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:
