X-Git-Url: http://www.chiark.greenend.org.uk/ucgi/~mdw/git/tripe/blobdiff_plain/90d03a8531584ba5467d13f28737e36d4afac658..96e12a8165129bfe385e6ec233317ada460a271d:/doc/wrestlers.tex diff --git a/doc/wrestlers.tex b/doc/wrestlers.tex index 033f9fea..c175b1a9 100644 --- a/doc/wrestlers.tex +++ b/doc/wrestlers.tex @@ -1,6 +1,6 @@ %%% -*-latex-*- %%% -%%% $Id: wrestlers.tex,v 1.3 2001/06/22 19:41:31 mdw Exp $ +%%% $Id: wrestlers.tex,v 1.5 2002/01/13 14:55:31 mdw Exp $ %%% %%% Description of the Wrestlers Protocol %%% @@ -10,6 +10,12 @@ %%%----- Revision history --------------------------------------------------- %%% %%% $Log: wrestlers.tex,v $ +%%% Revision 1.5 2002/01/13 14:55:31 mdw +%%% More incomplete stuff. +%%% +%%% Revision 1.4 2001/06/29 19:36:05 mdw +%%% Some progress made on laptop. +%%% %%% Revision 1.3 2001/06/22 19:41:31 mdw %%% Restart with different structure and rather more formal objectives. %%% @@ -20,11 +26,26 @@ %%% Initial versions of documentation. %%% -\documentclass{article} -\usepackage{amssymb} -\usepackage{amstext} +\newif\iffancystyle\fancystylefalse + +\iffancystyle + \documentclass + [a4paper, article, 10pt, numbering, noherefloats, notitlepage] + {strayman} + \usepackage[palatino, helvetica, courier, maths=cmr]{mdwfonts} + \usepackage[margin]{amsthm} +\else + \documentclass[a4paper]{article} + \usepackage{a4wide} + \usepackage{amsthm} +\fi + +\usepackage{amssymb, amstext} +\usepackage{mdwtab, mathenv} \errorcontextlines=999 +\showboxbreadth=999 +\showboxdepth=999 \makeatletter \title{The Wrestlers Protocol: proof-of-receipt and secure key exchange} @@ -33,14 +54,27 @@ \bibliographystyle{alpha} \newtheorem{theorem}{Theorem} -\newenvironment{proof}[1][Proof]{% - \par\noindent\textbf{#1.} % -}{% - \penalty\@M\hfill\vadjust{}% - \penalty\z@\relax\vadjust{}% - \penalty\@M\hfill$\square$% - \par% -} +\renewcommand{\qedsymbol}{$\square$} + +\newcommand{\getsr}{\stackrel{\scriptscriptstyle R}{\gets}} +\newcommand{\inr}{\in_{\scriptscriptstyle R}} +\newcommand{\oracle}[1]{\mathcal{#1}} + +\newcommand{\expt}[2]{\mathbf{Exp}^{\text{\normalfont#1}}_{#2}} +\newcommand{\adv}[2]{\mathbf{Adv}^{\text{\normalfont#1}}_{#2}} +\newcommand{\ord}{\mathop{\operator@font ord}} +\newcommand{\poly}[1]{\mathop{\operator@font poly}({#1})} +\newcommand{\negl}[1]{\mathop{\operator@font negl}({#1})} +\newcommand{\compind}{\stackrel{c}{\approx}} + +\newcolumntype{G}{p{0pt}} + +\newcommand{\xor}{\oplus} +\renewcommand{\epsilon}{\varepsilon} + +\newenvironment{program} + {\begin{tabular}[C]{|G|}\hlx{hv[][\tabcolsep]}\begin{tabbing}} + {\end{tabbing}\\\hlx{v[][\tabcolsep]h}\end{tabular}} \begin{document} @@ -65,19 +99,285 @@ % authentication oracle is useless if the hash function has appropriate % properties. -Suppose that $G$ is some cyclic group of order $q$, generated by an element -$g$, in which the decision Diffie-Hellman problem \cite{Boneh:1998:DDP} is -hard. Alice can choose a private key $1 < \alpha < q$ and publish her -corresponding public key $A = g^\alpha$. Later, Bob can verify that he's -talking to Alice by choosing a random $1 < \beta < q$ and sending Alice a -\emph{challenge} $B = g^\beta$. If she replies with $B^\alpha$, Bob accepts -that he's talking to Alice, otherwise he doesn't. +We begin by introducing a simple proof-of-identity protocol, based on the +Diffie-Hellman key-exchange protocol \cite{Diffie:1976:NDC} and prove some of +its useful properties. -%%%-------------------------------------------------------------------------- +\subsection{Introduction} + +We start by selecting a security parameter $k$. We choose a cyclic group $G += \langle g \rangle$ with $q = |G|$ prime, in which the decision +Diffie-Hellman problem is assumed to be hard \cite{Boneh:1998:DDP}; we +require that $q$ is a $k$-bit number. Suitable groups include elliptic +curves over finite fields and prime-order subgroups of prime fields. +Throughout, we shall write the group operation as multiplication, and we +shall assume that group elements are interchangeable with their binary +representations. + +Alice can choose a private key $\alpha$ from the set $\{ 1, 2, \ldots, q - 1 +\}$ and publish her corresponding public key $a = g^\alpha$. + +A simplistic proof-of-identity protocol might then proceed as follows: +\begin{enumerate} +\item Bob chooses a random exponent $\beta$, also from $\{ 1, 2, \ldots, q - + 1 \}$, and sends a \emph{challenge} $b = g^\beta$ to Alice. +\item Alice computes her \emph{response} $b^\alpha$ and returns it to Bob. +\item If Alice's response is equal to $a^\beta$ then Bob is satisfied. +\end{enumerate} +If Bob always plays by the rules then this protocol is evidently secure, +since computing $g^{\alpha\beta}$ given $g^\alpha$ and $g^\beta$ is precisely +the Diffie-Hellman problem. + +This protocol has a flaw, though: by using Alice as an oracle for the +function $x \mapsto x^\alpha$, he could conceivably acquire information about +Alice's private key which he could later use to impersonate her. As an +example, Bob can submit $-g^\beta$ as a challenge: if the reply +$(-g^\beta)^\alpha = g^{\alpha\beta}$, Bob can conclude that $\alpha$ is +even. + +We fix the protocol by requiring that Bob prove that he already knows the +answer to his challenge. We introduce a hash function $h$ whose properties +we shall investigate later. The \emph{Wrestlers Authentication Protocol} is +then: +\begin{enumerate} +\item Bob chooses a secret $\beta \getsr \{ 1, 2, \ldots, q - 1 \}$. He + computes his \emph{challenge} $b \gets g^\beta$ and also a \emph{check + value} $c \gets \beta \xor h(a^\beta)$. He sends the pair $(b, c)$. +\item Alice receives $(b', c')$. She computes $r \gets b'^\alpha$ and + $\gamma \gets c' \xor h(R)$. If $b' = g^\gamma$, she sends $r$ back as her + \emph{response}; otherwise she sends the distinguished value $\bot$. +\item Bob receives $r'$. If $r' = a^\beta$ then he accepts that he's talking + to Alice. +\end{enumerate} +We show that the introduction of the check value has indeed patched up the +weakness described above, and that the check value itself hasn't made an +impersonator's job much easier. + +\subsection{Analysis in the random oracle model} + +Here, we prove that the Wrestlers Authentication Protocol is secure if $h$ is +implemented by a public \emph{random oracle} \cite{Bellare:1993:ROP}. + +Fix a private key $\alpha \inr \{ 1, 2, \ldots, q - 1\}$ and the corresponding public +key $a = g^\alpha$, and consider a probabilistic polynomial-time adversary +$A$, equipped with two oracles. One is a random oracle $\oracle{H}\colon +\{0, 1\}^* \to \{0, 1\}^k$, which implements a function $h$ randomly selected +from the set of all functions with that signature; the other is an +authentication oracle $\oracle{O}\colon G \times \{0, 1\}^k \to G \cup \{ +\bot \}$ defined by: +\[ +\oracle{O}^(b, c) = \begin{cases} + b^\alpha & if $b = g^{c \xor h(b^\alpha)}$ \\ + \bot & otherwise +\end{cases} +\] +The authentication oracle will play the part of Alice in the following. We +first show that access to this authentication oracle can't help an adversary +learn how to impersonate Alice. + +\begin{theorem} + The Wrestlers Authentication Protocol with random oracle is computational + zero knowledge \cite{Brassard:1989:SZK}. + \label{thm:wap-czk} +\end{theorem} +\begin{proof} + \newcommand{\Hlist}{\textit{$\oracle{H}$-list}} + \newcommand{\Hsim}{\textit{$\oracle{H}$-sim}} + \newcommand{\Osim}{\textit{$\oracle{O}$-sim}} + \renewcommand{\H}{\oracle{H}} + \renewcommand{\O}{\oracle{O}} + % + Let $A$ be any probabilistic polynomial-time adversary with access to a + random. We shall construct a probabilistic polynomial-time adversary $A'$ + without either oracle such that the probability distributions on the output + of $A$ and $A'$ are equal. Specifically, $A'$ runs $A$ with simulated + random and authentication oracles whose behaviour is computationally + indistinguishable from the originals. The algorithm for $A'$ and the + simulated oracles is shown in figure~\ref{fig:wap-czk-sim}. -\section{An MT-authenticator} -% Use the protocol of the previous section as an MT-authenticator, within the -% meaning of [Canetti:2001:AKE]. + \begin{figure} + \begin{program} + \quad \= \kill + Adversary $A'^{\H(\cdot)}$: \+ \\ + $\Hlist \gets \emptyset$; \\ + $r \gets A^{\Hsim(\cdot), \Osim(\cdot, \cdot)}$; \\ + \textbf{return} $r$; \- \\[\medskipamount] + % + Simulated random oracle $\Hsim(q)$: \+ \\ + \textbf{if} $\exists r : (q, r) \in \Hlist$ \textbf{then} + \textbf{return} $r$; \\ + $r \getsr \H(q)$; \\ + $\Hlist \gets \Hlist \cup \{ (q, r) \}$; \\ + \textbf{return} $r$; \- \\[\medskipamount] + % + Simulated authentication oracle $\Osim(b, c)$: \+ \\ + \textbf{if} $\exists q, r : (q, r) \in \Hlist \wedge + b = g^{c \xor r} \wedge + q = a^{c \xor r}$ \textbf{then} + \textbf{return} $a^{c \xor r}$; \\ + \textbf{else} \textbf{return} $\bot$; + \end{program} + % + \caption{The algorithm and simulated oracles for the proof of + theorem~\ref{thm:wap-czk}.} + \label{fig:wap-czk-sim} + \end{figure} + + Firstly we show that $A'$ runs in polynomial time. The only modifications + to $\Hlist$ are in $\Hsim$, which adds at most one element for each oracle + query. Since $\Hlist$ is initially empty and $A$ can only make $\poly{k}$ + queries to $\Hsim$, this implies that $|\Hlist|$ is always + polynomially-bounded. Each query can be answered in polynomial time: + indeed, the search implied by the test $\exists r : (q, r) \in \Hlist$ can + be performed in $O(\log q)$ time by indexing $\Hlist$ on $q$ using a radix + tree. Now, since $|\Hlist|$ is polynomially bounded, the test $\exists q, + r : (q, r) \in \Hlist \wedge b = g^{c \xor r} \wedge q = a^{c \xor r}$ at + the start of $\Osim$ runs in polynomial time even though it requires an + exhaustive search of the history of $A$'s queries to its simulated random + oracle. + + Next, we examine the behaviour of the simulated oracles. Since $\Hsim$ is + implemented in terms of the public random oracle $\H$, and returns the same + answers to queries, its probability distribution must be identical. + + We turn our attention to the simulation of the authentication oracle. + Consider a query $(b, c)$ made by $A$ to its authentication oracle. + + Firstly, suppose that there is a $\beta$ such that $b = g^\beta$ and $c = + \beta \xor r$ where $r$ is the response to a previous query made by $A$ to + its random oracle on $a^\beta$. Then the true authentication oracle $\O$ + is satisfied because $b = g^\beta = g^{c \xor r} = g^{c \xor h(a^\beta)} = + g^{c \xor h(b^\alpha)}$, and returns $b^\alpha$. Similarly, the simulated + authentication oracle $\Osim$ will find the pair $(a^\beta, r) \in \Hlist$, + recover $\beta = c \xor r$, confirm that $b = g^\beta$, and return + $a^\beta$ as required. These events all occur with probability 1. + + Conversely, suppose that there is no such $\beta$. Then the simulated + oracle $\Osim$ will reject the query, returning $\bot$, again with + probability 1. However, a genuine authentication oracle $\O$ will succeed + and return $b^\alpha$ with probability $2^{-k}$. To see this, note that $b + = g^\gamma$ for some $\gamma \in \{ 0, 1, \ldots, q - 1 \}$, and the + probability that $h(b^\alpha) = c \xor \gamma$ is precisely $2^{-k}$, since + $c$ and $\gamma$ are $k$-bit strings. Since this probability is + negligible, we have shown that the distributions of the simulated oracles + $\Hsim$ and $\Osim$ are computationally indistinguishable from the genuine + oracles $\H$ and $\O$. The theorem follows. +\end{proof} + +Our next objective is to show that pretending to be Alice is hard without +knowledge of her secret $\alpha$. We say that an adversary $A$ +\emph{impersonates} in the Wrestlers Authentication Protocol with probability +$\epsilon$ if +\[ \Pr[A^{\oracle{H}(\cdot)}(g^\alpha, g^\beta, + \beta \xor h(g^{\alpha\beta})) + = g^{\alpha\beta}] + = \epsilon +\] +where the probability is taken over all choices of $\alpha, \beta \in \{ 1, +2, \ldots, q - 1 \}$ and random oracles $\oracle{H}$. + +\begin{theorem} + Assuming that the Diffie-Hellman problem is hard in $G$, no probabilistic + polynomial-time adversary can impersonate in the Wrestlers Authentication + Protocol with random oracle with better than negligible probability. + \label{thm:wap-dhp} +\end{theorem} +\begin{proof} + \newcommand{\Hlist}{\textit{$\oracle{H}$-list}} + \newcommand{\Hqueries}{\textit{$\oracle{H}$-queries}} + \newcommand{\Hsim}{\textit{$\oracle{H}$-sim}} + \renewcommand{\H}{\oracle{H}} + % + We prove the theorem by contradiction. Given a polynomial-time adversary + $A$ which impersonates Alice with non-negligible probability $\epsilon$, we + construct an adversary which solves the Diffie-Hellman problem with + probability no less than $\epsilon \poly{k}$, i.e., + \[ \Pr[A'(g^\alpha, g^\beta) = g^{\alpha\beta}] \ge \epsilon \poly{k}. \] + Thus, if there is any polynomial-time $A$ which impersonates with better + than negligible probability, then there is a polynomial-time $A'$ which + solves Diffie-Hellman with essentially the same probability, which would + violate our assumption of the hardness of Diffie-Hellman. + + The idea behind the proof is that the check value is only useful to $A$ + once it has discovered the correct response to the challenge, which it must + have done by solving the Diffie-Hellman problem. Hence, our Diffie-Hellman + + \begin{figure} + \begin{program} + \quad \= \kill + Adversary $A'(a, b)$: \+ \\ + $\Hlist \gets \emptyset$; \\ + $\Hqueries \gets \emptyset$; \\ + $r \getsr \{ 0, 1 \}^k$; \\ + $x \gets A^{\Hsim(\cdot)}(a, b, r)$; \\ + $c \getsr (\Hlist \cup \{ x \}) \cap G$; \\ + \textbf{return} $c$; \- \\[\medskipamount] + % + Simulated random oracle $\Hsim(q)$: \+ \\ + \textbf{if} $\exists r : (q, r) \in \Hlist$ + \textbf{then} \textbf{return} $r$; \\ + $r \gets \{ 0, 1 \}^k$; \\ + $\Hlist \gets \Hlist \cup \{ (q, r) \}$; \\ + $\Hqueries \gets \Hqueries \cup \{ q \}$; \\ + \textbf{return} $r$; + \end{program} + % + \caption{The algorithm for the proof of theorem~\ref{thm:wap-dhp}} + \label{fig:wap-dhp-rdc} + \end{figure} + + The adversary $A'$ is shown in figure~\ref{fig:wap-dhp-rdc}. It generates + a random check-value and runs the impersonator $A$. + + The simulated random oracle $\Hsim$ gathers together the results of all of + the random oracle queries made by $A$. The final result returned by $A'$ + is randomly chosen from among all of the `plausible' random oracle queries + and $A$'s output, i.e., those queries which are actually members of the + group. The check value given to $A$ is most likely incorrect (with + probability $1 - 2^{-k}$): the intuition is that $A$ can only notice if it + actually computes the right answer and then checks explicitly. + + If $A$ does compute the correct answer $g^{\alpha\beta}$ and either returns + it or queries $\Hsim$ on it, then $A'$ succeeds with probability $\epsilon + / |(\Hlist \cup \{ x \}) \cap G| = \epsilon \poly{k}$, since $|\Hlist|$ is + no greater than the number of random oracle queries made by $A$ (which must + be polynomially bounded).