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\documentclass[a4paper,10pt]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{listings}
\lstdefinelanguage{pseudo}
{morekeywords={if, then, else, end, let, and, for, in, while}}
\lstset{escapechar=?, mathescape=true, language=pseudo, frame=single,
basicstyle=\sffamily\small, keywordstyle=\sffamily\bfseries}
\title{Pacemaker}
\begin{document}
\maketitle
\section{Base protocol}
\subsection{Principle}
\begin{itemize}
\item{\bf Phase 1:} The server generates a unique seed and initiates
a seed pulse which is diffused through the network. Upon receiving the seed,
each peer signs it with its private key and hashes it. This hash is called the
\emph{token} of the peer.
\item{\bf Phase 2:} Each peer starts collecting the tokens of its
neighbors and add its own hash to the collected hashes, obtaining a
peer-token map. The hash of this map yields a new token which is sent back to
the neighbors. This progressively builds a hash tree until the root
server itselfs hashes the tokens of its children.
\item{\bf Phase 3:} The server initiates a new pulse with the map it has
collected from its childrens. Upon receiving the pulse, each peer adds to the
pulse the map associated with the token that has been collected during
Phase 2. Thus, each peer receives a list of hash maps
corresponding to a branch from the root to itself in the hash tree computed during phase 2.
\end{itemize}
\subsection{Messages}
It is assumed that each peer $p$ has a pair of public/private key
\textsf{(K$^p_{pub}$, K$^p_{priv}$)} and has access to the 3
following cryptographic primitives:
\begin{itemize}
\item{\sf sign(data, K$_{priv}$):} Returns a signature for \textsf{data}
using the private key \textsf{K$_{priv}$}.
\item{\sf verify(S, data, K$_{pub}$):} Verifies that \textsf{S} is a
signature for \textsf{data} that was created using the private key
\textsf{K$_{priv}$} associated with \textsf{K$_{pub}$}.
\item{\sf hash(data):} Returns the hash of \textsf{data}.
\end{itemize}
The key pair of the server will be denoted by \textsf{(KS$_{pub}$,
KS$_{priv}$)}. \textsf{KS$_{pub}$} is assumed to be known by every peer.
The following messages are used in the protocol :
\begin{itemize}
\item{\sf Seed($i$, seed$^i$, S$^i_{seed}$):} used to diffuse the seed
pulse during phase 1, where:
\begin{itemize}
\item $i$ is the round number.
\item \textsf{seed$^i$} is the seed of round $i$.
\item \textsf{S$^i_{pulse}$} is \textsf{sign(<$i$, seed$^i$>, KS$_{priv}$)}.
\end{itemize}
\item{\sf SeedReply($i$, H$_p$):} used to build the hash tree during Phase 2,
where \textsf{H$_p$} is the token received from peer $p$, that is, the hash of
the collected map of the tokens received by peer $p$.
\item{\sf Pulse($i$, seed$^i$, H$^i$, level, branch$^i$, S$^i_{pulse}$):} used
during Phase 3. Where:
\begin{itemize}
\item \textsf{H$^i$} is the hash of the map built by the server during
phase 2.
\item \textsf{level} is the current level in the branch built from the root
server to the current peer.
\item \textsf{branch$^i$} is a branch of hash maps going from the server
to the current peer. This branch is a part of the hash tree built during
phase 2.
\item \textsf{S$^i_{pulse}$} is \textsf{sign(<i, seed$^i$, H$^i$>, KS$_{priv}$)}
\end{itemize}
\end{itemize}
The protocol also provides the following request/reply logic to challenge the
availability of a peer:
\begin{itemize}
\item{\sf Availability($i$, bits):} used by a peer to send its avaibility
history up to round $i$. \textsf{bits} is a bit array where a $0$ (resp. 1)
bit at index $j$ means that the peer was absent (resp. available) during round
$j$.
\item{\sf Challenge($i$):} used to ask a peer the proof of its presence
during round $i$.
\item{\sf Proof($i$, seed$^i$, branch$^i$, S$^i_{pulse}$, S$^i_p$):} to answer
a challenge, where \textsf{S$^i_p$} is \textsf{sign(<i, seed$^i$>,
K$^p_{priv}$)}.
