,\tau}=\pdlabel_{z,i,j}
  \]
\item for a \texttt{"Pedersen"} item corresponding to trustees
  $\mathcal{T}_{z_1},\dots,\mathcal{T}_{z_\mu}$:
  \[
    F_{i,j,\tau}=\prod_{\delta\in\mathcal{I}}(\pdlabel_{z_\delta,i,j})^{\lambda_\delta^{\mathcal{I}}}
  \]
  where $\mathcal{I}$ is the set of ($t+1$) indexes of supplied
  partial decryptions, relative to
  $\mathcal{T}_{z_1},\dots,\mathcal{T}_{z_\mu}$ (i.e. $\mathcal{I}\subseteq\{1,\dots,\mu\}$), and
  $\lambda_\delta^{\mathcal{I}}$ are the Lagrange coefficients:
  \[
    \lambda_\delta^{\mathcal{I}}=\prod_{k\in\mathcal{I}\backslash\{\delta\}}\frac{k}{k-\delta}\mod q
  \]
\end{itemize}
The synthetic factor is then computed as the product of all sub-factors:
\[
  F_{i,j}=\prod_\tau F_{i,j,\tau}
\]

The $\resultlabel$ field of the $\result$ structure is then computed
as follows:
\begin{itemize}
\item if question $i$ is homomorphic,
  \[
    \resultlabel_{i,j}=\log_g\left(\frac{\betalabel(a'_{i,j})}{F_{i,j}}\right)
  \]
  where $j$ represents an answer. The discrete logarithm can be easily
  computed because it is bounded by the sum of all weights;
\item if question $i$ is non-homomorphic,
  \[
    \resultlabel_{i,j}=\textsf{to\_ints}\left(\frac{\betalabel(a'_{i,j})}{F_{i,j}}\right)
  \]
  where $j$ represents a ballot.
\end{itemize}

\section{Groups}
\label{default-group}

A group is identified by a short string. In addition to the usual
mathematical group definition, it must support the following
operations:
\begin{itemize}
\item \textsf{check}: checking that an element of the surrounding set
  is indeed a group element;
\item \textsf{to\_string}, \textsf{of\_string}: (de-)serialization
  to/from strings, used when serializing JSON structures or computing
  hashes.
\end{itemize}
Additionally, it may support the following operations, needed for
non-homomorphic questions (see section~\ref{nh-answers}):
\begin{itemize}
\item \textsf{to\_ints}, \textsf{of\_ints}: embedding of vectors of
  small integers into group elements,
\item \textsf{get\_generator}: a function mapping integers to group
  generators different from $g$.
\end{itemize}
Supported groups include:
\begin{itemize}
\item BELENIOS-2048
\item RFC-3526-2048
\item Ed25519
\end{itemize}

\subsection{Finite fields}

Here, groups are multiplicative subgroups of $\F^*_p$ described by the
following structure:
\begin{gather*}
  \texttt{group}=\left\{
    \begin{array}{rcl}
      \textsf{g}&:&\G\\
      \textsf{p}&:&\N\\
      \textsf{q}&:&\N\\
      \textsf{?embedding}&:&\texttt{embedding}
    \end{array}
  \right\}
\end{gather*}
When serialized as strings, group elements are written in base 10.

These groups may support non-homomorphic encoding described by the
following structure:
\begin{gather*}
  \texttt{embedding}=\left\{
    \begin{array}{rcl}
      \textsf{padding}&:&\I\\
      \textsf{bits\_per\_int}&:&\I
    \end{array}
  \right\}
\end{gather*}
The encoding works as follows:
\begin{itemize}
\item in the following, \textsf{bits\_per\_int} is denoted by $\kappa$
  and \textsf{padding} by $p$;
\item it is assumed that each $v_i$ is $\kappa$ bits (or less);
\item $[v_1,\dots,v_n]$ is encoded as:
  \[
    \xi=\textsf{of\_ints}([v_1,\dots,v_n])=(((v_1\times
    2^\kappa+v_2)\times 2^\kappa+\dotsb)\times 2^\kappa+v_n)\times
    2^p+\varepsilon
  \]
  where $\varepsilon$ (of $p$ bits or less) is chosen so that $\xi\in\G$;
\item the $\textsf{to\_ints}$ is the inverse of $\textsf{of\_ints}$, and
  takes as input a group element $\xi$ and the number $n$ of encoded
  integers.
\end{itemize}
The \textsf{get\_generator} function is the pure part of
\textsf{GetGenerator} defined in table~\ref{get-generator}.

\subsubsection{BELENIOS-2048}
\label{h-group}

This group is optimized for elections that have only homomorphic
questions. It has no \textsf{embedding}. Its parameters have been
generated by the \verb=fips.sage= script (available in Belenios
sources), which is itself based on FIPS 186-4.

\[
\begin{array}{lcr}
\textsf{p}&=&20694785691422546\\
&&401013643657505008064922989295751104097100884787057374219242\\
&&717401922237254497684338129066633138078958404960054389636289\\
&&796393038773905722803605973749427671376777618898589872735865\\
&&049081167099310535867780980030790491654063777173764198678527\\
&&273474476341835600035698305193144284561701911000786737307333\\
&&564123971732897913240474578834468260652327974647951137672658\\
&&693582180046317922073668860052627186363386088796882120769432\\
&&366149491002923444346373222145884100586421050242120365433561\\
&&201320481118852408731077014151666200162313177169372189248078\\
&&507711827842317498073276598828825169183103125680162072880719\\
\textsf{g}&=&2402352677501852\\
&&209227687703532399932712287657378364916510075318787663274146\\
&&353219320285676155269678799694668298749389095083896573425601\\
&&900601068477164491735474137283104610458681314511781646755400\\
&&527402889846139864532661215055797097162016168270312886432456\\
&&663834863635782106154918419982534315189740658186868651151358\\
&&576410138882215396016043228843603930989333662772848406593138\\
&&406010231675095763777982665103606822406635076697764025346253\\
&&773085133173495194248967754052573659049492477631475991575198\\
&&775177711481490920456600205478127054728238140972518639858334\\
&&115700568353695553423781475582491896050296680037745308460627\\
\textsf{q}&=&78571733251071885\\
&&079927659812671450121821421258408794611510081919805623223441
\end{array}
\]

The additional output of the generation algorithm is:
\[
\begin{array}{lcr}
\textsf{domain\_parameter\_seed}&=&478953892617249466\\
&&166106476098847626563138168027\\
&&716882488732447198349000396592\\
&&020632875172724552145560167746\\
\textsf{counter}&=&109
\end{array}
\]

\subsubsection{RFC-3526-2048}
\label{nh-group}

The group described in the previous section is not suitable for
encoding non-homomorphic answers. Therefore, we describe here a
different group for cases where the election has non-homomorphic
questions. This group is the 2048-bit one defined in RFC 3526:
\[
\begin{array}{lcr}
\textsf{p}&=&32317006071311007\\
&&300338913926423828248817941241140239112842009751400741706634\\
&&354222619689417363569347117901737909704191754605873209195028\\
&&853758986185622153212175412514901774520270235796078236248884\\
&&246189477587641105928646099411723245426622522193230540919037\\
&&680524235519125679715870117001058055877651038861847280257976\\
&&054903569732561526167081339361799541336476559160368317896729\\
&&073178384589680639671900977202194168647225871031411336429319\\
&&536193471636533209717077448227988588565369208645296636077250\\
&&268955505928362751121174096972998068410554359584866583291642\\
&&136218231078990999448652468262416972035911852507045361090559\\
\textsf{g}&=&2\\
\textsf{q}&=&16158503035655503\\
&&650169456963211914124408970620570119556421004875700370853317\\
&&177111309844708681784673558950868954852095877302936604597514\\
&&426879493092811076606087706257450887260135117898039118124442\\
&&123094738793820552964323049705861622713311261096615270459518\\
&&840262117759562839857935058500529027938825519430923640128988\\
&&027451784866280763083540669680899770668238279580184158948364\\
&&536589192294840319835950488601097084323612935515705668214659\\
&&768096735818266604858538724113994294282684604322648318038625\\
&&134477752964181375560587048486499034205277179792433291645821\\
&&068109115539495499724326234131208486017955926253522680545279
\end{array}
\]
Additionally, its \textsf{embedding} field is set to:
\[
  \left\{
    \begin{array}{rcl}
      \textsf{padding}&=&8\\
      \textsf{bits\_per\_int}&=&8
    \end{array}
  \right\}
\]

\subsection{Elliptic curves}

\subsubsection{Ed25519}

The Ed25519 group is a well-known group, defined in RFC 7748. It is
defined by the following parameters:
\begin{itemize}
\item $p=2^{255}-19$,
\item $E/\F_p$ is the twisted Edwards curve:
  \[
    -x^2+y^2=1-\frac{121665}{121666}x^2y^2,
  \]
\item $g$ is the unique point in $E(\F_p)$ whose $y$ coordinate is
  $4/5$ and whose $x$ coordinate is positive,
\item the order of $g$ is $q=2^{252}+27742317777372353535851937790883648493$.
\end{itemize}
In the above, \emph{positive} is defined in terms of bit-encoding:
\begin{itemize}
\item \emph{positive} coordinates are even coordinates (least significant bit is cleared),
\item \emph{negative} coordinates are odd coordinates (least significant bit is set).
\end{itemize}

Group elements are points represented by their coordinates $(x, y)$,
which can be \emph{compressed} into a single 256-bit number
$z=\textsf{compress}(x,y)$:
\begin{itemize}
\item let $b$ be the least significant bit of $x$ shifted to the left
  by 255 bits,
\item let $z$ be the binary XOR of $b$ and $y$.
\end{itemize}
This $z$ can be uncompressed into a single point
$\textsf{uncompress}(z)$. This compressed form is used for
serialization, written in hexadecimal, always padded with
leading-zeroes so that the resulting string is always 64 characters.

Additionally, a non-homomorphic encoding is defined as follows:
\begin{itemize}
\item let $\kappa=8$ and $p=14$;
\item it is assumed that each small integer is $\kappa$ bits (or less);
\item $[v_1,\dots,v_n]$ is encoded as:
  \[
    \xi=\textsf{of\_ints}([v_1,\dots,v_n])=\textsf{uncompress}((((v_1\times
    2^\kappa+v_2)\times 2^\kappa+\dotsb)\times 2^\kappa+v_n)\times
    2^p+\varepsilon)
  \]
  where $\varepsilon$ (of $p$ bits or less) is the smallest
  non-negative integer such that $\xi\in\G$;
\item the $\textsf{to\_ints}$ is the inverse of $\textsf{of\_ints}$, and
  takes as input a group element $\xi$ and the number $n$ of encoded
  integers.
\end{itemize}
The \textsf{get\_generator} function is the pure part of
\textsf{GetGenerator} defined in table~\ref{get-generator-ed25519}.

\section{Shuffle algorithms}
\label{shuffle-algorithms}

The algorithms $\textsf{GenShuffle}$ and $\textsf{GenShuffleProof}$
are referred to in section~\ref{shuffles}. They were taken from
version 1.3.2 of the CHVote System
Specification~\cite{CHVote},
%\footnote{\url{https://eprint.iacr.org/2017/325}},
and
are given here for self-completeness. We also give the
$\textsf{CheckShuffleProof}$ algorithm, used to check a proof produced
by $\textsf{GenShuffleProof}$. For more explanations on these
algorithms, please refer to the CHVote System Specification.

\begin{table}[h]
  \begin{framed}
    \noindent\paragraph{Input}
    \begin{itemize}
    \item $\textbf{e}=[e_1,\dots,e_N]\in\texttt{ciphertext}^N$:
      encrypted answers to one non-homomorphic question
    \item $y\in\G$: public key of the election
    \end{itemize}
    \noindent\paragraph{Algorithm}
    \begin{enumerate}
    \item $\psi\leftarrow\textsf{GenPermutation}(N)$\hfill//
      $\psi=[j_1,\dots,j_N]$, see table~\ref{gen-permutation}
    \item For $i=1,\dots,N$:
      \begin{itemize}
      \item
        $(e'_i,r'_i)\leftarrow\textsf{GenReEncryption}(e_i,y)$\hfill//
        see table~\ref{gen-re-encryption}
      \end{itemize}
    \item $\textbf{e}'\leftarrow[e'_{j_1},\dots,e'_{j_N}]$
    \item $\textbf{r}'\leftarrow[r'_1,\dots,r'_N]$
    \item Return $(\textbf{e}',\textbf{r}',\psi)$\hfill//
      $\textbf{e}'\in\texttt{ciphertext}^N$, $\textbf{r}'\in\Z_q^N$,
      $\psi\in\Psi_N$
    \end{enumerate}
  \end{framed}
  \caption{Function $\mathsf{GenShuffle}(\textbf{e},y)$}
  \label{gen-shuffle}
\end{table}

\begin{table}
  \begin{framed}
    \noindent\paragraph{Input}
    \begin{itemize}
    \item $N\in\N$: permutation size
    \end{itemize}
    \noindent\paragraph{Algorithm}
    \begin{enumerate}
    \item $I\leftarrow[1,\dots,N]$
    \item For $i=0,\dots,N-1$:
      \begin{enumerate}
      \item Pick $k$ uniformly at random in $\{i,\dots,N-1\}$
      \item $j_{i+1}\leftarrow I[k]$
      \item $I[k]\leftarrow I[i]$
      \end{enumerate}
    \item $\psi\leftarrow[j_1,\dots,j_N]$
    \item Return $\psi$\hfill// $\psi\in\Psi_N$
    \end{enumerate}
  \end{framed}
  \caption{Function $\mathsf{GenPermutation}(N)$}
  \label{gen-permutation}
\end{table}

\begin{table}
  \begin{framed}
    \noindent\paragraph{Input}
    \begin{itemize}
    \item $e\in\texttt{ciphertext}$: one encrypted answer to one
      non-homomorphic question
    \item $y\in\G$: public key of the election
    \end{itemize}
    \noindent\paragraph{Algorithm}
    \begin{enumerate}
    \item Pick $r'$ uniformly at random in $\Z_q$
    \item $\alpha'\leftarrow\alphalabel(e)\times g^{r'}$
    \item $\beta'\leftarrow\betalabel(e)\times y^{r'}$
    \item Let $e'$ be a new $\texttt{ciphertext}$ with
      $\alphalabel=\alpha'$ and $\betalabel=\beta'$
    \item Return $(e',r')$\hfill// $e'\in\texttt{ciphertext}$,
      $r'\in\Z_q$
    \end{enumerate}
  \end{framed}
  \caption{Function $\mathsf{GenReEncryption}(e,y)$}
  \label{gen-re-encryption}
\end{table}

\begin{table}
  \small
  \begin{framed}
    \noindent\paragraph{Input}
    \begin{itemize}
    \item $\textbf{e}=[e_1,\dots,e_N]\in\texttt{ciphertext}^N$:
      encrypted answers to one question; we will denote by
      $\alpha_i$ and $\beta_i$ the contents of $e_i$
    \item $\textbf{e}'=[e'_1,\dots,e'_N]\in\texttt{ciphertext}^N$:
      shuffled encrypted answers; we will denote by
      $\alpha'_i$ and $\beta'_i$ the contents of $e'_i$
    \item $\textbf{r}'=[r'_1,\dots,r'_N]\in\Z_q^N$: re-encryption
      randomizations
    \item $\psi=[j_1,\dots,j_N]\in\Psi_N$: permutation
    \item $pk\in\G$: the public key of the election
    \item $\varphi\in\jstring$: the fingerprint of the election
    \end{itemize}
    \noindent\paragraph{Algorithm}
    \begin{enumerate}
    \item $h\leftarrow\textsf{GetSecondaryGenerator}()$,
      $\textbf{h}\leftarrow\textsf{GetGenerators}(N)$\hfill// see
      tables~\ref{get-secondary-generator} and \ref{get-generators}
    \item
      $(\textbf{c},\textbf{r})\leftarrow\textsf{GenPermutationCommitment}(\psi,\textbf{h})$\hfill//
      see table~\ref{gen-permutation-commitment}
    \item
      $\textsf{str}_c\leftarrow\llbracket\textbf{e}\rrbracket\llbracket\textbf{e}'\rrbracket\llbracket\textbf{c}\rrbracket$\hfill//
      see table~\ref{serialize-array}
    \item
      $\textbf{u}\leftarrow\textsf{GetNIZKPChallenges}(N,\verb=shuffle-challenges|=\varphi\verb=|=\textsf{str}_c)$\hfill//
      see table~\ref{get-nizkp-challenges}
    \item For $i=1,\dots,N$: $u'_i\leftarrow u_{j_i}$
    \item $\textbf{u}'\leftarrow[u'_1,\dots,u'_N]$
    \item
      $(\hat{\textbf{c}},\hat{\textbf{r}})\leftarrow\textsf{GenCommitmentChain}(h,\textbf{u}')$\hfill//
      see table~\ref{gen-commitment-chain}
    \item For $i=1,\dots,4$: pick $\omega_i$ at random in $\Z_q$
    \item For $i=1,\dots,N$: pick $\hat{\omega}_i$ and $\omega'_i$ at random in $\Z_q$
    \item $t_1\leftarrow g^{\omega_1}$, $t_2\leftarrow g^{\omega_2}$,
      $t_3\leftarrow g^{\omega_3}\prod_{i=1}^N h_i^{\omega'_i}$
    \item
      $(t_{4,1},t_{4,2})\leftarrow(pk^{-\omega_4}\prod_{i=1}^N(\beta'_i)^{\omega'_i},g^{-\omega_4}\prod_{i=1}^N(\alpha'_i)^{\omega'_i})$
    \item $\hat{c}_0\leftarrow h$
    \item For $i=1,\dots,N$:
      $\hat{t}_i\leftarrow
      g^{\hat{\omega}_i}\hat{c}_{i-1}^{\omega'_i}$
    \item
      $t\leftarrow(t_1,t_2,t_3,(t_{4,1},t_{4,2}),[\hat{t}_1,\dots,\hat{t}_N])$,
      $\textsf{str}_t\leftarrow\llbracket[t_1,t_2,t_3,t_{4,1},t_{4,2}]\rrbracket\llbracket[\hat{t}_1,\dots,\hat{t}_N]\rrbracket$
    \item
      $y\leftarrow(\textbf{e},\textbf{e}',\textbf{c},\hat{\textbf{c}},pk)$,
      $\textsf{str}_y\leftarrow\textsf{str}_c\llbracket\hat{\textbf{c}}\rrbracket
      pk$\hfill
    \item
      $c\leftarrow\textsf{GetNIZKPChallenge}(\verb=shuffle-challenge|=\varphi\verb=|=\textsf{str}_t\textsf{str}_y)$\hfill//
      see table~\ref{get-nizkp-challenge}
    \item $\bar{r}\leftarrow\sum_{i=1}^N r_i\mod q$,
      $s_1\leftarrow\omega_1+c\times\bar{r}\mod q$
    \item $v_N\leftarrow 1$
    \item For $i=N-1,\dots,1$: $v_i\leftarrow u'_{i+1}v_{i+1}\mod q$
    \item $\hat{r}\leftarrow\sum_{i=1}^N \hat{r}_iv_i\mod q$, $s_2\leftarrow\omega_2+c\times\hat{r}\mod q$
    \item $\tilde{r}\leftarrow\sum_{i=1}^N r_iu_i\mod q$, $s_3\leftarrow\omega_3+c\times\tilde{r}\mod q$
    \item $r'\leftarrow\sum_{i=1}^N r'_iu_i\mod q$, $s_4\leftarrow\omega_4+c\times r'\mod q$
    \item For $i=1,\dots,N$:
      $\hat{s}_i\leftarrow\hat{\omega}_i+c\times\hat{r}_i\mod q$,
      $s'_i\leftarrow\omega'_i+c\times u'_i\mod q$
    \item
      $s\leftarrow(s_1,s_2,s_3,s_4,[\hat{s}_1,\dots,\hat{s}_N],[s'_1,\dots,s'_N])$
    \item $\pi\leftarrow(t,s,\textbf{c},\hat{\textbf{c}})$
    \item Return $\pi$\hfill// $\pi\in\texttt{shuffle\_proof}$
    \end{enumerate}
  \end{framed}
  \caption{Function $\mathsf{GenShuffleProof}(\textbf{e},\textbf{e}',\textbf{r}',\psi,pk,\varphi)$}
  \label{gen-shuffle-proof}
\end{table}

\begin{table}
  \small
  \begin{framed}
    \noindent\paragraph{Input}
    \begin{itemize}
    \item $\pi\in\texttt{shuffle\_proof}$: shuffle proof
    \item $\textbf{e}=[e_1,\dots,e_N]\in\texttt{ciphertext}^N$:
      encrypted answers to one question; we will denote by
      $\alpha_i$ and $\beta_i$ the contents of $e_i$
    \item $\textbf{e}'=[e'_1,\dots,e'_N]\in\texttt{ciphertext}^N$:
      shuffled encrypted answers; we will denote by
      $\alpha'_i$ and $\beta'_i$ the contents of $e'_i$
    \item $pk\in\G$: the public key of the election
    \item $\varphi\in\jstring$: the fingerprint of the election
    \end{itemize}
    \noindent\paragraph{Algorithm}
    \begin{enumerate}
    \item $(t,s,\textbf{c},\hat{\textbf{c}})\leftarrow\pi$
    \item $(t_1,t_2,t_3,(t_{4,1},t_{4,2}),[\hat{t}_1,\dots,\hat{t}_N])\leftarrow t$
    \item $(s_1,s_2,s_3,s_4,[\hat{s}_1,\dots,\hat{s}_N],[s'_1,\dots,s'_N])\leftarrow s$
    \item $[c_1,\dots,c_N]\leftarrow\textbf{c}$,
      $[\hat{c}_1,\dots,\hat{c}_N]\leftarrow\hat{\textbf{c}}$
    \item $h\leftarrow\textsf{GetSecondaryGenerator}()$,
      $\textbf{h}\leftarrow\textsf{GetGenerators}(N)$\hfill// see
      tables~\ref{get-secondary-generator} and \ref{get-generators}
    \item
      $\textsf{str}_c\leftarrow\llbracket\textbf{e}\rrbracket\llbracket\textbf{e}'\rrbracket\llbracket\textbf{c}\rrbracket$\hfill//
      see table~\ref{serialize-array}
    \item
      $\textbf{u}\leftarrow\textsf{GetNIZKPChallenges}(N,\verb=shuffle-challenges|=\varphi\verb=|=\textsf{str}_c)$\hfill//
      see table~\ref{get-nizkp-challenges}
    \item
      $\textsf{str}_t\leftarrow\llbracket[t_1,t_2,t_3,t_{4,1},t_{4,2}]\rrbracket\llbracket[\hat{t}_1,\dots,\hat{t}_N]\rrbracket$
    \item
      $\textsf{str}_y\leftarrow\textsf{str}_c\llbracket\hat{\textbf{c}}\rrbracket
      pk$\hfill
    \item
      $c\leftarrow\textsf{GetNIZKPChallenge}(\verb=shuffle-challenge|=\varphi\verb=|=\textsf{str}_t\textsf{str}_y)$\hfill//
      see table~\ref{get-nizkp-challenge}
    \item $\bar{c}\leftarrow\prod_{i=1}^N c_i/\prod_{i=1}^N h_i$
    \item $u\leftarrow\prod_{i=1}^N u_i\mod q$
    \item $\hat{c}_0\leftarrow h$
    \item $\hat{c}\leftarrow\hat{c}_N/h^u$
    \item $\tilde{c}\leftarrow\prod_{i=1}^N c_i^{u_i}$
    \item $(\alpha',\beta')\leftarrow(\prod_{i=1}^N \alpha_i^{u_i},\prod_{i=1}^N \beta_i^{u_i})$
    \item $t'_1\leftarrow\bar{c}^{-c}\times g^{s_1}$
    \item $t'_2\leftarrow\hat{c}^{-c}\times g^{s_2}$
    \item $t'_3\leftarrow\tilde{c}^{-c}\times g^{s_3}\prod_{i=1}^N h_i^{s'_i}$
    \item
      $(t'_{4,1},t'_{4,2})\leftarrow((\beta')^{-c}\times
      pk^{-s_4}\prod_{i=1}^N(\beta'_i)^{s'_i},(\alpha')^{-c}\times
      g^{-s_4}\prod_{i=1}^N(\alpha'_i)^{s'_i})$
    \item For $i=1,\dots,N$: $\hat{t'_i}\leftarrow\hat{c}_i^{-c}\times g^{\hat{s}_i}\times\hat{c}_{i-1}^{s'_i}$
    \item Return
      $(t_1=t'_1)\land(t_2=t'_2)\land(t_3=t'_3)\land(t_{4,1}=t'_{4,1})\land(t_{4,2}=t'_{4,2})\land\left[\bigwedge_{i=1}^N(\hat{t_i}=\hat{t'_i})\right]$
    \end{enumerate}
  \end{framed}
  \caption{Function $\mathsf{CheckShuffleProof}(\pi,\textbf{e},\textbf{e}',pk,\varphi)$}
  \label{check-shuffle-proof}
\end{table}

\begin{table}
  \begin{framed}
    \noindent\paragraph{Algorithm}
    \begin{enumerate}
    \item $h\leftarrow\textsf{GetGenerator}(-1)$\hfill// see
      table~\ref{get-generator}
    \item Return $h$\hfill// $h\in\G^N$
    \end{enumerate}
  \end{framed}
  \caption{Function $\mathsf{GetSecondaryGenerator}()$}
  \label{get-secondary-generator}
\end{table}

\begin{table}
  \begin{framed}
    \noindent\paragraph{Input}
    \begin{itemize}
    \item $N\in\N$: number of independent generators to get
    \end{itemize}
    \noindent\paragraph{Algorithm}
    \begin{enumerate}
    \item For $i=0,\dots,N-1$:
      $h_i\leftarrow\textsf{GetGenerator}(i)$\hfill// see
      table~\ref{get-generator}
    \item $\textbf{h}\leftarrow[h_0,\dots,h_{N-1}]$
    \item Return $\textbf{h}$\hfill// $\textbf{h}\in\G^N$
    \end{enumerate}
  \end{framed}
  \caption{Function $\mathsf{GetGenerators}(N)$}
  \label{get-generators}
\end{table}

\begin{table}
  \begin{framed}
    \noindent\paragraph{Input}
    \begin{itemize}
    \item $i\in\Z$: number of the independent generator to get
    \end{itemize}
    \noindent\paragraph{State (shared between all runs)}
    \begin{itemize}
    \item $\mathcal{X}\in\mathcal{P}(\N\times\G)$ (initialized to
      $\emptyset$): generators to avoid
    \end{itemize}
    \noindent\paragraph{Algorithm}
    \begin{enumerate}
    \item $c\leftarrow(p-1)/q$\hfill// typically, $c=2$
    \item $x\leftarrow\shatwo(\verb=ggen|=i)$\hfill// $i$ in base 10,
      output as a big-endian number
    \item $h\leftarrow x^c$
    \item If $h\in\{0,1,g\}$, abort
    \item If $\exists j\neq i, (j,h)\in\mathcal{X}$, abort
    \item $\mathcal{X}\leftarrow\mathcal{X}\cup\{(i,h)\}$
    \item Return $h$\hfill// $h\in\G$
    \end{enumerate}
  \end{framed}
  \caption{Function $\mathsf{GetGenerator}(i)$ (for a multiplicative subgroup of a finite field)}
  \label{get-generator}
\end{table}

\begin{table}
  \begin{framed}
    \noindent\paragraph{Input}
    \begin{itemize}
    \item $i\in\Z$: number of the independent generator to get
    \end{itemize}
    \noindent\paragraph{State (shared between all runs)}
    \begin{itemize}
    \item $\mathcal{X}\in\mathcal{P}(\N\times\G)$ (initialized to
      $\emptyset$): generators to avoid
    \end{itemize}
    \noindent\paragraph{Algorithm}
    \begin{enumerate}
    \item $x\leftarrow\shatwo(\verb=ggen|=i)>>2$\hfill// $i$ in base 10,
      output as a big-endian number
    \item $b\leftarrow\textsf{uncompress}(x+\varepsilon)$
      \hfill// $\varepsilon$: smallest non-negative integer such that
      $b\in E$
    \item $h\leftarrow b^8$
    \item If $h\in\{1,g\}$, abort
    \item If $\exists j\neq i, (j,h)\in\mathcal{X}$, abort
    \item $\mathcal{X}\leftarrow\mathcal{X}\cup\{(i,h)\}$
    \item Return $h$\hfill// $h\in\G$
    \end{enumerate}
  \end{framed}
  \caption{Function $\mathsf{GetGenerator}(i)$ (for Ed25519)}
  \label{get-generator-ed25519}
\end{table}

\begin{table}
  \begin{framed}
    \noindent\paragraph{Input}
    \begin{itemize}
    \item $\psi=[j_1,\dots,j_N]\in\Psi_N$: permutation
    \item $\textbf{h}=[h_1,\dots,h_N]\in\G^N$: independent generators
    \end{itemize}
    \noindent\paragraph{Algorithm}
    \begin{enumerate}
    \item For $i=1,\dots,N$:
      \begin{itemize}
      \item Pick $r_{j_i}$ at random in $\Z_q$
      \item $c_{j_i}\leftarrow g^{r_{j_i}}\times h_i$
      \end{itemize}
    \item $\textbf{c}\leftarrow[c_1,\dots,c_N]$
    \item $\textbf{r}\leftarrow[r_1,\dots,r_N]$
    \item Return $(\textbf{c},\textbf{r})$\hfill//
      $\textbf{c}\in\G^N$, $\textbf{r}\in\Z_q^N$
    \end{enumerate}
  \end{framed}
  \caption{Function $\mathsf{GenPermutationCommitment}(\psi,\textbf{h})$}
  \label{gen-permutation-commitment}
\end{table}

\begin{table}
  \begin{framed}
    \noindent\paragraph{Input}
    \begin{itemize}
    \item $\textbf{e}=[e_1,\dots,e_N]\in\texttt{ciphertext}^N$: array
      of ciphertexts, or
    \item $\textbf{c}=[c_1,\dots,c_N]\in\G^N$: array of group elements
    \end{itemize}
    \noindent\paragraph{Algorithm}
    \begin{enumerate}
    \item set $S$ to the empty string
    \item For $i=1,\dots,N$:
      \begin{itemize}
      \item append $\alphalabel(e_i)$, a comma, $\betalabel(e_i)$ and a comma to $S$, or\hfill
      \item append $c_i$ and a comma to $S$\hfill
      \end{itemize}
    \item Return $S$\hfill// $S\in\jstring$
    \end{enumerate}
  \end{framed}
  \caption{Functions $\llbracket\textbf{e}\rrbracket$ and $\llbracket\textbf{c}\rrbracket$}
  \label{serialize-array}
\end{table}

\begin{table}
  \begin{framed}
    \noindent\paragraph{Input}
    \begin{itemize}
    \item $N\in\N$: number of ciphertexts
    \item $S\in\jstring$: challenge string
    \end{itemize}
    \noindent\paragraph{Algorithm}
    \begin{enumerate}
    \item $H\leftarrow\shatwo(S)$\hfill// output interpreted as an
      hexadecimal string
    \item For $i=0,\dots,N-1$:
      \begin{enumerate}
      \item $T\leftarrow\shatwo(i)$\hfill// input taken as decimal,
        output interpreted as hexadecimal
      \item $u_i\leftarrow\shatwo(HT)\mod q$\hfill// output
        interpreted as big-endian
      \end{enumerate}
    \item $\textbf{u}\leftarrow[u_0,\dots,u_{N-1}]$
    \item Return $\textbf{u}$\hfill// $\textbf{u}\in\Z_q^N$
    \end{enumerate}
  \end{framed}
  \caption{Function $\mathsf{GetNIZPKChallenges}(N,S)$}
  \label{get-nizkp-challenges}
\end{table}

\begin{table}
  \begin{framed}
    \noindent\paragraph{Input}
    \begin{itemize}
    \item $c_0\in\G$: initial commitment
    \item $\textbf{u}=[u_1,\dots,u_N]\in\Z_q^N$: public challenges
    \end{itemize}
    \noindent\paragraph{Algorithm}
    \begin{enumerate}
    \item For $i=1,\dots,N$:
      \begin{enumerate}
      \item Pick $r_i$ at random in $\Z_q$
      \item $c_i\leftarrow g^{r_i}\times c_{i-1}^{u_i}$
      \end{enumerate}
    \item $\textbf{c}\leftarrow[c_1,\dots,c_N]$
    \item $\textbf{r}\leftarrow[r_1,\dots,r_N]$
    \item Return $(\textbf{c},\textbf{r})$\hfill//
      $\textbf{c}\in\G^N$, $\textbf{r}\in\Z_q^N$
    \end{enumerate}
  \end{framed}
  \caption{Function $\mathsf{GenCommitmentChain}(c_0,\textbf{u})$}
  \label{gen-commitment-chain}
\end{table}

\begin{table}
  \begin{framed}
    \noindent\paragraph{Input}
    \begin{itemize}
    \item $S\in\jstring$: challenge string
    \end{itemize}
    \noindent\paragraph{Algorithm}
    \begin{enumerate}
    \item $c\leftarrow\shatwo(S)\mod q$\hfill// output interpreted as
      a big-endian number
    \item Return $c$\hfill// $c\in\Z_q$
    \end{enumerate}
  \end{framed}
  \caption{Function $\mathsf{GetNIZPKChallenge}(S)$}
  \label{get-nizkp-challenge}
\end{table}

\FloatBarrier

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