\documentclass{llncs}
%\usepackage[margin=1in,a4paper]{geometry}
\usepackage[T1]{fontenc}
\usepackage{palatino}
\usepackage{xspace}
\usepackage{microtype}
\usepackage{tikz,eurosym}
\usepackage{amsmath,amssymb}
\usepackage{enumitem}
\usetikzlibrary{shapes,arrows}
\usetikzlibrary{positioning}
\usetikzlibrary{calc}
% Relate to:
% http://fc14.ifca.ai/papers/fc14_submission_124.pdf
% Terminology:
% - SEPA-transfer -- avoid 'SEPA transaction' as we use
% 'transaction' already when we talk about taxable
% transfers of Taler coins and database 'transactions'.
% - wallet = coins at customer
% - reserve = currency entrusted to exchange waiting for withdrawal
% - deposit = SEPA to exchange
% - withdrawal = exchange to customer
% - spending = customer to merchant
% - redeeming = merchant to exchange (and then exchange SEPA to merchant)
% - refreshing = customer-exchange-customer
% - dirty coin = coin with exposed public key
% - fresh coin = coin that was refreshed or is new
% - coin signing key = exchange's online key used to (blindly) sign coin
% - message signing key = exchange's online key to sign exchange messages
% - exchange master key = exchange's key used to sign other exchange keys
% - owner = entity that knows coin private key
% - transaction = coin ownership transfer that should be taxed
% - sharing = coin copying that should not be taxed
\def\mathcomma{,}
\def\mathperiod{.}
\title{Offline Taler}
\begin{document}
\mainmatter
\author{Jeffrey Burdges}
\institute{Intria / GNUnet / Taler}
\maketitle
% \begin{abstract}
% \end{abstract}
% \section{Introduction}
% \section{Taler's refresh protocol}
\def\Nu{N}
\def\newmathrm#1{\expandafter\newcommand\csname #1\endcsname{\mathrm{#1}}}
\newmathrm{FDH}
We shall describe Taler's refresh protocol in this section.
All notation defined here persists throughout the remainder of
the article.
We let $\kappa$ denote the exchange's taxation security parameter,
meaning the highest marginal tax rate is $1/\kappa$. Also, let
$n_\mu$ denote the maximum number of coins returned by a refresh.
\smallskip
Let $\iota$ denote a coin idetity paramater that
links together the different commitments but must reemain secret
from the exchange.
Let $n_\nu$ denote the identity security paramater.
An online coin's identity commitment $\Nu$ is the empty string.
In the offline coin case, we begin with a reserve public key $R$
and a private identity commitment seed $\nu$.
For $k \le n_\nu$, we define
\[ \begin{aligned}
\nu_{k,0} &= H(\nu || i) \mathcomma \\
\nu_{k,1} &= H(\nu || i) \oplus R \mathcomma \\
\Nu_k &= H(\nu_{k,0} || \nu_{k,1} || H(\iota || k) ) \mathperiod \\
\end{aligned} \]
% We define $\Nu = H( \Nu_i \quad\textrm{for $k \le n_\nu$})$ finally.
\smallskip
A coin $(C,\Nu,S)$ consists of
a Ed25519 public key $C = c G$,
an optional set of offline identity commitments $\Nu = \{\Nu_k | k \in \Gamma \}$
an RSA-FDH signature $S = S_d(\FDH(C) * \Pi_{k \in \Gamma} \FDH(\Nu_k))$ by a denomination key $d$.
A coin is spent by signing a contract with $C$. The contract must
specify the recipiant merchant and what portion of the value denoted
by the denomination $d$ they recieve.
There was of course a blinding factor $b$ used in the creation of
the coin's signature $S$. In addition, there was a private seed $s$
used to generate $c$ and $b$ but we need not retain $s$
outside the refresh protocol.
$$ c = H(\textrm{"Ed25519"} || s)
\qquad b = H(\textrm{"Blind"} || s) $$
We generate $\nu = H("Offline" || s)$ from $s$ as well,
but only for offline coins.
\smallskip
We begin refresh with a possibly tainted coin $(C,S)$ whose value
we wish to save by refreshing it into untainted coins.
In the change sitaution, our coin $(C,\Nu,S)$ was partially spent and
retains only a part of the value determined by the denominaton $d$.
For $x$ amongst the symbols $c$, $C$, $b$, and $s$,
we let $x_{j,i}$ denote the value normally denoted $x$ of
the $j$th cut of the $i$th new coin being created.
% So $C_{j,i} = c_{j,i} G$, $\Nu_{j,i}$, $m_{j,i}$, and $b^{j,i}$
% must be derived from $s^{j,i}$ as above.
We need only consider one such new coin at a time usually,
so let $x'$ denote $x_{j,i}$ when $i$ and $j$ are clear from context.
In other words, $c'$, and $b_j$ are derived from $s_j$,
and both $C' = c' G$.
\paragraph{Wallet phase 1.}
\begin{itemize}
\item For $i = 1 \cdots n$, create random coin ids $\iota_i$.
\item For $j = 1 \cdots \kappa$:
\begin{itemize}
\item Create random $\zeta_j$ and $l_j$.
\item Also compute $L_j = l_j G$.
\item Set $k_j = H(l_j C || \eta_j)$.
\end{itemize}
\smallskip
\item For $i = 1 \cdots n$:
\begin{itemize}
\item Create random pre-coin id $\iota'_i$.
\item Set $\iota_i = H("Id" || \iota'_i)$.
\item $j = 1 \cdots \kappa$:
\begin{itemize}
\item Set $s' = H(\zeta_j || i)$.
\item Derive $c'$ and $b'$from $s'$ as above.
\item Compute $C' = c' G$ too.
\item Compute $B_{j,i} = B_{b'}(C' || H(\iota_i || H(s')))$.
\item Encrypt $\Gamma'_{j,i} = E_{k_j}(s')$.
\item Set the coin commitments $\Gamma_{j,i} = (\Gamma'_{j,i},B_{j,i})$.
\end{itemize}
\item For $k = 1 \cdots 2 n_\nu$:
\begin{itemize}
\item Set $\nu_k = H(\iota'_i || k)$.
\item Generate $\Nu_k$ from $\nu_k$ and $H(\iota_i || k)$.
\item Set the coin commitment $\Gamma_{\kappa+k,i} = B_{b'}(\Nu_{i,k})$.
\end{itemize}
\end{itemize}
\smallskip
\item Save $\zeta_*$ and $\iota'_*$.
\item Send $(C,S)$ and the signed commitments
$\Gamma_* = S_C( \Gamma_{j,i} \quad\textrm{for $j=1\cdots\kappa+2n_\nu, i=0 \cdots n$} )$.
\end{itemize}
\paragraph{Exchange phase 1.}
\begin{itemize}
\item Verify the signature $S$ by $d$ on $C$.
\item Verify the signatures by $C$ on the $\Gamma_{j,i}$ in $\Gamma_*$.
\item Pick random $\gamma \in \{1 \cdots \kappa\}$.
\item Pick random $\Gamma \subset \{1,\ldots,2 n_\nu\}$ with $|\Gamma| = n_\nu$.
\item Mark $C$ as spent by saving $(C,\gamma,\Gamma,\Gamma_*)$.
\item Send $(\gamma,\Gamma)$ as $S(C,\gamma)$.
\end{itemize}
\paragraph{Wallet phase 2.}
\begin{itemize}
\item Save $S(C,\gamma,\Gamma)$.
\item For $j = 1 \cdots \kappa$ except $\gamma$:
\begin{itemize}
\item Send $S_C(l_j)$.
\item Send $S_C(H(\iota_i || H(s_{j,i})) \quad\textrm{for $i = 1 \cdots n$})$.
\end{itemize}
\item For $i = 1 \cdots n$ and $k \not\in \Gamma$:
\begin{itemize}
\item Send $S_C( \nu_{k,i}, H(\iota_i || k) )$.
\end{itemize}
\end{itemize}
\paragraph{Exchange phase 2.}
\begin{itemize}
\item Verify the signature by $C$.
\item For $j = 1 \cdots \kappa$ except $\gamma$:
\begin{itemize}
\item Set $k_j = H(l_j C)$.
\item For $i=1 \cdots n$:
\begin{itemize}
\item Decrypt $s' = D_{k_j}(\Gamma'_{j,i})$.
\item Compute $c'$, $m'$, and $b'$ from $s_j$.
\item Compute $C' = c' G$ too.
\item Verify $B' = B_{b'}(C' || H(\iota_i || H(s_{j,i})))$.
\end{itemize}
\end{itemize}
\item For $i=1 \cdots n$ and $k \not\in \Gamma$:
\begin{itemize}
\item Generate $\Nu_k$ from $\nu_k$ and $H(\iota_i || k)$.
\item Verify the coin commitment $\Gamma_{\kappa+k,i} = B_{b'}(\Nu_{i,k})$.
\end{itemize}
\item If verifications all pass then send $S_{d_i}(B_\gamma * \Pi_{k \in \Gamma} B_k)$.
\end{itemize}
!!! PLEASE READ CHAUM BEFORE USING THIS !!!
There are several really deadly attacks that require careful defenses.
Also, one must find a proof of security that works for this product.
And Brands might do better anyways.
\bibliographystyle{alpha}
\bibliography{taler,rfc}
% \newpage
% \appendix
% \section{}
\end{document}