From 414abf9cdaeddf888fbdc19cb22d1482389dd8fc Mon Sep 17 00:00:00 2001 From: Jeff Burdges Date: Mon, 9 May 2016 01:37:41 +0200 Subject: Initial discriotion of offline payments The notation here gets horrible because I started from the post-quantum write up. And it has slightly different needs. --- doc/paper/offline.tex | 228 ++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 228 insertions(+) create mode 100644 doc/paper/offline.tex (limited to 'doc') diff --git a/doc/paper/offline.tex b/doc/paper/offline.tex new file mode 100644 index 000000000..7913ceb94 --- /dev/null +++ b/doc/paper/offline.tex @@ -0,0 +1,228 @@ +\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} + + + + + +\bibliographystyle{alpha} +\bibliography{taler,rfc} + +% \newpage +% \appendix + +% \section{} + + + +\end{document} + -- cgit v1.2.3