\end{itemize}
A peer can also choose to send its availability history up to round $i$
\subsection{Protocol}
\subsubsection{Proof building}
\begin{lstlisting}[title=Server at round $i$]
let phase$^i$ = Seeding;
let seed$^i$ = random_seed();
let S$^i_{seed}$ = sign(<$i$,seed$^i$>, KS$_{priv}$);
let M$^i_{seed}$ = Seed($i$, seed$^i$, S$^i_{seed}$);
let map$^i$ = $\emptyset$
$\forall q \in$ NS$_{server}$, send($q$, M$^i_{seed}$);
wait $T$;
phase$^i$ = Idle;
H$^i$ = Hash(map$^i$);
let S$^i_{pulse}$ = sign(<$i$,seed$^i$,H$^i$>, KS$_{priv}$);
let M$^i_{pulse}$ = Pulse($i$, seed$^i$, H$^i$, 0,
$\{$ 0 $\rightarrow$ map$^i$ $\}$, S$^i_{pulse}$);
$\forall q \in$ NS$_{server}$, send($q$, M$^i_{pulse}$);
\end{lstlisting}
\begin{lstlisting}[title={Server or peer receiving \textsf{SeedReply($i$,
X$_q$)} from peer $q$}]
if phase$^i$ = Seeding then
map$^i$ = $\{ q \rightarrow $X$_q \} \oplus$map$^i$;
end if
\end{lstlisting}
\begin{lstlisting}[title={Every $\Delta << T$ seconds, on peer $p$}]
if phase$^i$ = Seeding then
let H$^i$ = H(map$^i$);
let M$^i_{seedreply}$ = SeedReply($i$, H$^i$);
nreplies$^i$ = nreplies$^i$ + 1;
replies$^i$[nreplies$^i$] = (H$^i$, map$^i$);
$\forall q \in$ NS$_p$, send($q$, M$^i_{seedreply}$);
end if
\end{lstlisting}
\begin{lstlisting}[title={Peer $p$ receiving M$^i_{seed}$ = \textsf{Seed($i$,
seed$^i$, S$^i_{seed}$)}}]
if
verify_sig(S$^i_{seed}$, <$i$,seed$^i$>, KS$_{pub}$)
then
$\forall q \in$ NS$_p$, send($q$, M$^i_{seed}$);
let S$^i_p$ = sign(<$i$,seed$^i$>, K$^p_{priv}$);
let map$^i$ = $\{p \rightarrow H(S^i_p)\}$;
nreplies$^i$ = 0;
phase$^i$ = Seeding;
end if
\end{lstlisting}
\begin{lstlisting}[title={Peer $p$ receiving \textsf{Pulse($i$, seed$^i$, H$^i$,
level, tree$^i$, S$^i_{pulse}$)}}]
if
verify_sig(S$^i_{pulse}$, <$i$,seed$^i$,H$^i$>, KS$_{pub}$)
then
phase$^i$ = Idle;
included$^i$ = $\bot$;
if
$\exists$ $\{$ $p$ $\rightarrow$ H$^i$ $\}$ $\in$ tree$^i$[level] and
$\exists$ $n$ $|$ replies[$n$] = (H$^i$, oldmap$^i$) and
included$^i$ $<$ $n$ and
verify_tree(H$^i$, level, tree$^i$)
then
let tree'$^i$ = tree$^i$$\oplus$$\{$level+1 $\rightarrow$ oldmap$^i$$\}$;
let M$^i_{pulse}$ = Pulse($i$, seed$^i$, H$^i$, level+1, tree'$^i$, S$^i_{pulse}$);
History[$i$] = M$^i_{pulse}$;
included$^i$ = $n$;
$\forall q \in$ children$^p$, send($q$, M$^i_{pulse}$);
end if
end if
\end{lstlisting}
\subsubsection{Challenges}
\begin{lstlisting}[title=Peer $p$ sending to $q$ its availability at round $i$]
let bits$^i$ = new bitfield[$N_t$];
for $x$ in [1..$N_t$]
if History[$i-N_t+x$] = $\emptyset$ then
bits$^i$[$x$] := 0;
else
bits$^i$[$x$] := 1;
end if
end for
let M$^i_{avail}$ = Availability($i$, bits);
send($q$, M$^i_{avail}$)
\end{lstlisting}
\begin{lstlisting}[title={Peer $p$ receiving \textsf{Challenge($i$)} from peer
$q$}]
let M$^i_{pulse}$ = History[$i$];
send($q$, Proof M$^i_{pulse}$)
\end{lstlisting}
\begin{lstlisting}[title={Peer $q$ receiving from peer $p$ \textsf{Proof($i$, nonce, K$^i_{pub}$, trees$^i$, S$^i_p$, S$^i_{pulse}$, S$_{proof}$)}}]
if
($i$, nonce, $p$) $\in$ challenges and
verify_sig(S$^i_{pulse}$, <$i$,seed$^i$,$K^i_{pub}$,trees$^i$[0]>,KS$_{pub}$) and
verify_sig(S$^i_p$, <$i$, seed$^i$ >, K$^p_{pub}$) and
S$^i_p$ $\in$ trees$^i$ and
$\forall n>0$, H(trees$^i$[n]) $\in$ trees$^i$[n-1] and
verify_sig(S$_{proof}$, < nonce, H$^p$, H$^q$ >, K$^i_{pub}$)
then
good_reply($p$)
else
bad_reply($p$)
end if
\end{lstlisting}
\end{document}
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