\footnote{% + This polynomial factor introduces a loss in the perceived security of the + authentication protocol. It appears that this loss is only caused by the + possibility of the adversary $A$ being deliberately awkward and checking + the value of $c$ after having computed the answer. However, we can't see + a way to improve the security bound on the scheme without imposing + artificial requirements on $A$.} % + + It remains to show that if $A$ never queries its random oracle on + $g^{\alpha\beta}$ then it cannot distinguish the random check value $r$ + from the correct one $\beta \xor \H(g^{\alpha\beta})$, and hence won't + notice that $A'$ is lying to it. But this is obvious: $A$'s probability of + guessing the random value that would have been returned had it actually + queried $\Hsim$ on the input $g^{\alpha\beta}$ is equal to the probability + of it guessing $\beta \xor r$ instead, since $\Hsim$ chooses its responses + uniformly at random. This completes the proof. +\end{proof} + +We conclude here that the Wrestlers Authentication Protocol is a secure +authentication protocol in the random oracle model: impersonation is hard if +the Diffie-Hellman problem is hard, and proving one's identity doesn't leak +secret key information. + +\subsection{Requirements on hash functions} + +Having seen that the Wrestlers Authentication Protocol is secure in the +random oracle model, we now ask which properties we require from the hash +function. This at least demonstrates that the protocol isn't deeply flawed, +and suggests an efficient implementation in terms of conventional hash +functions. + +Here we investigate more carefully the properties required of the hash +function, and provide a more quantitative analysis of the protocol. + +Looking at the proofs of the previous two sections, we see that the random +oracles are mainly a device which allow our constructions to `grab hold' of +the hashing operations performed by the adversary. %%%-------------------------------------------------------------------------- @@ -87,6 +387,25 @@ that he's talking to Alice, otherwise he doesn't. % authenticated model using a much simpler protocol. Show that the simpler % protocol is SK-secure. + +% messages +% +% pre-challenge: g^b +% cookie: g^b, h(cookie, g^b') +% challenge: g^b, h(g^b'), b + h(reply-check, g^b, g^b', g^ab') +% reply: g^b, h(g^b'), b + h(reply-check, g^b, g^b', g^ab'), E_k(g^a'b) +% switch: h(g^b), h(g^b'), E_k(g^a'b, h(switch-request, g^a, g^a')) +% switch-ok: E_k(g^ab, h(switch-confirm, g^a, g^a'))begin{ + + +We now describe the full Wrestlers Protocol, in a multi-party setting. Fix a +cyclic group $G = \langle g \rangle$ of order $q = |G|$. Each player $P_i$ +chooses a private key $\alpha_i \in \{ 1, 2, \ldots, q - 1 \}$ and publishes +the corresponding public key $a_i = g^{\alpha_i}$ to the other players. + + + + %%%----- That's all, folks -------------------------------------------------- \bibliography{cryptography,mdw-crypto} @@ -95,4 +414,4 @@ that he's talking to Alice, otherwise he doesn't. %%% Local Variables: %%% mode: latex %%% TeX-master: "wrestlers" -%%% End: +%%% End: