From a9f5958d16a6467c763f7b4d1b4872a8006572cb Mon Sep 17 00:00:00 2001 From: Jeff Burdges Date: Fri, 29 Apr 2016 04:20:59 +0200 Subject: [PATCH] Preliminary work on integrating key exchanges with Merkle trick I still need to work out exactly what proerties are needed. And it won't tex yet. --- doc/paper/postquantum_melt.tex | 317 ++++++++++++++++++++++++++++----- 1 file changed, 275 insertions(+), 42 deletions(-) diff --git a/doc/paper/postquantum_melt.tex b/doc/paper/postquantum_melt.tex index 2dfefeeed..2740fd37a 100644 --- a/doc/paper/postquantum_melt.tex +++ b/doc/paper/postquantum_melt.tex @@ -128,7 +128,29 @@ risks regardless. \smallskip -We could add an existing post-quantum key exchange, but these all +We propose two variations on Taler's refresh protocol that offer +resistane to a quantum adversary. + +First, we describe attaching contemporary post-quantum key exchanges, +based on either super-singular eliptic curve isogenies \cite{SIDH} or +ring learning with errors (Ring-LWE) \cite{Peikert14,NewHope}. +These provide strong post-quantum security so long as the underlying +scheme retain their post-quantum security. + +Second, we propose a hash based scheme that + +Merkle tree based scheme that provides a + query complexity bound suitable for current deployments, and + depends only upon the strength of the hash function used. + + + + +much smaller + + + +but these all incur significantly larger key sizes, requiring more badwidth and storage space for the exchange, and take longer to run. In addition, the established post-quantum key exchanges based on @@ -168,12 +190,261 @@ In these systems, anonymity is not post-quantum due to the zero-knowledge proofs they employ. +\section{Taler's refresh protocol} + +We first describe Taler's refresh protocol adding place holders +$\eta$, $\lambda$, $\Lambda$, $\mu$, and $\Mu$ for key material +involved in post-quantum operations. We view $\Lambda$ and $\Mu$ +as public keys with respective private keys $\lambda$ and $\mu$, +and $\eta$ as the symetric key resulting from the key exchange +between them. + +We require there be effeciently computable + $\CPK$, $\CSK$, $\LPK$, $\LSK$, $\KEX_2$ and $\KEX_3$ such that +\begin{itemize} +\item $\mu = \CSK(s)$ for a random bitstring $s$, + $\Mu = \CPK(\mu)$, +\item $\lambda = \LSK(t,\mu)$ and $\Lambda = \LPK(t,\mu)$ + for a random bitstring $t$, and +\item $\eta = \KEX_2(\lambda,\Mu) = \KEX_3(\Lambda,\mu)$. +\end{itemize} +In particular, if $\KEX_3(\Lambda,\mu)$ would fail + then $\KEX_2(\lambda,\Mu)$ must fail too. + +% Talk about assumption that if KEX_2 works then KEX_3 works? +If these are all read as empty, then our description below reduces +to Taler's existing refresh protocol. + +\smallskip + +We let $\kappa$ denote the exchange's taxation security parameter, +meaning the highest marginal tax rate is $1/\kappa$. Also, let +$\theta$ denote the maximum number of coins returned by a refresh. + +A coin $(C,\Mu,S)$ consists of + a Ed25519 public key $C = c G$, + a post-quantum public key $\Mu$, and + an RSA-FDH signature $S = S_d(C || \Mu)$ 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. +If $\Mu$ is large, we may replace it by $H(C || \Mu)$ to make signing +contracts more efficent. + +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 $c$, $b$, and $\mu$, but we need not retain $s$ +outside the refresh protocol. +$$ c = H(\textr{"Ed25519"} || s) +\qquad \mu = \CSK(s) +\qquad b = H(\textr{"Blind"} || s) $$ + +\smallskip + +We begin refresh with a possibly tainted coin $(C,\Mu,S)$ that +we wish to refresh into $n \le \theta$ untainted coins. + +In the change sitaution, our coin $(C,M,S)$ was partially spent and +retains only a part of the value determined by the denominaton $d$. +There is usually no denomination that matchets this risidual value +so we must refresh from one coin into $n \le \theta$. + +For $x$ amongst the symbols $c$, $C$, $\mu$, $\Mu$, $b$, 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 $C_{j,i} = c_{j,i} G$, $\Mu_{j,i}$, $m_{j,i}$, 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. +So as above $c'$, $\mu'$, and $b_j$ are derived from $s_j$, + and both $C' = c' G$ and $\Mu' = \CSK(s')$. + +\paragraph{Wallet phase 1.} +\begin{itemize} +\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 Generate $\lambda_j$, $\Lambda_j$, and + $\eta_j = \KEX_2(\lambda,\Mu)$ as appropriate + using $\mu$. % or possibly $\Mu$. + \item Set the linking commitment $\Gamma_{j,0} = (L_j,\Lambda_j)$. + \item Set $k_j = H(l_j C || \eta_j)$. +\smallskip + \item For $i=1 \cdots n$: + \begin{itemize} + \item Set $s' = H(\zeta_j || i)$. + \item Derive $c'$, $m'$, and $b'$ from $s'$ as above. + \item Compute $C' = c' G$ and $\Mu' = \CPK(m')$ too. + \item Compute $B_{j,i} = B_{b'}(C' || \Mu')$. + \item Encrypt $\Eta_{j,i} = E_{k_j}(s')$. + \item Set the coin commitments $\Gamma_{j,i} = (\Eta_{j,i},B_{j,i})$ +\end{itemize} +\smallskip +\end{itemize} +\item Send $(C,\Mu,S)$ and the signed commitments + $\Gamma_* = S_C( \Gamma_{j,i} \quad\textrm{for}\quad j=1\cdots\kappa, i=0 \cdots n )$. +\end{itemize} + +\paragraph{Exchange phase 1.} +\begin{itemize} +\item Verify the signature $S$ by $d$ on $(C || \Mu)$. +\item Verify the signatures by $C$ on the $\Gamma_{j,i}$ in $\Gamma_*$. +\item Pick random $\gamma \in \{1 \cdots \kappa\}$. +\item Mark $C$ as spent by saving $(C,\gamma,\Gamma_*)$. +\item Send $\gamma$ as $S(C,\gamma)$. +\end{itemize} + +\paragraph{Wallet phase 2.} +\begin{itemize} +\item Save $S(C,\gamma)$. +\item For $j = 1 \cdots \kappa$ except $\gamma$: + \begin{itemize} + \item Create a proof $\lambda_j^{\textrm{proof}}$ that + $\lambda_j$ is compatable with $\Lambda_j$ and $\Mu$. + \item Set a responce tuple + $R_j = (\zeta_j,l_j,\lambda_j,\lambda_j^{\textrm{proof}})$. + \end{itemize} +\item Send $S_C(R_j \quad\textrm{for}\quad j \ne \gamma )$. +\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 Compute $\eta_j = \KEX_2(\lambda_j,\Mu)$. + \item Verify that $\Lambda_j = \LPK(???)$ + \item Set $k_j = H(l_j C || \eta_j)$. + \item For $i=1 \cdots n$: + \begin{itemize} + \item Decrypt $s' = D_{k_j}(\Eta_{j,i})$. + \item Compute $c'$, $m'$, and $b'$ from $s_j$. + \item Compute $C' = c' G$ too. + \item Verify $B' = B_{b'}(C' || \Mu')$. + \end{itemize} + \end{itemize} +\item If verifications all pass then send $S_{d_i}(B_\gamma)$. +\end{itemize} + +We could optionally save long-term storage space by +replacing $\Gamma_*$ with both $\Gamma_{\gamma,0}$ and + $S_C(\Eta_{j,i} \quad\textrm{for}\quad j \ne \gamma )$. +It's clear this requires the wallet send that signature in some phase, +but also the wallet must accept a phase 2 responce to a phase 1 request. + + +\section{Post-quantum key exchanges} + +In \cite{SIDH}, there is a Diffie-Helman like key exchange (SIDH) +based on computing super-singular eliptic curve isogenies which +functions as a drop in replacement, or more likely addition, for +Taler's refresh protocol. + +In SIDH, private keys are the kernel of an isogeny in the 2-torsion +or the 3-torsion of the base curve. Isogenies based on 2-torsion can +only be paired with isogenies based on 3-torsion, and visa versa. +This rigidity makes constructing signature schemes with SIDH hard +\cite{}, but does not impact our use case. + +We let $\mu$ and $\Mu$ be the SIDH 2-torsion private and public keys, +repectively. We simlarly let $\lambda_j$ and $\Lambda_j$ be the +SIDH 3-torsion private and public keys. +% DO IT : +We define $\CPK$, $\CSK$, $\LPK$, $\LSK$, $\KEX_2$ and $\KEX_3$ + as appropriate from \cite{SIDH} too. + +\smallskip + +Ring-LWE based key exchanges like \cite{Peikert14,NewHope} require +that both Alice and Bob's keys be ephemeral because the success or +failure of the key exchange leaks one bit about both keys\cite{}. +As a result, authentication with Ring-LWE based schemes remains +problematic\cite{}. + +We observe however that the Taler wallet controls both sides during +the refresh protocol, so the wallet can ensure that the key exchange +always succeeds. In fact, the Ring-LWE paramaters could be tunned to +make the probability of failure arbitrarily high, saving the exchange +bandwidth, storage, and verification time. + + +We let $\mu$ and $\Mu$ be Alice (initator) side the private and public +keys, repectively. We simlarly let $\lambda_j$ and $\Lambda_j$ be the +Bob (respondent) private and public keys. +% DO IT : +Again now, $\CPK$, $\CSK$, $\LPK$, $\LSK$, $\KEX_2$ and $\KEX_3$ +can be defined from \cite{Peikert14,NewHope}. % DO IT + + +\section{Hashed-based one-sided public keys} + +We now define our hash-based encryption scheme. +Let $\delta$ denote our query security paramater and + let $\mu$ be a bit string. +For $j \le \kappa$, we define a Merkle tree $T_j$ of height $\delta$ +with leaves $\eta_{j,t} = H(\mu || "YeyCoins!" || t || j)$ + for $t \le 2^\delta$. +Let $\Lambda_j$ denote the root of $T_j$, making + $\LPK(j,\mu)$ the Merkle tree root function. +Set $\Mu = H(\Lambda_1 || \cdots || \Lambda_\kappa)$, + which defines $\CPK(\mu)$. + +Now let $\lambda_{j,t}$ consist of $(j,t,\eta_{j,t})$ along with +both the Merkle tree path that proves $\eta_{j,i}$ is a leaf of $T_j$, +and $(\Lambda_1,\ldots,\Lambda_\kappa)$, + making $\LSK(t,\mu)$ an embelished Merkle tree path function. + +We define $\KEX_2(\lambda_{j,t},\Mu) = \eta_{j,t}$ + if $\lambda_{j,t}$ proves that $\eta_{j,t}$ is a leaf for $\Mu$, +or empty otherwise. + + +$\Mu = H(\Lambda_1 || \cdots || \Lambda_\kappa)$ + +$\KEX_3(\Lambda,\mu)$ + + + +$H(\eta_{j,i})$ along with a path + +$\eta$, $\lambda$, $\Lambda$, $\mu$, and $\Mu$ for key material + + +We require there be effeciently computable + $\CPK$, $\CSK$, $\LPK$, $\LSK$, $\KEX_2$ and $\KEX_3$ such that +\begin{itemize} +\item $\mu = \CSK(s)$ for a random bitstring $s$, + $\Mu = \CPK(\mu)$, +\item $\lambda = \LSK(t,\mu)$ and $\Lambda = \LPK(t,\mu)$ + for a random bitstring $t$, and +\item $\eta = \KEX_2(\lambda,\Mu) = \KEX_3(\Lambda,\mu)$. +\end{itemize} +In particular, if $\KEX_3(\Lambda,\mu)$ would fail + then $\KEX_2(\lambda,\Mu)$ must fail too. + +\begin{itemize} +\item +\item +\end{itemize} + + +\bibliographystyle{alpha} +\bibliography{taler,rfc} + +% \newpage +% \appendix + +% \section{} + + + +\end{document} + -\section{Background} -\section{Refresh} Let $\kappa$ and $\theta$ denote @@ -210,6 +481,7 @@ In addition, there was a value $s$ such that but we try not to retain $s$ if possible. + We have a tainted coin $(C,M,S)$ that we wish to refresh into $n \le \theta$ untained coins. For simplicity, we allow $x'$ to stand for the component @@ -261,45 +533,6 @@ Exchange phase 2. \section{Withdrawal} -\bibliographystyle{alpha} -\bibliography{taler,rfc} - -% \newpage -% \appendix - -% \section{} - - - -\end{document} - - - -$l$ denotes Merkle tree levels -yields $2^l$ leaves -costs $2^{l+1}$ hashing operations - -$a$ denotes number of leaves used -yields $2^{a l}$ outcomes - - - - - - -commit H(h) and h l C and E_{l C)(..) -reveal h and l - - - -x_n ... x_1 c G - - - - - - -waiting period of 10 min