| 1 | %%% -*-latex-*- |
| 2 | %%% |
| 3 | %%% The Wrestlers Protocol: secure, deniable key-exchange |
| 4 | %%% |
| 5 | %%% (c) 2006 Mark Wooding |
| 6 | %%% |
| 7 | |
| 8 | \newif\iffancystyle\fancystylefalse |
| 9 | \newif\ifshort\shortfalse |
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| 11 | \InputIfFileExists{wr.cfg}\relax\relax |
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| 15 | |
| 16 | \iffancystyle |
| 17 | \documentclass |
| 18 | [a4paper, article, 10pt, numbering, noherefloats, notitlepage, |
| 19 | runinsubsubsec] |
| 20 | {strayman} |
| 21 | \usepackage[T1]{fontenc} |
| 22 | \usepackage[palatino, helvetica, courier, maths=cmr]{mdwfonts} |
| 23 | \usepackage[within = subsection, mdwmargin]{mdwthm} |
| 24 | \usepackage{mdwlist} |
| 25 | \usepackage{sverb} |
| 26 | \if0\ifx\pdfoutput\xxundefined0\else\the\pdfoutput\fi |
| 27 | \PassOptionsToPackage{dvips}{xy} |
| 28 | \fi |
| 29 | \else |
| 30 | \documentclass[a4paper]{llncs} |
| 31 | \usepackage{a4wide} |
| 32 | \fi |
| 33 | |
| 34 | \PassOptionsToPackage{show}{slowbox} |
| 35 | %\PassOptionsToPackage{hide}{slowbox} |
| 36 | \usepackage{mdwtab, mathenv, mdwmath, crypto} |
| 37 | \usepackage{slowbox} |
| 38 | \usepackage{amssymb, amstext} |
| 39 | \usepackage{url, multicol} |
| 40 | \usepackage{tabularx} |
| 41 | \DeclareUrlCommand\email{\urlstyle{tt}} |
| 42 | \ifslowboxshow |
| 43 | \usepackage[all]{xy} |
| 44 | \turnradius{4pt} |
| 45 | \fi |
| 46 | |
| 47 | \iffancystyle |
| 48 | \def\next{\title[The Wrestlers Protocol]} |
| 49 | \else |
| 50 | \def\next{\title} |
| 51 | \fi |
| 52 | \next |
| 53 | {The Wrestlers Protocol \\ |
| 54 | A simple, practical, secure, deniable protocol for key-exchange} |
| 55 | \iffancystyle |
| 56 | \author{Mark Wooding \\ \email{mdw@distorted.org.uk}} |
| 57 | \else |
| 58 | \author{Mark Wooding} |
| 59 | \institute{\email{mdw@distorted.org.uk}} |
| 60 | \fi |
| 61 | |
| 62 | \iffancystyle |
| 63 | \bibliographystyle{mdwalpha} |
| 64 | \let\epsilon\varepsilon |
| 65 | \let\emptyset\varnothing |
| 66 | \let\le\leqslant\let\leq\le |
| 67 | \let\ge\geqslant\let\geq\ge |
| 68 | \numberwithin{table}{section} |
| 69 | \numberwithin{figure}{section} |
| 70 | \numberwithin{equation}{subsection} |
| 71 | \let\random\$ |
| 72 | \else |
| 73 | \bibliographystyle{plain} |
| 74 | \expandafter\let\csname claim*\endcsname\claim |
| 75 | \expandafter\let\csname endclaim*\endcsname\endclaim |
| 76 | \fi |
| 77 | |
| 78 | \iffancystyle |
| 79 | \newcommand{\Nupto}[1]{\N_{<{#1}}} |
| 80 | \else |
| 81 | \newcommand{\Nupto}[1]{\{0, 1, \ldots, #1 - 1\}} |
| 82 | \fi |
| 83 | \let\Bin\Sigma |
| 84 | \let\emptystring\lambda |
| 85 | \edef\Pr{\expandafter\noexpand\Pr\nolimits} |
| 86 | \newcommand{\bitsto}{\mathbin{..}} |
| 87 | \newcommand{\E}{{\mathcal{E}}} |
| 88 | \newcommand{\M}{{\mathcal{M}}} |
| 89 | \iffancystyle |
| 90 | \def\description{% |
| 91 | \basedescript{% |
| 92 | \let\makelabel\textit% |
| 93 | \desclabelstyle\multilinelabel% |
| 94 | \desclabelwidth{1in}% |
| 95 | }% |
| 96 | } |
| 97 | \fi |
| 98 | \def\fixme#1{\marginpar{FIXME}[FIXME: #1]} |
| 99 | \def\hex#1{\texttt{#1}_{x}} |
| 100 | |
| 101 | \def\dbox#1{% |
| 102 | \vtop{% |
| 103 | \def\\{\unskip\egroup\hbox\bgroup\strut\ignorespaces}% |
| 104 | \hbox{\strut#1}% |
| 105 | }% |
| 106 | } |
| 107 | |
| 108 | \def\Wident{\Xid{W}{ident}} |
| 109 | \def\Wkx{\Xid{W}{kx}} |
| 110 | \def\Wdkx{\Xid{W}{dkx}} |
| 111 | \def\Func#1#2{\mathcal{F}[#1\to#2]} |
| 112 | \def\diff#1#2{\Delta_{#1, #2}} |
| 113 | \def\Hid{H_{\textit{ID}}} |
| 114 | |
| 115 | %% protocol run diagrams |
| 116 | \def\PRaction#1{#1\ar[r]} |
| 117 | \def\PRcreatex#1{\PRaction{\textsf{Create session\space}#1}} |
| 118 | \def\PRcreate#1#2#3{\PRcreatex{(\text{#1},\text{#2},#3)}} |
| 119 | \def\PRreceivex#1{\PRaction{\textsf{Receive\space}#1}} |
| 120 | \def\PRreceive#1#2{\PRreceivex{\msg{#1}{#2}}} |
| 121 | \def\PRsession#1{\relax\mathstrut#1\ar[r]} |
| 122 | \def\msg#1#2{(\cookie{#1},#2)} |
| 123 | \def\PRresult#1{#1} |
| 124 | \def\PRsendx#1{\PRresult{\textsf{Send\space}#1}} |
| 125 | \def\PRsend#1#2{\PRsendx{\msg{#1}{#2}}} |
| 126 | \def\PRcomplete#1{\textsf{Complete:\space}#1} |
| 127 | \def\PRstate#1{\textsf{State:\space}#1} |
| 128 | \def\PRignore{\textsf{(ignored)}} |
| 129 | \def\PRreveal{\textsf{Session-state reveal}\ar[r]} |
| 130 | \def\protocolrun#1{\[\xymatrix @R=0pt @C=2em {#1}\]} |
| 131 | |
| 132 | %% class-ids for proof of extractor lemma |
| 133 | \let\Cid=\Lambda |
| 134 | \let\Csession=S |
| 135 | \let\Cindex=r |
| 136 | \let\Cquery=Q |
| 137 | \let\Chash=H |
| 138 | \let\Ccheck=c |
| 139 | \let\Ccheckset=V |
| 140 | \let\Ccount=\nu |
| 141 | |
| 142 | \def\HG#1{\mathbf{H}_{#1}} |
| 143 | |
| 144 | \begin{document} |
| 145 | |
| 146 | %%%-------------------------------------------------------------------------- |
| 147 | |
| 148 | \maketitle |
| 149 | \iffancystyle \thispagestyle{empty} \fi |
| 150 | |
| 151 | \begin{abstract} |
| 152 | We describe and prove (in the random-oracle model) the security of a simple |
| 153 | but efficient zero-knowledge identification scheme, whose security is based |
| 154 | on the computational Diffie-Hellman problem. Unlike other recent proposals |
| 155 | for efficient identification protocols, we don't need any additional |
| 156 | assumptions, such as the Knowledge of Exponent assumption. |
| 157 | |
| 158 | From this beginning, we build a simple key-exchange protocol, and prove |
| 159 | that it achieves `SK-security' -- and hence security in Canetti's Universal |
| 160 | Composability framework. |
| 161 | |
| 162 | Finally, we show how to turn the simple key-exchange protocol into a |
| 163 | slightly more complex one which provides a number of valuable `real-life' |
| 164 | properties, without damaging its security. |
| 165 | \end{abstract} |
| 166 | |
| 167 | \iffancystyle |
| 168 | \newpage |
| 169 | \thispagestyle{empty} |
| 170 | \columnsep=2em \columnseprule=0pt |
| 171 | \tableofcontents[\begin{multicols}{2}\raggedright][\end{multicols}] |
| 172 | %%\listoffigures[\begin{multicols}{2}\raggedright][\end{multicols}] |
| 173 | %% \listoftables[\begin{multicols}{2}\raggedright][\end{multicols}] |
| 174 | \newpage |
| 175 | \fi |
| 176 | |
| 177 | %%%-------------------------------------------------------------------------- |
| 178 | |
| 179 | \section{Introduction} |
| 180 | |
| 181 | |
| 182 | This paper proposes protocols for \emph{identification} and |
| 183 | \emph{authenticated key-exchange}. |
| 184 | |
| 185 | An identification protocol allows one party, say Bob, to be sure that he's |
| 186 | really talking to another party, say Alice. It assumes that Bob has some way |
| 187 | of recognising Alice; for instance, he might know her public key. Our |
| 188 | protocol requires only two messages -- a challenge and a response -- and has |
| 189 | a number of useful properties. It is very similar to, though designed |
| 190 | independently of, a recent protocol by Stinson and Wu; we discuss their |
| 191 | protocol in section~\ref{sec:stinson-ident}. |
| 192 | |
| 193 | Identification protocols are typically less useful than they sound. As Shoup |
| 194 | \cite{Shoup:1999:OFM} points out, it provides a `secure ping', by which Bob |
| 195 | can know that Alice is `there', but provides no guarantee that any other |
| 196 | communication was sent to or reached her. However, there are situations |
| 197 | where this an authentic channel between two entities -- e.g., a device and a |
| 198 | smartcard -- where a simple identification protocol can still be useful. |
| 199 | |
| 200 | An authenticated key-exchange protocol lets Alice and Bob agree on a shared |
| 201 | secret, known to them alone, even if there is an enemy who can read and |
| 202 | intercept all of their messages, and substitute messages of her own. Once |
| 203 | they have agreed on their shared secret, of course, they can use standard |
| 204 | symmetric cryptography techniques to ensure the privacy and authenticity of |
| 205 | their messages. |
| 206 | |
| 207 | |
| 208 | \subsection{Desirable properties of our protocols} |
| 209 | |
| 210 | Our identification protocol has a number of desirable properties. |
| 211 | \begin{itemize} |
| 212 | \item It is \emph{simple} to understand and implement. In particular, it |
| 213 | requires only two messages. |
| 214 | \item It is fairly \emph{efficient}, requiring two scalar multiplications by |
| 215 | each of the prover and verifier. |
| 216 | \item It is provably \emph{secure} (in the random oracle model), assuming the |
| 217 | intractability of the computational Diffie-Hellman problem. |
| 218 | \end{itemize} |
| 219 | |
| 220 | Our key-exchange protocol also has a number of desirable |
| 221 | properties. |
| 222 | \begin{itemize} |
| 223 | \item It is fairly \emph{simple} to understand and implement, though there |
| 224 | are a few subtleties. In particular, it is \emph{symmetrical}. We have |
| 225 | implemented a virtual private network system based on this protocol. |
| 226 | \item It is \emph{efficient}, requiring four scalar multiplications by each |
| 227 | participant. The communication can be reduced to three messages by |
| 228 | breaking the protocol's symmetry. |
| 229 | \item It is provably \emph{secure} (again, in the random oracle model), |
| 230 | assuming the intractability of the computational Diffie-Hellman problem, |
| 231 | and the security of a symmetric encryption scheme. |
| 232 | \item It provides \emph{perfect forward secrecy}. That is, even if a user's |
| 233 | long-term secrets are compromised, past sessions remain secure. |
| 234 | \item It is \emph{deniable}. It is possible to construct simulated |
| 235 | transcripts of protocol executions between any number of parties without |
| 236 | knowing any of their private keys. The simulated transcripts are (almost) |
| 237 | indistinguishable from real protocol transcripts. Hence, a transcript |
| 238 | does not provide useful evidence that a given party was really involved in |
| 239 | a given protocol execution. |
| 240 | \end{itemize} |
| 241 | |
| 242 | \subsection{Asymptotic and concrete security results} |
| 243 | |
| 244 | Most security definitions for identification (particularly zero-knowledge) |
| 245 | and key-exchange in the literature are \emph{asymptotic}. That is, they |
| 246 | consider a family of related protocols, indexed by a \emph{security |
| 247 | parameter}~$k$; they that any \emph{polynomially-bounded} adversary has only |
| 248 | \emph{negligible} advantage. A polynomially-bounded adversary is one whose |
| 249 | running time is a bounded by some polynomial~$t(k)$. The security definition |
| 250 | requires that, for any such polynomially-bounded adversary, and any |
| 251 | polynomial $p(k)$, the adversary's advantage is less than $p(k)$ for all |
| 252 | sufficiently large values of $k$. |
| 253 | |
| 254 | Such asymptotic notions are theoretically interesting, and have obvious |
| 255 | connections to complexity theory. Unfortunately, such an asymptotic result |
| 256 | tells us nothing at all about the security of a particular instance of a |
| 257 | protocol, or what parameter sizes one needs to choose for a given level of |
| 258 | security against a particular kind of adversary. Koblitz and Menezes |
| 259 | \cite{Koblitz:2006:ALP} (among other useful observations) give examples of |
| 260 | protocols, proven to meet asymptotic notions of security, whose security |
| 261 | proofs guarantee nothing at all for the kinds of parameters typically used in |
| 262 | practice. |
| 263 | |
| 264 | Since, above all, we're interested in analysing a practical and implemented |
| 265 | protocol, we follow here the `practice-oriented provable security' approach |
| 266 | largely inspired by Bellare and Rogaway, and exemplified by |
| 267 | \cite{Bellare:1994:SCB,Bellare:1995:XMN,Bellare:1995:OAE,Bellare:1996:ESD,% |
| 268 | Bellare:1996:KHF,Bellare:2000:CST}; see also \cite{Bellare:1999:POP}. |
| 269 | Rather than attempting to say, formally, whether or not a protocol is |
| 270 | `secure', we associate with each protocol an `insecurity function' which |
| 271 | gives an upper bound on the advantage of any adversary attacking the protocol |
| 272 | within given resource bounds. |
| 273 | |
| 274 | |
| 275 | \subsection{Formal models for key-exchange} |
| 276 | |
| 277 | Many proposed key-exchange protocols have turned out to have subtle security |
| 278 | flaws. The idea of using formal methods to analyse key-exchange protocols |
| 279 | begins with the logic of Burrows, Abadi and Needham \cite{Burrows:1989:LA}. |
| 280 | Their approach requires a `formalising' step, in which one expresses in the |
| 281 | logic the contents of the message flows, and the \emph{beliefs} of the |
| 282 | participants. |
| 283 | |
| 284 | Bellare and Rogaway \cite{Bellare:1994:EAK} describe a model for studying the |
| 285 | computational security of authentication and key-exchange protocols in a |
| 286 | concurrent setting, i.e., where multiple parties are running several |
| 287 | instances of a protocol simultaneously. They define a notion of security in |
| 288 | this setting, and show that several simple protocols achieve this notion. |
| 289 | Their original paper dealt with pairs of parties using symmetric |
| 290 | cryptography; they extended their definitions in \cite{Bellare:1995:PSS} to |
| 291 | study three-party protocols involving a trusted key-distribution centre. |
| 292 | |
| 293 | Blake-Wilson, Johnson and Menezes \cite{Blake-Wilson:1997:KAP} applied the |
| 294 | model of \cite{Bellare:1994:EAK} to key-exchange protocols using asymmetric |
| 295 | cryptography, and Blake-Wilson and Menezes \cite{Blake-Wilson:1998:EAA} |
| 296 | applied it to protocols based on the Diffie-Hellman protocol. |
| 297 | |
| 298 | The security notion of \cite{Bellare:1994:EAK} is based on a \emph{game}, in |
| 299 | which an adversary nominates a \emph{challenge session}, and is given either |
| 300 | the key agreed by the participants of the challenge session, or a random |
| 301 | value independently sampled from an appropriate distribution. The |
| 302 | adversary's advantage -- and hence the insecurity of the protocol -- is |
| 303 | measured by its success probability in guessing whether the value it was |
| 304 | given is really the challenge key. This challenge-session notion was also |
| 305 | used by the subsequent papers described above. |
| 306 | |
| 307 | Bellare, Canetti and Krawczyk \cite{Bellare:1998:MAD} described a pair of |
| 308 | models which they called the \textsc{am} (for `authenticated links model') |
| 309 | and \textsc{um} (`unauthenticated links model'). They propose a modular |
| 310 | approach to the design of key-exchange protocols, whereby one first designs a |
| 311 | protocol and proves its security in the \textsc{am}, and then applies a |
| 312 | authenticating `compiler' to the protocol which they prove yields a protocol |
| 313 | secure in the realistic \textsc{um}. Their security notion is new. They |
| 314 | define an `ideal model', in which an adversary is limited to assigning |
| 315 | sessions fresh, random and unknown keys, or matching up one session with |
| 316 | another, so that both have the same key. They define a protocol to be secure |
| 317 | if, for any adversary~$A$ in the \textsc{am} or \textsc{um}, there is an |
| 318 | ideal adversary~$I$, such that the outputs of $A$ and $I$ are computationally |
| 319 | indistinguishable. |
| 320 | |
| 321 | In \cite{Shoup:1999:OFM}, Shoup presents a new model for key-exchange, also |
| 322 | based on the idea of simulation. He analyses the previous models, |
| 323 | particularly \cite{Bellare:1994:EAK} and \cite{Bellare:1998:MAD}, and |
| 324 | highlights some of their inadequacies. |
| 325 | |
| 326 | Canetti and Krawczyk \cite{Canetti:2001:AKE} describe a new notion of |
| 327 | security in the basic model of \cite{Bellare:1998:MAD}, based on the |
| 328 | challenge-session notion of \cite{Bellare:1994:EAK}. The security notion, |
| 329 | called `SK-security', seems weaker in various ways than those of earlier |
| 330 | works such as \cite{Bellare:1994:EAK} or \cite{Shoup:1999:OFM}. However, the |
| 331 | authors show that their notion suffices for constructing `secure channel' |
| 332 | protocols, which they also define. |
| 333 | |
| 334 | In \cite{Canetti:2001:UCS}, Canetti describes the `universal composition' |
| 335 | framework. Here, security notions are simulation-based: one defines security |
| 336 | notions by presenting an `ideal functionality'. A protocol securely |
| 337 | implements a particular functionality if, for any adversary interacting with |
| 338 | parties who use the protocol, there is an adversary which interacts with |
| 339 | parties using the ideal functionality such that no `environment' can |
| 340 | distinguish the two. The environment is allowed to interact freely with the |
| 341 | adversary throughout, differentiating this approach from that of |
| 342 | \cite{Bellare:1998:MAD} and \cite{Shoup:1999:OFM}, where the distinguisher |
| 343 | was given only transcripts of the adversary's interaction with the parties. |
| 344 | With security defined in this way, it's possible to prove a `universal |
| 345 | composition theorem': one can construct a protocol, based upon various ideal |
| 346 | functionalities, and then `plug in' secure implementations of the ideal |
| 347 | functionalities and appeal to the theorem to prove the security of the entire |
| 348 | protocol. The UC framework gives rise to very strong notions of security, |
| 349 | due to the interactive nature of the `environment' distinguisher. |
| 350 | |
| 351 | Canetti and Krawczyk \cite{Canetti:2002:UCN} show that the SK-security notion |
| 352 | of \cite{Canetti:2001:AKE} is \emph{equivalent} to a `relaxed' notion of |
| 353 | key-exchange security in the UC framework, and suffices for the construction |
| 354 | of UC secure channels. |
| 355 | |
| 356 | The result of \cite{Canetti:2002:UCN} gives us confidence that SK-security is |
| 357 | the `right' notion of security for key-exchange protocols. Accordingly, |
| 358 | SK-security is the standard against which we analyse our key-exchange |
| 359 | protocol. |
| 360 | |
| 361 | |
| 362 | \subsection{Outline of the paper} |
| 363 | |
| 364 | The remaining sections of this paper are as follows. |
| 365 | \begin{itemize} |
| 366 | \item Section \ref{sec:prelim} provides the essential groundwork for the rest |
| 367 | of the paper. It introduces important notation, and describes security |
| 368 | notions and intractability assumptions. |
| 369 | \item Section \ref{sec:zk-ident} describes our zero-knowledge identification |
| 370 | protocol and proves its security. |
| 371 | \item Section \ref{sec:kx} describes the simple version of our key-exchange |
| 372 | protocol, and proves its security and deniability. It also describes some |
| 373 | minor modifications which bring practical benefits without damaging |
| 374 | security. |
| 375 | \item Finally, section \ref{sec:conc} presents our conclusions. |
| 376 | \end{itemize} |
| 377 | |
| 378 | %%%-------------------------------------------------------------------------- |
| 379 | |
| 380 | \section{Preliminaries} |
| 381 | \label{sec:prelim} |
| 382 | |
| 383 | \subsection{Miscellaneous notation} |
| 384 | |
| 385 | We write $\Func{D}{R}$ for the set of all functions with domain $D$ and range |
| 386 | $R$. |
| 387 | \iffancystyle |
| 388 | We write $\Nupto{n} = \{\, i \in \Z \mid 0 \le i < n \,\} = \{0, 1, \ldots, n |
| 389 | - 1\}$ for the set of nonnegative integers less than $n$. |
| 390 | \fi |
| 391 | |
| 392 | |
| 393 | \subsection{Groups} |
| 394 | |
| 395 | Let $(G, +)$ be a cyclic group\footnote{ |
| 396 | We find that additive group notation is easier to read. In particular, in |
| 397 | multiplicative groups, one ends up with many interesting things tucked away |
| 398 | in little superscripts.}% |
| 399 | of prime order $q$, and generated by an element $P$. We shall write the |
| 400 | identity of $G$ as $0_G$, or simply as $0$ when no ambiguity is likely to |
| 401 | arise. Thus, we have $\langle P \rangle = G$ and $q P = 0$. Any $X \in G$ |
| 402 | can be written as $X = x P$ for some $x \in \Nupto{q}$. |
| 403 | |
| 404 | |
| 405 | \subsection{Bit strings and encodings} |
| 406 | \label{sec:bitenc} |
| 407 | |
| 408 | Let $\Bin = \{0, 1\}$ be the set of binary digits. Then $\Bin^n$ is the set |
| 409 | of $n$-bit strings, and $\Bin^*$ the set of all (finite) bit strings. If $x |
| 410 | \in \Bin^n$ is a bit string, we write its length as $|x| = n$. For a bit |
| 411 | string $x \in \Bin^n$, and for $0 \le i < n$, we write $x[i]$ as the $i$th |
| 412 | bit of $x$. The empty string is denoted $\emptystring$. |
| 413 | |
| 414 | Let $x$ and $y$ be two bit strings. If $|x| = |y| = n$, we write $x \xor y$ |
| 415 | to mean the bitwise exclusive-or of $x$ and $y$: if $z = x \xor y$ then $|z| |
| 416 | = n$, and $z[i] = (x[i] + y[i]) \bmod 2$ for $0 \le i < n$. We write $x \cat |
| 417 | y$ to mean the concatenation of $x$ and $y$: if $z = x \cat y$ then $|z| = |
| 418 | |x| + |y|$ and $z[i] = x[i]$ if $0 \le i < |x|$ and $z[i] = y[i - |x|]$ if |
| 419 | $|x| < i \le |x| + |y|$. |
| 420 | |
| 421 | Finally, we let $\bot$ be a value distinct from any bit string. |
| 422 | |
| 423 | We shall want to encode group elements $X \in G$ and indices $x \in I = |
| 424 | \Nupto{|G|}$ as bit strings. To this end, we shall assume the existence of |
| 425 | integers $\ell_G, \ell_I > 0$ and functions |
| 426 | \[ |
| 427 | e_S\colon S \to \Bin^{\ell_S} |
| 428 | \quad \textrm{and} \quad |
| 429 | d_S\colon \Bin^{\ell_S} \to S \cup \{ \bot \} |
| 430 | \qquad |
| 431 | \textrm{for } S \in \{ G, I \}. |
| 432 | \] |
| 433 | with the following properties. |
| 434 | \begin{itemize} |
| 435 | \item The functions are \emph{unique} and \emph{unambiguous}, i.e., for any |
| 436 | $t \in \Bin^{\ell_S}$, we have |
| 437 | \[ d_S(t) = \begin{cases} |
| 438 | s & if there is some $s \in S$ such that $t = e_S(s)$, or \\ |
| 439 | \bot & if no such $s$ exists. |
| 440 | \end{cases} |
| 441 | \] |
| 442 | \item The functions should be \emph{efficient} to compute. Indeed, we shall |
| 443 | be assuming that the time taken for encoding and decoding is essentially |
| 444 | trivial. |
| 445 | \end{itemize} |
| 446 | Note that, as we have defined them, all encodings of group elements are the |
| 447 | same length, and similarly for encodings of indices. This is necessary for |
| 448 | the security of our protocols. |
| 449 | |
| 450 | We shall frequently abuse notation by omitting the encoding and decoding |
| 451 | functions where it is obvious that they are required. |
| 452 | |
| 453 | |
| 454 | \subsection{Games, adversaries, and oracles} |
| 455 | \label{sec:games} |
| 456 | |
| 457 | Many of the security definitions and results given here make use of |
| 458 | \emph{games}, played with an \emph{adversary}. An adversary is a |
| 459 | probabilistic algorithm. In some games, the adversary is additionally |
| 460 | equipped with \emph{oracles}, which perform computations with values chosen |
| 461 | by the adversary and secrets chosen by the game but not revealed to the |
| 462 | adversary. We impose limits on the adversary's resource usage: in |
| 463 | particular, the total time it takes, and the number of queries it makes to |
| 464 | its various oracles. Throughout, we include the size of the adversary's |
| 465 | program as part of its `time', in order to model adversaries which contain |
| 466 | large precomputed tables. |
| 467 | |
| 468 | The games provide models of someone trying to attack a construction or |
| 469 | protocol. For security, we will either define a notion of `winning' the |
| 470 | game, and require that all adversaries have only a very small probability of |
| 471 | winning, or we consider two different games and require that no adversary can |
| 472 | distinguish between the two except with very small probability. |
| 473 | |
| 474 | Our proofs make frequent use of sequences of games; see |
| 475 | \cite{Shoup:2004:SGT,Bellare:2004:CBG}. The presentation owes much to Shoup |
| 476 | \cite{Shoup:2004:SGT}. We begin with a game $\G0$ based directly on a |
| 477 | relevant security definition, and construct a sequence of games $\G1$, $\G2$, |
| 478 | \dots, each slightly different from the last. We define all of the games in |
| 479 | a sequence over the same underlying probability space -- the random coins |
| 480 | tossed by the algorithms involved -- though different games may have slightly |
| 481 | differently-defined events and random variables. Our goal in doing this is |
| 482 | to bound the probability of the adversary winning the initial game $\G0$ by |
| 483 | simultaneously (a) relating the probability of this event to that of |
| 484 | corresponding events in subsequent games, and (b) simplifying the game until |
| 485 | the probability of the corresponding event can be computed directly. |
| 486 | |
| 487 | The following simple lemma from \cite{Shoup:2001:OR} will be frequently |
| 488 | useful. |
| 489 | \begin{lemma}[Difference Lemma] |
| 490 | \label{lem:shoup} |
| 491 | Let $S$, $T$, $F$ be events. Suppose $\Pr[S|\bar F] = \Pr[T|\bar F]$. |
| 492 | Then $|\Pr[S] - \Pr[T]| \le \Pr[F]$. |
| 493 | \end{lemma} |
| 494 | \begin{proof} |
| 495 | A simple calculation: |
| 496 | \begin{eqnarray*}[rl] |
| 497 | |\Pr[S] - \Pr[T]| |
| 498 | & = |(\Pr[S|F]\Pr[F] + \Pr[S|\bar F]\Pr[\bar F]) - |
| 499 | (\Pr[T|F]\Pr[F] + \Pr[T|\bar F]\Pr[\bar F])| \\ |
| 500 | & = \Pr[F] \cdot |\Pr[S|F] - \Pr[T|F]| \\ |
| 501 | & \le \Pr[F] |
| 502 | \end{eqnarray*} |
| 503 | and we're done! |
| 504 | \end{proof} |
| 505 | |
| 506 | |
| 507 | \subsection{The random oracle model} |
| 508 | \label{sec:ro} |
| 509 | |
| 510 | In particular, most of our results will make use of the \emph{random oracle} |
| 511 | model \cite{Bellare:1993:ROP}, in which all the participants, including the |
| 512 | adversary, have access to a number of `random oracles'. A random oracle with |
| 513 | domain $D$ and range $R$ is an oracle which computes a function chosen |
| 514 | uniformly at random from the set of all such functions. (In the original |
| 515 | paper \cite{Bellare:1993:ROP}, random oracles are considered having domain |
| 516 | $\Bin^*$ and range $\Bin^\omega$; we use finite random oracles here, because |
| 517 | they're easier to work with.) |
| 518 | |
| 519 | Given a protocol proven secure in the random oracle model, we can instantiate |
| 520 | each random oracle by a supposedly-secure hash function and be fairly |
| 521 | confident that either our protocol will be similarly secure, or one of the |
| 522 | hash functions we chose has some unfortunate property. |
| 523 | |
| 524 | Proofs in the random oracle must be interpreted carefully. For example, |
| 525 | Canetti, Goldreich and Halevi \cite{Canetti:2004:ROM} show that there are |
| 526 | schemes which can be proven secure in the random oracle model but provably |
| 527 | have no secure instantiation in the standard model. |
| 528 | |
| 529 | The random oracle model is useful for constructing reductions and simulators |
| 530 | for two main reasons. |
| 531 | \begin{enumerate} |
| 532 | \item One can use the transcript of an adversary's queries to random oracles |
| 533 | in order to extract knowledge from it. |
| 534 | \item One can `program' a random oracle so as to avoid being bound by prior |
| 535 | `commitments', or to guide an adversary towards solving a selected instance |
| 536 | of some problem. |
| 537 | \end{enumerate} |
| 538 | Our proofs only make use of the first feature. This becomes particularly |
| 539 | important when we consider issues of zero-knowledge and deniability in a |
| 540 | concurrent setting, because we want to be able to claim that we retain these |
| 541 | features when the random oracle is instantiated using a cryptographic hash |
| 542 | function, and hash functions definitely aren't `programmable' in this way! |
| 543 | The former property seems rather more defensible -- one would indeed hope |
| 544 | that the only sensible way of working out (anything about) the hash of a |
| 545 | particular string is to actually compute the hash function, and the random |
| 546 | oracle model is, we hope, just giving us a `hook' into this process. |
| 547 | |
| 548 | (Our protocols can be modified to make use of bilinear pairings so as to |
| 549 | provide identity-based identification and key-exchange, using the techniques |
| 550 | of \cite{Boneh:2003:IBE}. Proving the security of the modifications we |
| 551 | discuss would involve `programming' random oracles, but this doesn't affect |
| 552 | the zero-knowledge or deniability of the resulting protocols.) |
| 553 | |
| 554 | |
| 555 | \subsection{Notation for algorithms} |
| 556 | |
| 557 | We shall have occasion to describe algorithms by means of a pseudocode. Our |
| 558 | choice of pseudocode is unlikely to be particularly controversial. We let $x |
| 559 | \gets y$ denote the action of setting $x$ to the value $y$; similarly, $x |
| 560 | \getsr Y$ denotes the action of sampling $x$ from the set $Y$ uniformly at |
| 561 | random. |
| 562 | |
| 563 | The expression $a \gets A^{O(\cdot, x)}(y)$ means `assign to $a$ the value |
| 564 | output by algorithm $A$ on input $y$, and with oracle access to the algorithm |
| 565 | which, given input $z$, computes $O(z, x)$'. |
| 566 | |
| 567 | We make use of conditional (\IF-\ELSE) and looping (\FOR-\DO and \WHILE-\DO) |
| 568 | constructions; in order to reduce the amount of space taken up, the bodies of |
| 569 | such constructions are shown by indentation only. |
| 570 | |
| 571 | We don't declare the types of our variables explicitly, assuming that these |
| 572 | will be obvious by inspection; also, we don't describe our variables' scopes |
| 573 | explicitly, leaving the reader to determine these from context. |
| 574 | |
| 575 | Finally, the notation $\Pr[\textit{algorithm} : \textit{condition}]$ denotes |
| 576 | the probability that \textit{condition} is true after running the given |
| 577 | \textit{algorithm}. |
| 578 | |
| 579 | |
| 580 | \subsection{Diffie-Hellman problems} |
| 581 | \label{sec:dhp} |
| 582 | |
| 583 | The security of our protocols is related to the hardness of the |
| 584 | computational, decisional, and gap Diffie-Hellman problems in the group $G$. |
| 585 | We define these problems and what it means for them to be `hard' here. |
| 586 | |
| 587 | The \emph{computational} Diffie-Hellman problem (CDH) is as follows: given |
| 588 | two group elements $X = x P$ and $Y = y P$, find $Z = x y P$. |
| 589 | \begin{definition}[The computational Diffie-Hellman problem] |
| 590 | Let $(G, +)$ be a cyclic group generated by $P$. For any adversary $A$, we |
| 591 | say that $A$'s \emph{success probability} at solving the computational |
| 592 | Diffie-Hellman problem in $G$ is |
| 593 | \[ \Succ{cdh}{G}(A) = |
| 594 | \Pr[ x \getsr I; y \getsr \Nupto{|G|} : A(x P, y P) = x y P ] |
| 595 | \] |
| 596 | where the probability is taken over the random choices of $x$ and $y$ and |
| 597 | any random decisions made by $A$. We say that the \emph{CDH insecurity |
| 598 | function} of $G$ is |
| 599 | \[ \InSec{cdh}(G; t) = \max_A \Succ{cdh}{G}(A) \] |
| 600 | where the maximum is taken over adversaries which complete in time $t$. |
| 601 | \end{definition} |
| 602 | Certainly, if one can compute discrete logarithms in the group $G$ (i.e., |
| 603 | given $x P$, find $x$), then one can solve the computational Diffie-Hellman |
| 604 | problem. The converse is not clear, though. Shoup \cite{Shoup:1997:LBD} |
| 605 | gives us some confidence in the difficulty of the problem by showing that a |
| 606 | \emph{generic} adversary -- i.e., one which makes no use of the specific |
| 607 | structure of a group -- has success probability no greater than $q^2/|G|$. |
| 608 | |
| 609 | This isn't quite sufficient for our purposes. Our proofs will be able to |
| 610 | come up with (possibly) a large number of guesses for the correct answer, and |
| 611 | at most one of them will be correct. Unfortunately, working out which one is |
| 612 | right seems, in general, to be difficult. This is the \emph{decision} |
| 613 | Diffie-Hellman problem (DDH), which \cite{Shoup:1997:LBD} shows, in the |
| 614 | generic group model, is about as hard as CDH. (See \cite{Boneh:1998:DDP} for |
| 615 | a survey of the decision Diffie-Hellman problem.) |
| 616 | |
| 617 | Our reference problem will be a `multiple-guess computational Diffie-Hellman |
| 618 | problem' (MCDH), which is captured by a game as follows. An adversary is |
| 619 | given a pair of group elements $(x P, y P)$, and an oracle $V(\cdot)$ which |
| 620 | accepts group elements as input. The adversary wins the game if it queries |
| 621 | $V(x y P)$. |
| 622 | \begin{definition}[The multiple-guess computational Diffie-Hellman problem] |
| 623 | \label{def:mcdh} |
| 624 | Let $(G, +)$ be a cyclic group generated by $P$. For some adversary $A$, |
| 625 | we say that $A$'s \emph{success probability} at solving the multiple-guess |
| 626 | computational Diffie-Hellman problem in $G$ is |
| 627 | \[ \Succ{mcdh}{G}(A) = \Pr[\Game{mcdh}{G}(A) = 1] \] |
| 628 | where $\Game{mcdh}{G}(A)$ is shown in figure~\ref{fig:mcdh}. We say that |
| 629 | the \emph{MCDH insecurity function of $G$} is |
| 630 | \[ \InSec{mcdh}(G; t, q_V) = \max_A \Succ{mcdh}{G}(A) \] |
| 631 | where the maximum is taken over adversaries which complete in time $t$ and |
| 632 | make at most $q_V$-oracle queries. |
| 633 | \end{definition} |
| 634 | |
| 635 | \begin{figure} |
| 636 | \begin{program} |
| 637 | $\Game{mcdh}{G}(A)$: \+ \\ |
| 638 | $w \gets 0$; \\ |
| 639 | $x \getsr \Nupto{|G|}$; $y \getsr \Nupto{|G|}$; \\ |
| 640 | $A^{V(\cdot)}(x P, y P)$; \\ |
| 641 | \RETURN $w$; |
| 642 | \next |
| 643 | Function $V(Z)$: \+ \\ |
| 644 | \IF $Z = x y P$ \THEN \\ \ind |
| 645 | $w \gets 1$; \\ |
| 646 | \RETURN $1$; \- \\ |
| 647 | \RETURN $0$; |
| 648 | \end{program} |
| 649 | |
| 650 | \caption{The multiple-guess computational Diffie-Hellman problem: |
| 651 | $\Game{mcdh}{G}(A)$} |
| 652 | \label{fig:mcdh} |
| 653 | \end{figure} |
| 654 | |
| 655 | Note that our MCDH problem is not quite the `gap Diffie-Hellman problem' |
| 656 | (GDH). The gap problem measures the intractibility of solving CDH even with |
| 657 | the assistance of an oracle for solving (restricted) decision Diffie-Hellman |
| 658 | problems in the group. Specifically, the adversary is given $(X, Y) = (x P, |
| 659 | y P)$ and tries to find $Z = x y P$, as for CDH, but now it has access to an |
| 660 | oracle $D(R, S)$ which answers $1$ if $S = x R$ and $0$ otherwise. |
| 661 | |
| 662 | Clearly MCDH is at least as hard as GDH, since our simple verification oracle |
| 663 | $V(Z)$ can be simulated with the gap problem's DDH oracle, as $D(Y, Z)$. |
| 664 | However, we can (loosely) relate the difficulty of MCDH to the difficulty of |
| 665 | CDH. |
| 666 | \begin{proposition}[Comparison of MCDH and CDH security] |
| 667 | For any cyclic group $(G, +)$, |
| 668 | \[ \InSec{mcdh}(G; t, q_V) \le q_V\,\InSec{cdh}(G; t + O(q_V)). \] |
| 669 | \end{proposition} |
| 670 | \begin{proof} |
| 671 | Let $A$ be an adversary attacking the multiple-guess computational |
| 672 | Diffie-Hellman problem in $G$, and suppose that it runs in time $t$ and |
| 673 | issues $q_V$ queries to its verification oracle. |
| 674 | |
| 675 | We use a sequence of games. Game $\G0$ is the original MCDH attack game. |
| 676 | In each game $\G{i}$, we let the event $S_i$ be the probability that the |
| 677 | adversary wins the game. |
| 678 | |
| 679 | Game $\G1$ is the same as $\G0$, except that we change the behaviour of the |
| 680 | verification oracle. Specifically, we make the oracle always return $0$. |
| 681 | We claim that this doesn't affect the adversary's probability of winning, |
| 682 | i.e., $\Pr[S_1] = \Pr[S_0]$. To see this, note that if none of the |
| 683 | adversary's $V(\cdot)$ queries was correct, then there is no change in the |
| 684 | game; conversely, if any query was correct, then the adversary will have |
| 685 | won regardless of its subsequent behaviour (which may differ arbitrarily |
| 686 | between the two games). |
| 687 | |
| 688 | We are now ready to construct from $A$ an adversary $B$ attacking the |
| 689 | standard computational Diffie-Hellman problem. |
| 690 | \begin{program} |
| 691 | Adversary $B(X, Y)$: \+ \\ |
| 692 | $n \gets 0$; \\ |
| 693 | \FOR $i \in \Nupto{q_V}$ \DO $Q_i \gets 0$; \\ |
| 694 | $A^{V(\cdot)}$; \\ |
| 695 | $r \getsr \Nupto{n}$; \\ |
| 696 | \RETURN $Q_r$; |
| 697 | \next |
| 698 | Function $D(Z')$: \+ \\ |
| 699 | $Q_n \gets Z'$; \\ |
| 700 | $n \gets n + 1$; \\ |
| 701 | \RETURN $0$; |
| 702 | \end{program} |
| 703 | Observe that $B$ provides $A$ with an accurate simulation of game $\G1$. |
| 704 | Moreover, at the end of the algorithm, we have $0 < n \le q_V$, the |
| 705 | output of $A$ is stored in $Q_{n-1}$ and the values $Q_0$, $Q_1$, \dots, |
| 706 | $Q_{n-1}$ are the values of $A$'s oracle queries. Hence, with probability |
| 707 | $Pr[S_1]$, at least of one of the $Q_i$ is the correct answer to the CDH |
| 708 | problem. Let $\epsilon = \Pr[S_1] = \Pr[S_0]$; we claim that $B$'s |
| 709 | probability of success is at least $\epsilon/q_V$. The proposition |
| 710 | follows directly from this claim and that, because $A$ was chosen |
| 711 | arbitrarily, we can maximize and count resources. |
| 712 | |
| 713 | We now prove the above claim. For $0 \le i < q_V$, let $W_i$ be the |
| 714 | event that $Q_i = x y P$, i.e., that $Q_i$ is the correct response. A |
| 715 | simple union bound shows that |
| 716 | \[ \sum_{0\le i<j} \Pr[W_i \mid n = j] \ge \epsilon. \] |
| 717 | We now perform a calculation: |
| 718 | \begin{eqnarray*}[rl] |
| 719 | \Succ{cdh}{G}(B) |
| 720 | & = \sum_{0\le i<q_V} \Pr[W_i \land r = i] \\ |
| 721 | & = \sum_{0<j\le q_V} \Pr[n = j] |
| 722 | \biggl( \sum_{0\le i<j} \Pr[W_i \land r = i \mid n = j] \biggr) \\ |
| 723 | & = \sum_{0<j\le q_V} \Pr[n = j] |
| 724 | \biggl( \frac{1}{j} \sum_{0\le i<j} \Pr[W_i \mid n = j] \biggr) \\ |
| 725 | &\ge \sum_{0<j\le q_V} \Pr[n = j] \frac{\epsilon}{j} \\ |
| 726 | &\ge \frac{\epsilon}{q_V} \sum_{0<j\le q_V} \Pr[n = j] \\ |
| 727 | & = \frac{\epsilon}{q_V}. |
| 728 | \end{eqnarray*} |
| 729 | which completes the proof. |
| 730 | \end{proof} |
| 731 | |
| 732 | |
| 733 | \subsection{Example groups and encodings} |
| 734 | |
| 735 | For nonnegative integers $0 \le n < 2^\ell$, there is a natural binary |
| 736 | encoding $N_\ell\colon \Nupto{2^\ell} \to \Bin^\ell$ which we can define |
| 737 | recursively as follows. |
| 738 | \[ N_0(0) = \emptystring \qquad |
| 739 | N_\ell(n) = \begin{cases} |
| 740 | N_{\ell-1}(n) \cat 0 & if $0 \le n < 2^{\ell-1}$ \\ |
| 741 | N_{\ell-1}(n - 2^{\ell-1}) \cat 1 & if $2^{\ell-1} \le n < 2^\ell$. |
| 742 | \end{cases} |
| 743 | \] |
| 744 | Given an encoding $a = N_\ell(n)$ we can recover $n$ as |
| 745 | \[ n = \sum_{0\le i<\ell} a[i] 2^i. \] |
| 746 | Hence, given some limit $L \le 2^\ell$, we can encode elements of $\Nupto{L}$ |
| 747 | using the functions $(e, d)$: |
| 748 | \[ e(L, \ell, n) = N_\ell(n) \qquad |
| 749 | d(L, \ell, a) = \begin{cases} |
| 750 | N_\ell(a) & if $N_\ell(a) < L$ \\ |
| 751 | \bot & otherwise |
| 752 | \end{cases} |
| 753 | \] |
| 754 | The reader can verify that the functions $e(L, \ell, \cdot)$ and $d(L, \ell, |
| 755 | \cdot)$ satisfy the requirements of section~\ref{sec:bitenc}. |
| 756 | |
| 757 | Given some $q < 2^{\ell_I}$ and $I = \Nupto{q}$, then, we can define an |
| 758 | encoding $(e_I, d_I)$ by $e_I(n) = e(q, \ell_I, n)$ and $d_I(a) = d(q, |
| 759 | \ell_I, a)$. |
| 760 | |
| 761 | Let $p$ and $q$ be primes, with $q|(p - 1)$. Then there is an order-$q$ |
| 762 | subgroup of $(\Z/p\Z)^*$. In practice, an order-$q$ element can be found |
| 763 | easily by taking elements $h \in (\Z/p\Z)^*$ at random and computing $g = |
| 764 | h^{(p-1)/2}$ until $g \ne 1$; then $G = \langle g \rangle$ is a group of $q$ |
| 765 | elements. Assuming that $p$ and $q$ are sufficiently large, the |
| 766 | Diffie-Hellman problems seem to be difficult in $G$. Some texts recommend |
| 767 | additional restrictions on $p$, in particular that $(p - 1)/2q$ be either |
| 768 | prime or the product of large primes. Primes of this form protect against |
| 769 | small-subgroup attacks; but our protocols are naturally immune to these |
| 770 | attacks, so such precautions are unnecessary here. Elements of $G$ can be |
| 771 | encoded readily, since each element $n + p\Z$ of $\Z/p\Z$ has an obvious |
| 772 | `representative' integer $n$ such that $0 \le n < p$, and given $2^{\ell_G} > |
| 773 | p$, we can encode $n$ as $e(p, \ell_G, n)$, as above. |
| 774 | |
| 775 | Alternatively, let $\F = \gf{p^f}$ be a finite field, and $E$ be an elliptic |
| 776 | curve defined over $\F$ such that the group $E(\F)$ of $\F$-rational points |
| 777 | of $E$ has a prime-order cyclic subgroup $G$. Elements of $G$ can be |
| 778 | represented as pairs of elements of $\F$. If $f = 1$, i.e., $\F = \Z/p\Z$ |
| 779 | then field elements can be encoded as above. If $p = 2$, we can represent |
| 780 | the field as $\F_2/(p(x))$ for some irreducible polynomial $p(x) \in \F_2[x]$ |
| 781 | of degree $f$. An element $r \in \F$ can then be represented by a polynomial |
| 782 | $r(x)$ with degree less than $f$, and coefficients $c_i \in \{0, 1\}$, i.e., |
| 783 | \[ r(x) = \sum_{0\le i<f} c_i x^i \] |
| 784 | and hence we can uniquely encode $r$ as an $f$-bit string $a$ such that $a[i] |
| 785 | = c_i$. |
| 786 | |
| 787 | |
| 788 | \subsection{Symmetric encryption} |
| 789 | \label{sec:sym-enc} |
| 790 | |
| 791 | Our key-exchange protocol requires a symmetric encryption scheme. Our |
| 792 | definition is fairly standard, except that, rather than specifying a |
| 793 | key-generation algorithm, we assume that key generation simply involves |
| 794 | selecting a string of a given length uniformly at random. |
| 795 | \begin{definition}[Symmetric encryption schemes] |
| 796 | A \emph{symmetric encryption scheme} $\E = (\kappa, E, D)$ consists of: |
| 797 | \begin{itemize} |
| 798 | \item an integer $\kappa \ge 0$, |
| 799 | \item a randomized \emph{encryption algorithm} $E$ which, on input $K \in |
| 800 | \Bin^\kappa$ and $p \in \Bin^*$ outputs some $c \in \Bin^*$, written $c |
| 801 | \gets E_K(p)$; |
| 802 | \item a \emph{decryption algorithm} $D$ which, on input $K \in \Bin^\kappa$ |
| 803 | and $c \in \Bin^*$ outputs some $p' \in \Bin^* \cup \{\bot\}$, written |
| 804 | $p' \gets D_K(c)$. |
| 805 | \end{itemize} |
| 806 | Furthermore, a symmetric encryption scheme must be \emph{sound}: that is, |
| 807 | if $c \gets E_K(p)$ for some $K \in \Bin^\kappa$ and $p \in \Bin^*$, and |
| 808 | $p' \gets D_K(c)$ then $p = p'$. |
| 809 | \end{definition} |
| 810 | Our security notion for symmetric encryption is the standard notion of |
| 811 | left-or-right indistinguishability of ciphertexts under chosen-ciphertext |
| 812 | attack. |
| 813 | \begin{definition} |
| 814 | [Indistinguishability under chosen-ciphertext attack (IND-CCA)] |
| 815 | \label{def:ind-cca} |
| 816 | Let $\E = (\kappa, E, D)$ be a symmetric encryption scheme, and $A$ be an |
| 817 | adversary. Let $\id{lr}_b(x_0, x_1) = x_b$ for $b \in \{0, 1\}$. Let |
| 818 | \[ P_b = |
| 819 | \Pr[K \getsr \Bin^\kappa; |
| 820 | b \gets A^{E_K(\id{lr}_b(\cdot, \cdot)), D_K(\cdot)}() : |
| 821 | b = 1] |
| 822 | \] |
| 823 | An adversary is \emph{valid} if |
| 824 | \begin{itemize} |
| 825 | \item for any query to its encryption oracle $E_K(\id{lr}_b(x_0, x_1))$ we |
| 826 | have $|x_0| = |x_1|$, and |
| 827 | \item no query to the decryption oracle $D_K(\cdot)$ is equal to any reply |
| 828 | from an encryption query. |
| 829 | \end{itemize} |
| 830 | If $A$ is valid, then we define its \emph{advantage} in attacking the |
| 831 | security of $\E$ as follows |
| 832 | \[ \Adv{ind-cca}{\E} = P_1 - P_0. \] |
| 833 | Further, we define the \emph{IND-CCA insecurity function of $\E$} to be |
| 834 | \[ \InSec{ind-cca}(\E; t, q_E, q_D) = \max_A \Adv{ind-cca}{\E}(A) \] |
| 835 | where the maximum is taken over all valid adversaries $A$ which run in time |
| 836 | $t$, and issue at most $q_E$ encryption and $q_D$ decryption queries. |
| 837 | \end{definition} |
| 838 | |
| 839 | |
| 840 | \subsection{Simulations} |
| 841 | \label{sec:sim} |
| 842 | |
| 843 | In section~\ref{sec:zk-ident}, we shall prove that our identification |
| 844 | protocol is zero-knowledge; in section~\ref{sec:denial}, we show that our |
| 845 | key-exchange protocol is deniable. In both of these proofs, we shall need to |
| 846 | demonstrate \emph{simulatability}. |
| 847 | |
| 848 | \subsubsection{General framework} |
| 849 | Consider a game in which an adversary~$A$ interacts with some `world'~$W$, |
| 850 | which we shall represent as a probabilistic algorithm. The world may in fact |
| 851 | represent a number of honest parties communicating in a concurrent fashion, |
| 852 | but we can consider them as a single algorithm for our present purposes. |
| 853 | |
| 854 | Initially the world and the adversary are both given the same \emph{common |
| 855 | input}~$c$; in addition, the world is given a \emph{private input}~$w$. |
| 856 | Both $c$ and~$w$ are computed by an \emph{initialization function}~$I$, which |
| 857 | is considered to be part of the definition of the game. Finally, the |
| 858 | adversary decides somehow that it has finished interacting, and outputs a |
| 859 | value~$a$. All this we notate as |
| 860 | \[ (w, c) \gets I(); a \gets A^{W(w, c)}(c). \] |
| 861 | This game is \emph{simulatable} if there is an algorithm~$S$ -- the |
| 862 | \emph{simulator} -- which can compute the same things as~$A$, but all by |
| 863 | itself without interacting with the world. That is, we run the simulator on |
| 864 | the common input~$c$, allowing it to interact in some way with the |
| 865 | adversary~$A$, and finally giving us an output~$s$. |
| 866 | \[ (w, c) \gets I(); s \gets S^A(c). \] |
| 867 | We shall say that the simulator is \emph{effective} if it's difficult to tell |
| 868 | whether a given string was output by the adversary after interacting with the |
| 869 | world, or by the simulator running by itself. That is, for any algorithm~$D$ |
| 870 | -- a \emph{distinguisher} -- running in some bounded amount of time, its |
| 871 | advantage |
| 872 | \begin{spliteqn*} |
| 873 | \Pr[(w, c) \gets I(); a \gets A^{W(w, c)}(c); |
| 874 | b \gets D(c, a) : b = 1] - {} \\ |
| 875 | \Pr[(w, c) \gets I(); s \gets S^A(c); b \gets D(c, s) : b = 1] |
| 876 | \end{spliteqn*} |
| 877 | is small. (Note that we gave the distinguisher the common input as well as |
| 878 | the output of the adversary or the simulator.) |
| 879 | |
| 880 | It's usual to study \emph{transcripts} of interactions in these kinds of |
| 881 | settings. We are considering arbitrary adversarial outputs here, so this |
| 882 | certainly includes adversaries which output a transcript of their |
| 883 | interactions. Indeed, for any adversary~$A$, we could construct an |
| 884 | adversary~$A_T$ which performs the same computation, and outputs the same |
| 885 | result, but also includes a complete transcript of $A$'s interaction with the |
| 886 | world. Therefore we're just providing additional generality. |
| 887 | |
| 888 | \subsubsection{Random oracles} |
| 889 | We shall be considering interactions in which all the parties have access to |
| 890 | several random oracles. We could simply say that the random oracles are part |
| 891 | of the world~$W$. In the setting described above, only the adversary |
| 892 | actually interacts with the world (and therefore would be able to query |
| 893 | random oracles). The simulator would be forced to `make up' its own random |
| 894 | oracle, and the distinguisher would have to study the distributions of the |
| 895 | random-oracle queries and their responses to make up its mind about which it |
| 896 | was given. |
| 897 | |
| 898 | However, this would be a poor model for the real world, since once we |
| 899 | instantiate the random oracle with a hash function, we know that everyone |
| 900 | would in actually be able to compute the hash function for themselves. Thus |
| 901 | a distinguisher in the real world would be able to tell the difference |
| 902 | immediately between a real interaction and the simulated transcript, since |
| 903 | the `random oracle' queries recorded in the latter would be wrong! |
| 904 | |
| 905 | Therefore we decide not to include the random oracles as part of the world, |
| 906 | but instead allow all the participants -- adversary, simulator and |
| 907 | distinguisher -- access to them. If we denote by~$\mathcal{H} = (H_0, H_1, |
| 908 | \ldots, H_{n-1})$ the collection of random oracles under consideration, the |
| 909 | expression for the distinguisher's advantage becomes |
| 910 | \begin{spliteqn*} |
| 911 | \Pr[(w, c) \gets I(); a \gets A^{W(w, c), \mathcal{H}}(c); |
| 912 | b \gets D^{\mathcal{H}}(c, a) : b = 1] - {} \\ |
| 913 | \Pr[(w, c) \gets I(); s \gets S^{A, \mathcal{H}}(c); |
| 914 | b \gets D^{\mathcal{H}}(c, s) : b = 1]. |
| 915 | \end{spliteqn*} |
| 916 | |
| 917 | \subsubsection{Auxiliary inputs} |
| 918 | If an adversary's output can be effectively simulated, then we can |
| 919 | confidently state that the adversary `learnt' very little of substance from |
| 920 | its interaction, and certainly very little it can \emph{prove} to anyone |
| 921 | else. However, as we have described the setting so far, we fix an adversary |
| 922 | before we choose inputs to the world, so our model says little about what an |
| 923 | adversary which has already acquired some knowledge might learn beyond that. |
| 924 | For example, an adversary might overhear some other conversation between |
| 925 | honest parties and be able to use this to its advantage. |
| 926 | |
| 927 | To this end, we give the adversary an \emph{auxiliary input}~$u$, computed by |
| 928 | an algorithm~$U$. We give $U$ both $c$ and $w$, in order to allow the |
| 929 | adversary to gain some (possibly partial) knowledge of the secrets of the |
| 930 | other parties. We also allow $U$ access to the random oracles~$\mathcal{H}$, |
| 931 | because clearly in the `real world' it would be ridiculous to forbid such an |
| 932 | algorithm from computing a publicly-known hash function. |
| 933 | |
| 934 | The simulator and distinguisher are also given the auxiliary input. The |
| 935 | simulator is meant to represent the adversary's ability to compute things on |
| 936 | its own, without interacting with the world, and since the adversary is given |
| 937 | the auxiliary input, the simulator must be too. The distinguisher must be |
| 938 | the auxiliary input because otherwise the simulator could just `make up' |
| 939 | plausible-looking inputs. |
| 940 | |
| 941 | \subsubsection{Resource limits} |
| 942 | We shall not allow our algorithms to perform completely arbitrary |
| 943 | computations and interactions. Instead, we impose limits on the amount of |
| 944 | time they are allowed to take, the number of random-oracle queries they make, |
| 945 | and so on. Specifically, we are interested in |
| 946 | \begin{itemize} |
| 947 | \item the time $t_A$ taken by the adversary and $t_D$ taken by the |
| 948 | distinguisher, |
| 949 | \item the number of oracle queries $\mathcal{Q}_A = (q_{A,0}, q_{A,1}, |
| 950 | \ldots, q_{A,n-1})$ made by the adversary, and $\mathcal{Q}_D$ made by the |
| 951 | distinguisher, |
| 952 | \item a number of resource bounds $\mathcal{R}$ on the adversary's |
| 953 | interaction with the world (e.g., number of messages of various kinds sent |
| 954 | and received), and |
| 955 | \item a number of bounds $\mathcal{U}$ on the contents of the adversary's |
| 956 | auxiliary input~$u$. |
| 957 | \end{itemize} |
| 958 | Sometimes we shall not be interested in proving simulatability of adversaries |
| 959 | with auxiliary inputs. We write $\mathcal{U} = 0$ to indicate that auxiliary |
| 960 | output is not allowed. |
| 961 | |
| 962 | \subsubsection{World syntax} |
| 963 | It will be worth our while being more precise about what a `world' actually |
| 964 | is, syntactically. We define a world to be a single, randomized algorithm |
| 965 | taking inputs $(\iota, \sigma, \tau, \mu) \in (\Bin^*)^4$; the algorithm's |
| 966 | output is a pair $(\sigma', \rho) \in (\Bin^*)^2$. We show how the |
| 967 | adversary's interaction is mapped on to this world algorithm in |
| 968 | figure~\ref{fig:sim-world}. |
| 969 | \begin{itemize} |
| 970 | \item The `input' $\iota$ is the result of the initialization function~$I$. |
| 971 | That is, it is the pair $(w, c)$ of the world's private input and the |
| 972 | common input. |
| 973 | \item The `state' $\sigma$ is empty on the world's first invocation; on each |
| 974 | subsequent call, the value of the world's output $\sigma'$ is passed back. |
| 975 | In this way, the world can maintain state. |
| 976 | \item The `type $\tau$ is a token giving the type of invocation this is. |
| 977 | \item The `message' $\mu$ is any other information passed in; its form will |
| 978 | typically depend on the type~$\tau$ of the invocation. |
| 979 | \item The `new state' $\sigma'$ is the value of $\sigma$ to pass to the next |
| 980 | invocation of the world. |
| 981 | \item The `reply $\rho$ is the actual output of the invocation. |
| 982 | \end{itemize} |
| 983 | There are two special invocation types. The adversary is \emph{forbidden} |
| 984 | from making special invocations. |
| 985 | \begin{itemize} |
| 986 | \item The special invocation type $\cookie{init}$ is used to allow the world to |
| 987 | prepare an initial state. The world is invoked as |
| 988 | \[ W^{\mathcal{H}}(\iota, \emptystring, \cookie{init}, \emptystring) \] |
| 989 | and should output an initial state $\sigma'$. The world's reply $\rho$ is |
| 990 | ignored. (Recall that $\emptystring$ represents the empty string.) |
| 991 | \item The special invocation type $\cookie{random}$ is used to inform the |
| 992 | world that the adversary has issued a random oracle query. The world is |
| 993 | invoked as |
| 994 | \[ W^{\mathcal{H}}(\iota, \sigma, \cookie{random}, (i, x, h)) \] |
| 995 | to indicate that the adversary has queried its random oracle $H_i(\cdot)$ |
| 996 | on the input $x$, giving output~$h$. The world may output an updated state |
| 997 | $\sigma'$; its reply $\rho$ is ignored. |
| 998 | \end{itemize} |
| 999 | The latter special query is a technical device used to allow the `fake-world' |
| 1000 | simulators we define below to be aware of the adversary's random oracle |
| 1001 | queries without being able to `program' the random oracle. Including it here |
| 1002 | does little harm, and simplifies the overall exposition. |
| 1003 | |
| 1004 | \begin{figure} |
| 1005 | \begin{program} |
| 1006 | Interaction $A^{W(w, c), \mathcal{H}}(c, u)$: \+ \\ |
| 1007 | $(\sigma, \rho) \gets |
| 1008 | W((w, c), \emptystring, \cookie{init}, \emptystring)$; \\ |
| 1009 | $a \gets A^{\id{world}(\cdot, \cdot), |
| 1010 | \id{random}(\cdot, \cdot)}(c, u)$; \\ |
| 1011 | \RETURN $a$; |
| 1012 | \newline |
| 1013 | Function $\id{world}(\tau, \mu)$: \+ \\ |
| 1014 | \IF $\tau \in \{\cookie{init}, \cookie{random}\}$ \THEN |
| 1015 | \RETURN $\bot$; \\ |
| 1016 | $(\sigma, \rho) \gets W((w, c), \sigma, \tau, \mu)$; \\ |
| 1017 | \RETURN $\rho$; \- |
| 1018 | \next |
| 1019 | Function $\id{random}(i, x)$: \+ \\ |
| 1020 | $h \gets H_i(x)$; \\ |
| 1021 | $(\sigma, \rho) \gets |
| 1022 | W((w, c), \sigma, \cookie{random}, (i, x, h))$; \\ |
| 1023 | \RETURN $h$; |
| 1024 | \end{program} |
| 1025 | |
| 1026 | \caption{Interacting with a world: Interaction $A^{W, \mathcal{H}}$} |
| 1027 | \label{fig:sim-world} |
| 1028 | \end{figure} |
| 1029 | |
| 1030 | \subsubsection{Definitions} |
| 1031 | We are now ready to begin making definitions. |
| 1032 | \begin{definition}[Simulation security] |
| 1033 | \label{def:sim} |
| 1034 | Consider the game described above, with the initialization function~$I$, |
| 1035 | and the world~$W$: let $A$ be an adversary, and let~$U$ be an |
| 1036 | auxiliary-input function; let $S$ be a simulator, and let $D$ be a |
| 1037 | distinguisher. We define $D$'s \emph{advantage against $S$'s simulation of |
| 1038 | $A$'s interaction with~$W$ with auxiliary inputs provided by~$U$} to be |
| 1039 | \[ \Adv{sim}{W, I, S}(A, U, D) = |
| 1040 | \Pr[\Game{real}{W, I, S}(A, U, D) = 1] - |
| 1041 | \Pr[\Game{sim}{W, I, S}(A, U, D)= 1] |
| 1042 | \] |
| 1043 | where the games are as shown in figure~\ref{fig:sim}. |
| 1044 | Furthermore, we define the \emph{simulator's insecurity function} to be |
| 1045 | \[ \InSec{sim}(W, I, S; |
| 1046 | t_D, t_A, \mathcal{Q}_D, \mathcal{Q}_A, \mathcal{R}, \mathcal{U}) = |
| 1047 | \max_{D, A, U} \Adv{sim}{W, I, S}(A, U, D) |
| 1048 | \] |
| 1049 | where the maximum is taken over all distinguishers~$D$ running in |
| 1050 | time~$t_D$ and making at most $\mathcal{Q}_D$ random-oracle queries, and |
| 1051 | all adversaries~$A$ running in time~$t_A$, making at most $\mathcal{Q}_A$ |
| 1052 | random-oracle queries, not exceeding the other stated resource |
| 1053 | bounds~$\mathcal{R}$ on its interaction with~$W$, and auxiliary-input |
| 1054 | functions producing output not exceeding the stated bounds~$\mathcal{U}$. |
| 1055 | \end{definition} |
| 1056 | \begin{remark} |
| 1057 | The usual definitions of zero-knowledge, for example, require the simulator |
| 1058 | to work for all choices of inputs (common, private and auxiliary), rather |
| 1059 | than for random choices. Our definition therefore looks weaker. Our proof |
| 1060 | of zero-knowledge actually carries through to the traditional |
| 1061 | stronger-looking definition. Critically, however, the standard |
| 1062 | universal quantification over inputs fails to capture deniability in the |
| 1063 | random oracle model, since the inputs can't therefore depend on the random |
| 1064 | oracle. Our formulation therefore actually gives \emph{stronger} |
| 1065 | deniability that the usual one. |
| 1066 | \end{remark} |
| 1067 | |
| 1068 | \begin{figure} |
| 1069 | \begin{program} |
| 1070 | $\Game{real}{W, I, S}(A, U, D)$: \+ \\ |
| 1071 | $(w, c) \gets I()$; \\ |
| 1072 | $u \gets U^{\mathcal{H}}(w, c)$; \\ |
| 1073 | $a \gets A^{W(w, c), \mathcal{H}}(c, u)$; \\ |
| 1074 | $b \gets D^{\mathcal{H}}(c, u, a)$; \\ |
| 1075 | \RETURN $b$; |
| 1076 | \next |
| 1077 | $\Game{sim}{W, I, S}(A, U, D)$: \+ \\ |
| 1078 | $(w, c) \gets I()$; \\ |
| 1079 | $u \gets U^{\mathcal{H}}(w, c)$; \\ |
| 1080 | $s \gets S^{A, \mathcal{H}}(c, u)$; \\ |
| 1081 | $b \gets D^{\mathcal{H}}(c, u, s)$; \\ |
| 1082 | \RETURN $b$; |
| 1083 | \end{program} |
| 1084 | |
| 1085 | \caption{Games for simulation: $\Game{real}{W, I}$ and $\Game{sim}{W, I}$} |
| 1086 | \label{fig:sim} |
| 1087 | \end{figure} |
| 1088 | |
| 1089 | \subsubsection{Fake-world simulators} |
| 1090 | The simulators we shall be considering in the present paper are of a specific |
| 1091 | type which we call `fake-world simulators'. They work by running the |
| 1092 | adversary in a fake `cardboard cut-out' world, and attempting to extract |
| 1093 | enough information from the adversary's previous interactions and random |
| 1094 | oracle queries to maintain a convincing illusion. |
| 1095 | |
| 1096 | That is, the behaviour of a fake-world simulator~$S$ is simply to allow the |
| 1097 | adversary to interact with a `fake world'~$W'$, which was not given the world |
| 1098 | private input. That is, there is some world $W'$ such that |
| 1099 | \[ S^{A, \mathcal{H}}(c, u) \equiv A^{W'(u, c), \mathcal{H}}(c, u) \] |
| 1100 | Fake-world simulators are convenient because they allow us to remove from |
| 1101 | consideration the distinguisher~$D$ as the following definition shows. |
| 1102 | \begin{definition}[Fake-world simulation security] |
| 1103 | \label{def:fakesim} |
| 1104 | Let $I$, $W$ and $U$ be as in definition~\ref{def:sim}. Let $A$ be an |
| 1105 | adversary which outputs a single bit. Let $S$ be a fake-world simulator. |
| 1106 | We define $A$'s \emph{advantage against $S$'s fake-world simulation of $W$ |
| 1107 | with auxiliary inputs provided by~$U$} to be |
| 1108 | \begin{spliteqn*} |
| 1109 | \Adv{fw}{W, I, S}(A, U) = |
| 1110 | \Pr[(w, c) \gets I(); u \gets U^{\mathcal{H}}(w, c); |
| 1111 | b \gets A^{W(w, c), \mathcal{H}}(c, u) : b = 1] - {} \\ |
| 1112 | \Pr[(w, c) \gets I(); u \gets U^{\mathcal{H}}(w, c); |
| 1113 | b \gets S^{A, \mathcal{H}}(c, u) : b = 1] |
| 1114 | \end{spliteqn*} |
| 1115 | Furthermore, we define the \emph{simulator's insecurity function} to be |
| 1116 | \[ \InSec{fw}(W, I, S; |
| 1117 | t_D, t, \mathcal{Q}, \mathcal{R}, \mathcal{U}) = |
| 1118 | \max_{A, U} \Adv{fw}{W, I, S}(A, U) |
| 1119 | \] |
| 1120 | where the maximum is taken over all adversaries~$A$ running in time~$t$, |
| 1121 | making at most $\mathcal{Q}$ random-oracle queries, not exceeding the other |
| 1122 | stated resource bounds~$\mathcal{R}$ on its interaction with~$W$, and |
| 1123 | auxiliary-input functions producing output not exceeding the stated |
| 1124 | bounds~$\mathcal{U}$. |
| 1125 | \end{definition} |
| 1126 | It remains for us to demonstrate that this is a valid way of analysing |
| 1127 | simulators; the following simple proposition shows that this is indeed the |
| 1128 | case. |
| 1129 | \begin{proposition}[Fake-world simulation] |
| 1130 | \label{prop:fakesim} |
| 1131 | Let $I$ be an initialization function and let $W$ be a world. Then, for |
| 1132 | any fake-world simulator~$S$, |
| 1133 | \[ \InSec{sim}(W, I, S; t_D, t_A, \mathcal{Q}_D, \mathcal{Q}_A, |
| 1134 | \mathcal{R}, \mathcal{U}) \le |
| 1135 | \InSec{fw}(W, I, S; t_A + t_D, \mathcal{Q}_D + \mathcal{Q}_A, |
| 1136 | \mathcal{R}, \mathcal{U}) |
| 1137 | \] |
| 1138 | (where addition of query bounds $\mathcal{Q}$ is done elementwise). |
| 1139 | \end{proposition} |
| 1140 | \begin{proof} |
| 1141 | Let $W$ and $I$ as in the proposition statement be given; also let a |
| 1142 | distinguisher~$D$ running in time~$t_D$ and making $\mathcal{Q}_D$ |
| 1143 | random-oracle queries, an adversary~$A$ running in time~$t_A$ and making |
| 1144 | $\mathcal{Q}_A$ random-oracle queries and interacting with its world within |
| 1145 | the stated bounds~$\mathcal{R}$, an auxiliary-input function~$U$ satisfying |
| 1146 | the constraints~$\mathcal{U}$ on its output, and a fake-world simulator~$S$ |
| 1147 | all be given. |
| 1148 | |
| 1149 | We construct an adversary~$B$ outputting a single bit as follows |
| 1150 | \begin{program} |
| 1151 | Adversary $B^{W, \mathcal{H}}(c, u)$: \+ \\ |
| 1152 | $a \gets A^{W, \mathcal{H}}(c, u)$; \\ |
| 1153 | $b \gets D^{\mathcal{H}}(c, u, a)$; \\ |
| 1154 | \RETURN $b$; |
| 1155 | \end{program} |
| 1156 | A glance at definitions \ref{def:sim} and~\ref{def:fakesim} and the |
| 1157 | resources used by $B$ shows that |
| 1158 | \[ \Adv{sim}{W, I, S}(A, U) = \Adv{fw}{W, I, S}(B, U) |
| 1159 | \le \InSec{fw}(W, I, S; t_D + t_A, \mathcal{Q}_D + \mathcal{Q}_A, |
| 1160 | \mathcal{R}, \mathcal{U}) |
| 1161 | \] |
| 1162 | as required. |
| 1163 | \end{proof} |
| 1164 | |
| 1165 | %%%-------------------------------------------------------------------------- |
| 1166 | |
| 1167 | \section{A zero-knowledge identification scheme} |
| 1168 | \label{sec:zk-ident} |
| 1169 | |
| 1170 | |
| 1171 | \subsection{Description} |
| 1172 | |
| 1173 | Here we present a simple zero-knowledge identification scheme. Fix some |
| 1174 | group $G$. Suppose Alice chooses a private key $x \inr \Nupto{|G|}$, and |
| 1175 | publishes the corresponding public key $X = x P$. Let $H_I\colon G^2 \to |
| 1176 | \Bin^{\ell_I}$ be a secure hash function. Here's a simple protocol which |
| 1177 | lets her prove her identity to Bob. |
| 1178 | \begin{enumerate} |
| 1179 | \item Bob selects a random $r \inr \Nupto{|G|}$, and computes $R = r P$, $Y = |
| 1180 | r X$, and $c = r \xor H_I(R, Y)$. He sends the pair $(R, c)$ to Alice as |
| 1181 | his \emph{challenge}. |
| 1182 | \item Alice receives $(R, c)$. She computes $Y' = x R$ and $r' = c \xor |
| 1183 | H_I(R', Y')$, and checks that $R = r' P$. If so, she sends $Y'$ as her |
| 1184 | \emph{response}; otherwise she sends $\bot$. |
| 1185 | \item Bob receives $Y'$ from Alice. He checks that $Y' = Y$. If so, he |
| 1186 | accepts that he's talking to Alice; otherwise he becomes suspicious. |
| 1187 | \end{enumerate} |
| 1188 | We name this the Wrestlers Identification Protocol in~$G$, $\Wident^G$ (we |
| 1189 | drop the superscript to refer to the protocol in general, or when no |
| 1190 | ambiguity is likely to result). A summary is shown in |
| 1191 | figure~\ref{fig:wident}. |
| 1192 | |
| 1193 | \begin{figure} |
| 1194 | \begin{description} |
| 1195 | \item[Setup] Group $G = \langle P \rangle$; $|G| = q$ is prime. |
| 1196 | $H_I(\cdot, \cdot)$ is a secure hash. |
| 1197 | \item[Private key] $x \inr \Nupto{q}$. |
| 1198 | \item[Public key] $X = x P$. |
| 1199 | \item[Challenge] $(R, c)$ where $r \inr \Nupto{q}$, $R = r P$, $c = r \xor |
| 1200 | H_I(R, r X)$. |
| 1201 | \item[Response] $x R = r X$ if $R = (c \xor H_I(R, x R)) P$; otherwise |
| 1202 | $\bot$. |
| 1203 | \end{description} |
| 1204 | |
| 1205 | \caption{Summary of the Wrestlers Identification Protocol, $\Wident$} |
| 1206 | \label{fig:wident} |
| 1207 | \end{figure} |
| 1208 | |
| 1209 | |
| 1210 | \subsection{Security} |
| 1211 | |
| 1212 | In order to evaluate the security of our protocol, we present a formal |
| 1213 | description of the algorithms involved in figure~\ref{fig:wident}. Here, the |
| 1214 | hash function $H_I(\cdot, \cdot)$ is modelled as a random oracle. |
| 1215 | |
| 1216 | \begin{figure} |
| 1217 | \begin{program} |
| 1218 | Function $\id{setup}()$: \+ \\ |
| 1219 | $x \getsr \Nupto{q}$; \\ |
| 1220 | $X \gets x P$; \\ |
| 1221 | \RETURN $(x, X)$; |
| 1222 | \next |
| 1223 | Function $\id{challenge}^{H_I(\cdot, \cdot)}(R, c, X)$: \+ \\ |
| 1224 | $r \getsr \Nupto{q}$; \\ |
| 1225 | $R \gets r P$; $Y \gets r X$; \\ |
| 1226 | $h \gets H_I(R, Y)$; $c \gets r \xor h$; \\ |
| 1227 | \RETURN $(Y, R, c)$; \- \\[\medskipamount] |
| 1228 | Function $\id{verify}(Y, Y')$: \+ \\ |
| 1229 | \IF $Y' = Y$ \THEN \RETURN $1$; \\ |
| 1230 | \RETURN $0$; |
| 1231 | \next |
| 1232 | Function $\id{response}^{H_I(\cdot, \cdot)}(R, c, x)$: \+ \\ |
| 1233 | $Y' \gets x R$; \\ |
| 1234 | $h \gets H_I(R', Y')$; $r' \gets c \xor h$; \\ |
| 1235 | \IF $R \ne r' P$ \THEN \RETURN $\bot$; \\ |
| 1236 | \RETURN $Y'$; |
| 1237 | \end{program} |
| 1238 | |
| 1239 | \caption{Functions implementing $\Wident$ in the random oracle model} |
| 1240 | \label{fig:wident-ro} |
| 1241 | \end{figure} |
| 1242 | |
| 1243 | \subsubsection{Completeness} |
| 1244 | Suppose that Bob really is talking to Alice. Note that $Y' = x R = x (r P) = |
| 1245 | r (x P) = r X = Y$. Hence $r' = c \xor H_I(R', Y') = c \xor H_I(R, Y) = r$, |
| 1246 | so $r' P = r P = R$, so Alice returns $Y' = Y$ to Bob. Therefore $\Wident$ |
| 1247 | is \emph{complete}: if Bob really is communicating with Alice then he |
| 1248 | accepts. |
| 1249 | |
| 1250 | \subsubsection{Soundness} |
| 1251 | We next show that impersonating Alice is difficult. The natural way to prove |
| 1252 | this would be to give an adversary a challenge and prove that its probability |
| 1253 | of giving a correct response is very small. However, we prove a stronger |
| 1254 | result: we show that if the adversary can respond correctly to any of a large |
| 1255 | collection of challenges then it can solve the MCDH problem. |
| 1256 | |
| 1257 | Consider the game $\Game{imp}{\Wident}$ shown in |
| 1258 | figure~\ref{fig:wident-sound}. An adversary's probability of successfully |
| 1259 | impersonating Alice in our protocol, by correctly responding to any one of |
| 1260 | $n$ challenges, is exactly its probability of winning the game (i.e., causing |
| 1261 | it to return $1$). |
| 1262 | |
| 1263 | \begin{figure} |
| 1264 | \begin{program} |
| 1265 | $\Game{imp-$n$}{\Wident}(A)$: \+ \\ |
| 1266 | $H_I \getsr \Func{G^2}{\Bin^{\ell_I}}$; \\ |
| 1267 | $(x, X) \gets \id{setup}()$; \\ |
| 1268 | $\id{win} \gets 0$; \\ |
| 1269 | $\Xid{R}{map} \gets \emptyset$; \\ |
| 1270 | $\mathbf{c} \gets \id{challenges}(n)$; \\ |
| 1271 | $(R', Y') \gets A^{H_I(\cdot, \cdot), \id{check}(\cdot, \cdot)} |
| 1272 | (X, \mathbf{c})$; \\ |
| 1273 | \RETURN $\id{win}$; |
| 1274 | \newline |
| 1275 | Function $\id{challenges}(n)$: \+ \\ |
| 1276 | \FOR $i \in \Nupto{n}$ \DO \\ \ind |
| 1277 | $(Y, R, c) \gets \id{challenge}^{H_I(\cdot, \cdot)}$; \\ |
| 1278 | $\Xid{R}{map} \gets \Xid{R}{map} \cup \{ R \mapsto Y \}$; \\ |
| 1279 | $\mathbf{c}[i] \gets (R, c)$; \- \\ |
| 1280 | \RETURN $\mathbf{c}$; \next |
| 1281 | Function $\id{check}(R', Y')$: \\ |
| 1282 | \IF $R' \notin \dom \Xid{R}{map}$ \THEN \RETURN $0$; \\ |
| 1283 | $Y \gets \Xid{R}{map}(R')$; \\ |
| 1284 | \IF $\id{verify}(Y, Y')$ \THEN \\ \ind |
| 1285 | $\id{win} \gets 1$; \\ |
| 1286 | \RETURN $1$; \- \\ |
| 1287 | \RETURN $0$; |
| 1288 | \end{program} |
| 1289 | |
| 1290 | \caption{Soundness of $\Wident$: $\Game{imp-$n$}{\Wident}(A)$} |
| 1291 | \label{fig:wident-sound} |
| 1292 | \end{figure} |
| 1293 | |
| 1294 | \begin{theorem}[Soundness of $\Wident$] |
| 1295 | \label{thm:wident-sound} |
| 1296 | Let $A$ be any adversary running in time $t$ and making $q_I$ queries to |
| 1297 | its random oracle, and $q_V$ queries to its verification oracle. Let $G$ |
| 1298 | be a cyclic group. Then |
| 1299 | \[ \Pr[\Game{imp-$n$}{\Wident^G}(A) = 1] \le |
| 1300 | \InSec{mcdh}(G; t', q_I + q_V) |
| 1301 | \] |
| 1302 | where $t' = t + O(q_I) + O(q_V)$. |
| 1303 | \end{theorem} |
| 1304 | \begin{remark} |
| 1305 | Note that the security bound here is \emph{independent} of the value of |
| 1306 | $n$. |
| 1307 | \end{remark} |
| 1308 | \begin{proof} |
| 1309 | We prove this by defining a sequence of games $\G{i}$. The first will be |
| 1310 | the same as the attack game $\Game{imp-$n$}{\Wident}(A)$ and the others |
| 1311 | will differ from it in minor ways. In each game $\G{i}$, let $S_i$ be the |
| 1312 | event that $A$ wins the game -- i.e., that it successfully impersonates the |
| 1313 | holder of the private key~$x$. |
| 1314 | |
| 1315 | Let game $\G0$ be the attack game $\Game{imp}{\Wident}(A)$, and let $(R', |
| 1316 | Y')$ be the output of $A$ in the game. |
| 1317 | |
| 1318 | We define a new game $\G1$ which is the same as $\G0$, except that we query |
| 1319 | the random oracle $H_I$ at $(R', Y')$ whenever the adversary queries |
| 1320 | $\id{check}(R', Y')$. (We don't charge the adversary for this.) This |
| 1321 | obviously doesn't affect the adversary's probability of winning, so |
| 1322 | $\Pr[S_1] = \Pr[S_0]$. |
| 1323 | |
| 1324 | Game $\G2$ is like $\G1$, except that we change the way we generate |
| 1325 | challenges and check their responses. Specifically, we new functions |
| 1326 | $\id{challenges}_2$ and $\id{check}_2$, as shown in |
| 1327 | figure~\ref{fig:wident-sound-2}. |
| 1328 | |
| 1329 | \begin{figure} |
| 1330 | \begin{program} |
| 1331 | Function $\id{challenges}_2(n)$: \+ \\ |
| 1332 | $r^* \getsr I$; $R^* \gets r^* P$; $Y^* \gets r^* X$; \\ |
| 1333 | \FOR $i \in \Nupto{n}$ \DO \\ \ind |
| 1334 | $r \getsr I$; $R \gets r R^*$; $Y \gets r Y^*$; \\ |
| 1335 | $h \gets H_I(R, Y)$; $c \gets r \xor h$; \\ |
| 1336 | $\Xid{R}{map} \gets \Xid{R}{map} \cup \{ R \mapsto r \}$; \\ |
| 1337 | $\mathbf{c}[i] \gets (R, c)$; \- \\ |
| 1338 | \RETURN $\mathbf{c}$; |
| 1339 | \next |
| 1340 | Function $\id{check}_2(R', Y')$: \+ \\ |
| 1341 | \IF $R' \notin \dom \Xid{R}{map}$ \THEN \RETURN $0$; \\ |
| 1342 | $r \gets \Xid{R}{map}(R')$; \\ |
| 1343 | \IF $\id{verify}(Y^*, Y'/r)$ \THEN \\ \ind |
| 1344 | $\id{win} \gets 1$; \\ |
| 1345 | \RETURN $1$; \- \\ |
| 1346 | \RETURN $0$; |
| 1347 | \end{program} |
| 1348 | |
| 1349 | \caption{Soundness of $\Wident$: $\id{challenges}_2$ and $\id{check}_2$} |
| 1350 | \label{fig:wident-sound-2} |
| 1351 | \end{figure} |
| 1352 | |
| 1353 | While we're generating and checking challenges in a more complicated way |
| 1354 | here, we're not actually changing the distribution of the challenges, or |
| 1355 | changing the winning condition. Hence $\Pr[S_2] = \Pr[S_1]$. |
| 1356 | |
| 1357 | Now we change the rules again. Let $\G3$ be the same as $\G2$ except that |
| 1358 | we change the winning condition. Instead, we say that the adversary wins |
| 1359 | if any of the queries to its random oracle $H_I(R', Y')$ would be a correct |
| 1360 | response -- i.e., $\id{check}_2(R', Y')$ would return $1$. Since we query |
| 1361 | the oracle on $(R', Y')$ on its behalf at the end of the game, no adversary |
| 1362 | can do worse in this game than it does in $\G2$, so we have $\Pr[S_3] \ge |
| 1363 | \Pr[S_2]$. (It's not hard to see that this only helps quite stupid |
| 1364 | adversaries. We can transform any adversary into one for which equality |
| 1365 | holds here.) |
| 1366 | |
| 1367 | Finally, let $\G4$ be the same as $\G3$ except that we change the way we |
| 1368 | generate challenges again: rather than computing $h$ and setting $c \gets h |
| 1369 | \xor r$, we just choose $c$ at random. Specifically, we use the new |
| 1370 | function, $\id{challenges}_4$, shown in figure~\ref{fig:wident-sound-4}. |
| 1371 | |
| 1372 | \begin{figure} |
| 1373 | \begin{program} |
| 1374 | Function $\id{challenges}_4(n)$: \+ \\ |
| 1375 | $r^* \getsr I$; $R^* \gets r^* P$; $Y^* \gets r^* X$; \\ |
| 1376 | \FOR $i \in \Nupto{n}$ \DO \\ \ind |
| 1377 | $r \getsr I$; $R \gets r R^*$; \\ |
| 1378 | $c \getsr \Bin^{\ell_I}$; \\ |
| 1379 | $\Xid{R}{map} \gets \Xid{R}{map} \cup \{ R \mapsto r \}$; \\ |
| 1380 | $\mathbf{c}[i] \gets (R, c)$; \- \\ |
| 1381 | \RETURN $\mathbf{c}$; |
| 1382 | \end{program} |
| 1383 | |
| 1384 | \caption{Soundness of $\Wident$: $\id{challenges}_4$} |
| 1385 | \label{fig:wident-sound-4} |
| 1386 | \end{figure} |
| 1387 | |
| 1388 | Since $H_I(\cdot, \cdot)$ is a random function, the adversary can only |
| 1389 | distinguish $\G4$ from $\G3$ if it queries its random oracle at some $(R, r |
| 1390 | Y^*)$. But if it does this, then by the rule introduced in $\G3$ it has |
| 1391 | already won. Therefore we must have $\Pr[S_4] = \Pr[S_3]$. |
| 1392 | |
| 1393 | Our $\id{challenges}_4$ function is interesting, since it doesn't actually |
| 1394 | make use of $r^*$ or $Y^*$ when generating its challenges. This gives us |
| 1395 | the clue we need to bound $\Pr[S_4]$: we can use adversary $A$ to solve the |
| 1396 | multiple-guess Diffie-Hellman problem in $G$ by simulating the game $\G4$. |
| 1397 | Specifically, we define the adversary $B$ as shown in |
| 1398 | figure~\ref{fig:wident-sound-cdh}. That is, for each query $A$ makes to |
| 1399 | its random oracle at a new pair $(R', Y')$, we see whether this gives us |
| 1400 | the answer we're looking for. We have $\Pr[S_0] \le \Pr[S_4] = |
| 1401 | \Succ{mcdh}{G}(B) \le \InSec{gdh}(G; t', q_I + q_V)$ as required. |
| 1402 | |
| 1403 | \begin{figure} |
| 1404 | \begin{program} |
| 1405 | Adversary $B^{V(\cdot)}(X, R^*)$: \+ \\ |
| 1406 | $F \gets \emptyset$; $\Xid{R}{map} \gets \emptyset$; \\ |
| 1407 | \FOR $i \in \Nupto{n}$ \DO \\ \ind |
| 1408 | $r \getsr I$; $R \gets r R^*$; $c \getsr \Bin^{\ell_I}$; \\ |
| 1409 | $\Xid{R}{map} \gets \Xid{R}{map} \cup \{ R \mapsto r \}$; \\ |
| 1410 | $\mathbf{c}[i] \gets (R, c)$; \- \\ |
| 1411 | $(R', Y') \gets A^{H_I(\cdot, \cdot), \id{check}(\cdot, \cdot)} |
| 1412 | (X, \mathbf{c})$; \\ |
| 1413 | \IF $Y' \neq \bot$ \THEN $H_I(R', Y')$; |
| 1414 | \next |
| 1415 | Oracle $H_I(R', Y')$: \+ \\ |
| 1416 | \IF $(R', Y') \in \dom F$ \THEN \\ \quad |
| 1417 | $h \gets F(x)$; \\ |
| 1418 | \ELSE \\ \ind |
| 1419 | $\id{check}(R', Y')$; \\ |
| 1420 | $h \getsr \Bin^{\ell_I}$; |
| 1421 | $F \gets F \cup \{ (R', Y') \mapsto h \}$; \- \\ |
| 1422 | \RETURN $h$; |
| 1423 | \- \\[\medskipamount] |
| 1424 | Oracle $\id{check}(R', Y')$: \+ \\ |
| 1425 | \IF $R' \in \dom \Xid{R}{map}$ \THEN |
| 1426 | $V(Y'/\Xid{R}{map}(R'))$; |
| 1427 | \end{program} |
| 1428 | |
| 1429 | \caption{Soundness of $\Wident$: reduction from MCDH} |
| 1430 | \label{fig:wident-sound-cdh} |
| 1431 | \end{figure} |
| 1432 | |
| 1433 | \end{proof} |
| 1434 | |
| 1435 | \subsubsection{Zero-knowledge} |
| 1436 | Finally we must prove that $\Wident$ is (statistical) zero-knowledge -- i.e., |
| 1437 | that, except with very small probability, Bob learns nothing of use to him |
| 1438 | except that he's interacting with Alice. To do this, we show that, for any |
| 1439 | algorithm $B$ which Bob might use to produce his challenge to the real Alice, |
| 1440 | there exists a simulator $S$ which produces transcripts distributed very |
| 1441 | similarly to transcripts of real conversations between $B$ and Alice, the |
| 1442 | difference being that $S$ doesn't know Alice's key. We shall show that the |
| 1443 | statistical difference between the two distributions is $2^{-\ell_I}$. |
| 1444 | |
| 1445 | The intuition here is that Bob ought to know what answer Alice is going to |
| 1446 | give him when he constructs his challenge. This is certainly true if he's |
| 1447 | honest: his challenge is $R = r P$ for some $r$ he knows, so he won't learn |
| 1448 | anything useful when Alice responds with $x R = r X$. However, if Bob sends |
| 1449 | a challenge $R$ when he doesn't know the corresponding $r$, he learns |
| 1450 | something potentially useful. The accompanying check value $c = r \xor |
| 1451 | H_I(R, r X)$ keeps him honest. |
| 1452 | |
| 1453 | To show this, we present an \emph{extractor} which, given any challenge $(R, |
| 1454 | c)$ Bob can construct, and his history of random-oracle queries, either |
| 1455 | returns a pair $(r, Y)$ such that $R = r P$ and $Y = r X$, or $\bot$; |
| 1456 | moreover, the probability that Alice returns a response $Y' \ne \bot$ given |
| 1457 | the challenge $(R, c)$ is $2^{-\ell}$. We can, of course, readily convert |
| 1458 | this extractor into a simulator to prove the zero-knowledge property of our |
| 1459 | protocol. |
| 1460 | |
| 1461 | We shall actually consider a slightly more complex setting. We grant Bob |
| 1462 | access to an oracle which produces random, correctly-formed challenges. We |
| 1463 | require this to model the legitimate challenges of other parties when we |
| 1464 | analyse the security of our key exchange protocol. The extractor is |
| 1465 | permitted to fail given |
| 1466 | |
| 1467 | \begin{definition}[Discrete-log extractors] |
| 1468 | Let $T$, $B$ be randomized algorithms. Define the game |
| 1469 | $\Game{dl-ext}{G}(T, B)$ as shown in figure~\ref{fig:dlext}. The |
| 1470 | \emph{success probability of $T$ as a discrete-log extractor against $B$} |
| 1471 | is defined as |
| 1472 | \[ \Succ{dl-ext}{G}(T, B) = \Pr[\Game{dl-ext}{G}(T, B) = 1]. \] |
| 1473 | \end{definition} |
| 1474 | Let's unpack this definition slightly. We make the following demands of our |
| 1475 | extractor. |
| 1476 | \begin{itemize} |
| 1477 | \item It is given a bare `transcript' of $B$'s execution. In particular, it |
| 1478 | is given only its output and a list of $B$'s random-oracle queries in no |
| 1479 | particular order. |
| 1480 | \item While the extractor is not given the private key~$x$, the adversary~$B$ |
| 1481 | is given the private key. |
| 1482 | \item We require that, if the extractor produces values $r, Y \ne \bot$ then |
| 1483 | $r$ and $Y$ are \emph{correct}; i.e., that $R = r P$ and $Y = x R$. |
| 1484 | \item The extractor is explicitly \emph{not} given the outputs of the |
| 1485 | challenge-generation oracle $C()$, nor of the random-oracle queries issued |
| 1486 | by $C()$. However, we allow the extractor to fail (i.e., return $\bot$) if |
| 1487 | $B$ simply parrots one of its $C$-outputs. |
| 1488 | \item The extractor is allowed -- indeed \emph{required} -- to fail if the |
| 1489 | challenge $(R, c)$ is \emph{invalid} (i.e., Alice would return $\bot$ given |
| 1490 | the challenge). |
| 1491 | \end{itemize} |
| 1492 | The resulting definition bears a striking similarity to the concept of |
| 1493 | \emph{plaintext awareness} in \cite{Bellare:1998:RAN}. |
| 1494 | |
| 1495 | \begin{figure} |
| 1496 | \begin{program} |
| 1497 | $\Game{dl-ext}{G}(T, B):$ \+ \\ |
| 1498 | $H_I \getsr \Func{G^2}{\Bin^{\ell_I}}$; |
| 1499 | $Q_H \gets \emptyset$; $Q_C \gets \emptyset$; \\ |
| 1500 | $(x, X) \gets \id{setup}()$; \\ |
| 1501 | $(R, c) \gets B^{\Xid{H_I}{trap}(\cdot, \cdot), C()}(x, X)$; \\ |
| 1502 | $(r, Y) \gets T(R, c, Q_H)$; \\ |
| 1503 | $Y' \gets x R$; $h' \gets H_I(R, Y')$; $r' \gets c \xor h'$; \\ |
| 1504 | \IF $r \ne \bot$ \THEN \\ \quad |
| 1505 | \IF $Y = \bot \lor R \ne r P \lor Y \ne Y'$ \THEN \RETURN $0$; \\ |
| 1506 | \IF $R = r' P$ \THEN $(r^*, Y^*) = (r', Y')$; \\ |
| 1507 | \ELSE $(r^*, Y^*) \gets (\bot, \bot)$; \\ |
| 1508 | \IF $(R, c) \in Q_C$ \THEN \RETURN $1$; \\ |
| 1509 | \IF $(r, Y) = (r', Y')$ \THEN \RETURN $1$; \\ |
| 1510 | \RETURN $0$; |
| 1511 | \next |
| 1512 | Oracle $\Xid{H_I}{trap}(R', Y')$: \+ \\ |
| 1513 | $h \gets H_I(R', Y')$; \\ |
| 1514 | $Q_H \gets Q_H \cup \{(R', Y', h)\}$; \\ |
| 1515 | \RETURN $h$; \- \\[\medskipamount] |
| 1516 | Oracle $C()$: \+ \\ |
| 1517 | $r \getsr \Nupto{q}$; \\ |
| 1518 | $R \gets r P$; $c \gets r \xor H_I(R, r X)$; \\ |
| 1519 | $Q_C \gets Q_C \cup \{(R, c)\}$; \\ |
| 1520 | \RETURN $(R, c)$ |
| 1521 | \end{program} |
| 1522 | |
| 1523 | \caption{Discrete log extraction game: $\Game{dl-ext}{G}(T, B)$} |
| 1524 | \label{fig:dlext} |
| 1525 | \end{figure} |
| 1526 | |
| 1527 | Such an extractor indeed exists, as the following lemma states. |
| 1528 | \begin{lemma}[Effectiveness of extractor $T_\Wident$] |
| 1529 | \label{lem:dlext} |
| 1530 | There exists a \emph{universal discrete-log extractor} $T_\Wident$ which, |
| 1531 | for any algorithm $B$, |
| 1532 | \[ \Succ{dl-ext}{G}(T_\Wident, B) \ge 1 - \frac{1}{2^{\ell_I}}. \] |
| 1533 | Moreover, if $B$ issues at most $q_H$ random-oracle queries, then the |
| 1534 | running time of $T_\Wident$ is $O(q_H)$. |
| 1535 | \end{lemma} |
| 1536 | We prove this result at the end of the section. For now, let us see how to |
| 1537 | prove that $\Wident$ is zero-knowledge. |
| 1538 | |
| 1539 | We use the set-up described in section~\ref{sec:sim}. Our initialization |
| 1540 | function~$I_\Wident$ just chooses a random $x \in \Nupto{q}$ as the world |
| 1541 | private input and sets $X = x P$ as the common input. In the `real world', |
| 1542 | the adversary is allowed to submit a (single) challenge to the prover; it is |
| 1543 | given the prover's response, and must then compute its output. This is shown |
| 1544 | on the left hand side of figure~\ref{fig:wident-sim}. |
| 1545 | |
| 1546 | The zero-knowledge property of the scheme is described by the following |
| 1547 | theorem. |
| 1548 | \begin{theorem}[Statistical zero-knowledge of $\Wident$] |
| 1549 | \label{thm:wident-zk} |
| 1550 | Let $I_\Wident$, $W_\Wident$ and $S_\Wident$ be the real-prover world and |
| 1551 | simulator shown in figure~\ref{fig:wident-sim}. Then, for any~$t$, |
| 1552 | $q_I$ and $q_C$, |
| 1553 | \[ \InSec{sim}(W_\Wident, I_\Wident, S_\Wident; t, q_I, q_C, 0) \le |
| 1554 | \frac{q_C}{2^\ell_I}. |
| 1555 | \] |
| 1556 | where $q_C$ is the maximum number of challenges allowed by the adversary. |
| 1557 | \end{theorem} |
| 1558 | \begin{proof} |
| 1559 | The simulator simply uses the extractor~$T_\Wident$ to extract the answer |
| 1560 | from the adversary's history of random oracle queries. Observe that |
| 1561 | $S_\Wident$ is a fake-world simulator. By lemma~\ref{lem:dlext}, the |
| 1562 | extractor fails with probability only $2^{-\ell_I}$. The theorem follows |
| 1563 | by a simple union bound and proposition~\ref{prop:fakesim}. |
| 1564 | \end{proof} |
| 1565 | |
| 1566 | \begin{figure} |
| 1567 | \begin{program} |
| 1568 | Initialization function $I_\Wident()$: \+ \\ |
| 1569 | $x \getsr \Nupto{|G|}$; \\ |
| 1570 | $X \gets x P$; \\ |
| 1571 | \RETURN $(x, X)$; |
| 1572 | \- \\[\medskipamount] |
| 1573 | Real-prover world $W_\Wident^{H_I(\cdot, \cdot)} |
| 1574 | ((x, X), \sigma, \tau, \mu)$: \+ \\ |
| 1575 | \IF $\tau = \cookie{challenge}$ \THEN \\ \ind |
| 1576 | $(R, c) \gets \mu$; \\ |
| 1577 | $Y \gets \id{response}^{H_I(\cdot, \cdot)}(R, c, x)$; \\ |
| 1578 | \RETURN $(1, Y)$; \- \\ |
| 1579 | \ELSE \\ \ind |
| 1580 | \RETURN $(\sigma, \bot)$; |
| 1581 | \next |
| 1582 | Simulator $S_\Wident$'s fake world \\ |
| 1583 | \hspace{1in} $W_{\text{sim}}^{H_I(\cdot, \cdot)} |
| 1584 | ((X, u), \sigma, \tau, \mu)$: \+ \\ |
| 1585 | \IF $\tau = \cookie{init}$ \THEN \\ \ind |
| 1586 | \RETURN $(\emptyset, \emptystring)$; \- \\ |
| 1587 | $Q_H \gets \sigma$; \\ |
| 1588 | \IF $\tau = \cookie{challenge}$ \THEN \\ \ind |
| 1589 | $(R, c) \gets \mu$; \\ |
| 1590 | $(r, Y) \gets T_\Wident(R, c, Q_H)$; \\ |
| 1591 | \RETURN $(Q_H, Y)$; \- \\ |
| 1592 | \ELSE \IF $\tau = \cookie{random}$ \THEN \\ \ind |
| 1593 | $(i, (R', Y'), h) \gets \mu$; \\ |
| 1594 | $Q_H \gets Q_H \cup \{(R', Y', h)\}$; \\ |
| 1595 | \RETURN $(Q_H, \emptystring)$; \- \\ |
| 1596 | \ELSE \\ \ind |
| 1597 | \RETURN $(\sigma, \bot)$; |
| 1598 | \end{program} |
| 1599 | |
| 1600 | \caption{Real-prover and simulator for zero-knowledge of $\Wident$} |
| 1601 | \label{fig:wident-sim} |
| 1602 | \end{figure} |
| 1603 | |
| 1604 | We now return to proving that our extractor exists and functions as claimed. |
| 1605 | The following two trivial lemmata will be useful, both now and later. |
| 1606 | \begin{lemma}[Uniqueness of discrete-logs] |
| 1607 | \label{lem:unique-dl} |
| 1608 | Let $G = \langle P \rangle$ be a cyclic group. For any $X \in G$ there is |
| 1609 | a unique integer $x$ where $0 \le x < |G|$ and $X = x P$. |
| 1610 | \end{lemma} |
| 1611 | \begin{proof} |
| 1612 | Certainly such an $x$ exists, since $G$ is cyclic and finite. Suppose $X = |
| 1613 | x P = x' P$: then $0 = x P - x' P = (x - x') P$. Hence $(x - x')$ is a |
| 1614 | multiple of $|G|$. |
| 1615 | \end{proof} |
| 1616 | \begin{lemma}[Uniqueness of check values] |
| 1617 | \label{lem:unique-c} |
| 1618 | Let $G = \langle P \rangle$ be a cyclic group; let $H_I$ be a function |
| 1619 | $H_I\colon \Bin^{2\ell_G} \to \Bin^{\ell_I}$. Fix some $x \in |
| 1620 | \Nupto{|G|}$ and define the set |
| 1621 | \[ V_x = \bigl\{\, (R, c) \in G \times \Bin^{\ell_I} \bigm| |
| 1622 | R = \bigl( c \xor H_I(R, x R) \bigr) P \,\bigr\}. |
| 1623 | \] |
| 1624 | Then, for any $R$, $c$, $c'$, if $(R, c) \in V_x$ and $(R, c') \in V_x$ |
| 1625 | then $c = c'$. |
| 1626 | \end{lemma} |
| 1627 | \begin{proof} |
| 1628 | From lemma~\ref{lem:unique-dl}, we see that there is a unique $r \in |
| 1629 | \Nupto{|G|}$ for which $R = r P$. Now, if $(R, c) \in V_x$, we must have |
| 1630 | $r = c \xor H_I(R, x R)$. It follows that $c = r \xor H_I(R, x R)$. |
| 1631 | \end{proof} |
| 1632 | |
| 1633 | \begin{proof}[Proof of lemma~\ref{lem:dlext}] |
| 1634 | Our extractor $T_\Wident$ is shown in figure~\ref{fig:twident}. |
| 1635 | |
| 1636 | \begin{figure} |
| 1637 | \begin{program} |
| 1638 | Extractor $T_\Wident(R, c, Q_H)$: \+ \\ |
| 1639 | \FOR $(R', Y', h)$ \IN $Q_H$ \DO \\ \ind |
| 1640 | $r \gets h \xor c$; \\ |
| 1641 | \IF $R = R' = r P \land Y' = r X$ \THEN \RETURN $(r, Y')$; \- \\ |
| 1642 | \RETURN $(\bot, \bot)$; |
| 1643 | \end{program} |
| 1644 | |
| 1645 | \caption{The discrete-log extractor $T_\Wident$} |
| 1646 | \label{fig:twident} |
| 1647 | \end{figure} |
| 1648 | |
| 1649 | Let $B$ be any randomized algorithm, and let $(R, c, Q_H)$ be as given to |
| 1650 | the extractor by $\Game{dl-ext}{G}(T_\Wident, B)$. Let the quantities |
| 1651 | $H_I$, $Q_C$, $r$, $r'$, $x$ and $X$ be as in that game. |
| 1652 | |
| 1653 | Suppose that the extractor returns values $(r, Y) \ne (\bot, \bot)$. Let |
| 1654 | $h = r \xor c$; then there must be a query $(R, Y, h) \in Q_H$, and we have |
| 1655 | $R = r P$ and $Y = r X = r (x P) = x (r P) = x R = Y'$, so the extractor's |
| 1656 | output must be correct unless it fails. |
| 1657 | |
| 1658 | Furthermore, in the case where the extractor did not fail, we have $h = |
| 1659 | H_I(R, Y) = H_I(R, Y')$ and $c = r \xor h$, so the challenge was valid. |
| 1660 | Therefore, if the challenge was invalid, the extractor will fail. |
| 1661 | |
| 1662 | We now deal with the challenge-generation oracle. Suppose that $(R, c') |
| 1663 | \in Q_C$ for some $c'$. Now, if $c = c'$ then $(R, c')$ is a repeat of |
| 1664 | some challenge from the challenge-generation oracle, and the extractor is |
| 1665 | permitted to fail. On the other hand, suppose $c \ne c'$; then, the |
| 1666 | challenge $(R, c)$ must be invalid by lemma~\ref{lem:unique-c}, so the |
| 1667 | extractor is required to fail, and we have established that indeed it will. |
| 1668 | From now on, suppose that $R$ is distinct from all the $R$-values returned |
| 1669 | by $C()$. |
| 1670 | |
| 1671 | Let $Y = x R$. Suppose that $B$ queried its random oracle at $(R, Y)$. |
| 1672 | Let $h = H_I(Y)$, so $r' = c \xor h$. If the challenge is valid then $R = |
| 1673 | r' P$; therefore $Y = x R = x r' P = r' X$, so we have $(R, Y, h) \in Q_H$ |
| 1674 | with $R = r P$ and $Y = r X$. Hence the extractor returns $r = r'$ as |
| 1675 | required. |
| 1676 | |
| 1677 | It remains to deal with the case where there is no random-oracle query at |
| 1678 | $(R, Y)$. But then $h = H_I(R, Y)$ is uniformly distributed, and |
| 1679 | independent of the entire game up to this point. Let $r$ be the correct |
| 1680 | discrete log of $R$; by lemma~\ref{lem:unique-dl} there is only one |
| 1681 | possible value. The extractor always fails under these circumstances, but |
| 1682 | a correct responder would reply with probability |
| 1683 | \[ \Pr[h = c \xor r] = \frac{1}{2^{\ell_I}}. \] |
| 1684 | This concludes the proof. |
| 1685 | \end{proof} |
| 1686 | \begin{remark} |
| 1687 | Note that the fact that the algorithm~$B$ was given the private key is |
| 1688 | irrelevant to the above argument. However, we shall need this property |
| 1689 | when we come to prove deniability for the key-exchange protocol. |
| 1690 | \end{remark} |
| 1691 | \begin{remark} |
| 1692 | It's easy to see from the above proof that the extractor works flawlessly |
| 1693 | on the `honest verifier' algorithm $\id{challenge}$ shown in |
| 1694 | figure~\ref{fig:wident-ro}. This shows that $\Wident$ is perfect |
| 1695 | zero-knowledge against honest verifiers. We're much more interested in |
| 1696 | dishonest verifiers, though. |
| 1697 | \end{remark} |
| 1698 | |
| 1699 | |
| 1700 | \subsection{An identity-based identification scheme} |
| 1701 | \label{sec:wident-id} |
| 1702 | |
| 1703 | Boneh and Franklin \cite{Boneh:2003:IBE} showed how to construct an |
| 1704 | identity-based encryption scheme using bilinear pairings. The resulting |
| 1705 | encryption scheme looks somewhat like a pairing-based version of ElGamal's |
| 1706 | encryption scheme \cite{ElGamal:1985:PKC}. We can easily apply their |
| 1707 | techniques to our identification protocol, and thereby obtain an |
| 1708 | identity-based identification scheme. Providing the necessary formalisms to |
| 1709 | prove theorems analogous to our theorems~\ref{thm:wident-sound} |
| 1710 | and~\ref{thm:wident-zk} would take us too far from our objectives; but given |
| 1711 | appropriate security notions, we can readily adapt our existing proofs to the |
| 1712 | new setting. |
| 1713 | |
| 1714 | \subsubsection{Bilinear pairings} |
| 1715 | Before we describe the necessary modifications to the protocol, we first give |
| 1716 | a (very brief!) summary of cryptographic pairings. (The Boneh-Franklin paper |
| 1717 | \cite{Boneh:2003:IBE} gives more detail; also \cite{Menezes:2005:IPB} |
| 1718 | provides a useful introduction to the topic.) |
| 1719 | |
| 1720 | Let $(G, +)$, $(G', +)$ and $(G_T, \times)$ be cyclic groups with prime order |
| 1721 | $q$; let $P \in G$ and $P' \in G'$ be elements of order $q$ in $G$ and $G'$ |
| 1722 | respectively. We say that a mapping $\hat{e}\colon G \times G' \to G_T$ is a |
| 1723 | \emph{non-degenerate bilinear pairing} if it satisfies the following |
| 1724 | properties. |
| 1725 | \begin{description} |
| 1726 | \item[Bilinearity] For all $R \in G$ and $S', T' \in G'$, we have $\hat{e}(R, |
| 1727 | S' + T') = \hat{e}(R, S')\,\hat{e}(R, T')$; and for all $R, S \in G$ and $T' |
| 1728 | \in G'$ we have $\hat{e}(R + S, T') = \hat{e}(R, T')\,\hat{e}(S, T')$. |
| 1729 | \item[Non-degeneracy] $\hat{e}(P, P') \ne 1$. |
| 1730 | \end{description} |
| 1731 | For practical use, we also want $\hat{e}(\cdot, \cdot)$ to be efficient to |
| 1732 | compute. The reader can verify that $\hat{e}(a P, b P') = \hat{e}(P, |
| 1733 | P')^{ab}$. It is permitted for the two groups $G$ and $G'$ to be equal. |
| 1734 | |
| 1735 | We require a different intractability assumption, specifically that the |
| 1736 | \emph{bilinear} Diffie-Hellman problem (BDH) -- given $(a P, b P, a P', c P') |
| 1737 | \in G^2 \times G'^2$, find $\hat{e}(P, P')^{abc} \in G_T$ -- is difficult. |
| 1738 | This implies the difficulty of the computational Diffie-Hellman problem in |
| 1739 | all three of $G$, $G'$ and~$G_T$. |
| 1740 | |
| 1741 | \subsubsection{The identity-based scheme} |
| 1742 | We need a trusted authority; following \cite{Schneier:1996:ACP} we shall call |
| 1743 | him Trent. Trent's private key is $t \in \Nupto{q}$; his public key is $T = |
| 1744 | t P$. |
| 1745 | |
| 1746 | Finally, we need cryptographic hash functions $H_I\colon G \times G_T \to |
| 1747 | \Bin^{\ell_I}$ and $\Hid\colon \Bin^* \to G'$; a formal security analysis |
| 1748 | would model these as random oracles. |
| 1749 | |
| 1750 | Alice's public key is $A = \Hid(\texttt{Alice}) \in G'$. Her private key is |
| 1751 | $K_A = t A \in G'$ -- she needs Trent to give this to her. Bob can interact |
| 1752 | with Alice in order to verify her identity as follows. |
| 1753 | \begin{enumerate} |
| 1754 | \item Bob computes $\gamma_A = \hat{e}(T, A) \in G_T$. (He can do this once |
| 1755 | and store the result if he wants, but it's not that onerous to work it out |
| 1756 | each time.) |
| 1757 | \item Bob chooses $r \inr \Nupto{q}$, and sets $R = r P$. He also computes |
| 1758 | $\psi = \gamma_A^r$, $h = H_I(R, \psi)$ and $c = r \xor h$. He sends his |
| 1759 | challenge $(R, c)$ to Alice. |
| 1760 | \item Alice receives $(R', c')$. She computes $\psi' = \hat{e}(R, K_A)$, $h' |
| 1761 | = H_I(R', \psi')$, and $r' = c' \xor h')$. She checks that $R' = r' P$; if |
| 1762 | so, she sends $\psi'$ back to Bob; otherwise she refuses to talk to him. |
| 1763 | \item Bob receives $\psi'$. If $\psi = \psi'$ then he accepts that he's |
| 1764 | talking to Alice. |
| 1765 | \end{enumerate} |
| 1766 | This works because $\psi = \gamma_A^r = \hat{e}(T, A)^r = \hat{e}(t P, A)^r = |
| 1767 | \hat{e}(r P, A)^t = \hat{e}(R, t A) = \psi'$. |
| 1768 | |
| 1769 | \subsubsection{Informal analysis} |
| 1770 | An analogue to lemma~\ref{lem:dlext} can be proven to show how to extract $r$ |
| 1771 | from a verifier's random-oracle queries; statistical zero knowledge would |
| 1772 | then follow easily, as in theorem~\ref{thm:wident-zk}. Soundness is |
| 1773 | intuitively clear, since an adversary must compute $\psi = \hat{e}(P, |
| 1774 | P')^{art}$ given $A = a P'$, $R = r P$ and $T = t P$, which is an instance of |
| 1775 | the BDH problem. An analogue of theorem~\ref{thm:wident-sound} would have to |
| 1776 | prove this for an adversary capable of making identity requests as well as |
| 1777 | obtaining challenges. Finally, our key-exchange protocol can be constructed |
| 1778 | out of this identity-based identification scheme, yielding an identity-based |
| 1779 | authenticated key-exchange protocol. We leave it to the reader to work |
| 1780 | through the details. |
| 1781 | |
| 1782 | |
| 1783 | \subsection{Comparison with the protocol of Stinson and Wu} |
| 1784 | \label{sec:stinson-ident} |
| 1785 | |
| 1786 | Our protocol is similar to a recent proposal by Stinson and Wu |
| 1787 | \cite{Stinson:2006:EST}. They restrict their attention to Schnorr groups $G |
| 1788 | \subset \F_p^*$. Let $\gamma$ be an element of order $q = |G|$. The |
| 1789 | prover's private key is $a \inr \Nupto{q}$ and her public key is $\alpha = |
| 1790 | \gamma^a$. In their protocol, the challenger chooses $r \inr \Nupto{q}$, |
| 1791 | computes $\rho = \gamma^r$ and $\psi = \alpha^r$, and sends a challenge |
| 1792 | $(\rho, H(\rho, \psi))$. The prover checks that $\rho^q \ne 1$, computes |
| 1793 | $\psi = \rho^a$, checks the hash, and sends $\psi$ back by way of response. |
| 1794 | They prove their protocol's security in the random-oracle model. |
| 1795 | |
| 1796 | Both the Wrestlers protocol and Stinson-Wu require both prover and verifier |
| 1797 | to compute two exponentiations (or scalar multiplications) each. The |
| 1798 | sizes of the messages used by the two protocols are also identical. |
| 1799 | |
| 1800 | (An earlier version of the Stinson-Wu protocol used a cofactor |
| 1801 | exponentiation: if we set $f = (p - 1)/q$, then we use $\psi = \alpha^{rf}) = |
| 1802 | \rho^{af} = \gamma^{afr}$. This is more efficient in typical elliptic curve |
| 1803 | subgroups, since the cofactor of such subgroups is usually small: indeed, |
| 1804 | \cite{SEC1} recommends rejecting groups with cofactor $f > 4$. However, in |
| 1805 | the Schnorr groups used by Stinson and Wu, the cofactor is much larger than |
| 1806 | $q$, and their new variant is more efficient.) |
| 1807 | |
| 1808 | We note that the zero-knowledge property of the Stinson-Wu protocol requires |
| 1809 | the Diffie-Hellman knowledge of exponent assumption (KEA). Very briefly: |
| 1810 | suppose $A$ is a randomized algorithm which takes as input $X \in G$ and |
| 1811 | outputs a pair $(r P, r X)$; intuitively, the KEA asserts $A$ must have done |
| 1812 | this by choosing $r$ somehow and then computing its output from it. |
| 1813 | Formally, it asserts the existence of an `extractor' algorithm which takes as |
| 1814 | input the element $X$ and the random coins used by $A$ and outputs $r$ with |
| 1815 | high probability. This is a very strong assumption, and one which is |
| 1816 | unnecessary for our protocol, since we can present an \emph{explicit} |
| 1817 | extractor. |
| 1818 | |
| 1819 | The KEA assumption as stated in \cite{Stinson:2006:EST} allows the extractor |
| 1820 | to fail with some negligible probability, over and above the probability that |
| 1821 | a dishonest verifier managed to guess the correct $h = H(\rho, \psi)$ without |
| 1822 | making this random-oracle query. Not only does our protocol achieve zero- |
| 1823 | knowledge without the KEA, our extractor is, in this sense, `perfect'. |
| 1824 | |
| 1825 | Our protocol is just as strong as Stinson-Wu under attack from active |
| 1826 | intruders: see table~\ref{tab:wident-active} for a very brief sketch of the |
| 1827 | case-analysis which would be the basis of a proof of this. |
| 1828 | |
| 1829 | \begin{table} |
| 1830 | \begin{tabular}[C]{|*{3}{c|}p{8cm}|} |
| 1831 | \hlx{hv[1]} |
| 1832 | \multicolumn{2}{|c|}{\textbf{Challenge}} & |
| 1833 | \textbf{Response} & |
| 1834 | \textbf{Security} |
| 1835 | \\ \hlx{v[1]hv} |
| 1836 | %% unpleasant hacking to make the R and c columns the same width :-( |
| 1837 | \settowidth{\dimen0}{\textbf{Challenge}}% |
| 1838 | \dimen0=.5\dimen0 |
| 1839 | \advance\dimen0by-\tabcolsep |
| 1840 | \advance\dimen0by-.5\arrayrulewidth |
| 1841 | \hbox to\dimen0{\hfil$R$\hfil} |
| 1842 | & $c$ & $Y$ & Nothing to prove. \\ \hlx{v} |
| 1843 | $R$ & $c'$ & --- & Prover rejects by lemma~\ref{lem:unique-c}; |
| 1844 | $Y'$ probably wrong by |
| 1845 | theorem~\ref{thm:wident-sound}. \\ \hlx{v} |
| 1846 | $R$ & $c$ & $Y'$ & Response is incorrect. \\ \hlx{v} |
| 1847 | $R'$ & --- & $Y$ & Response is incorrect. \\ \hlx{v} |
| 1848 | $R'$ & $c$ & $Y'$ & Prover rejects with probability $1 - 2^{-\ell_I}$; |
| 1849 | $Y'$ probably wrong by |
| 1850 | theorem~\ref{thm:wident-sound}. \\ \hlx{v} |
| 1851 | $R'$ & $c'$ & $Y'$ & Simulate prover using extractor |
| 1852 | (lemma~\ref{lem:dlext}); $Y'$ probably wrong by |
| 1853 | theorem~\ref{thm:wident-sound}. \\ \hlx{vh} |
| 1854 | \end{tabular} |
| 1855 | |
| 1856 | \caption{Security of $\Wident$ against active intruders (summary)} |
| 1857 | \label{tab:wident-active} |
| 1858 | \end{table} |
| 1859 | |
| 1860 | An identity-based analogue of Stinson-Wu can be defined using a bilinear |
| 1861 | pairing, just as we did in section~\ref{sec:wident-id}. However, to prove |
| 1862 | the zero-knowledge property, one needs to make a bilinear analogue of the |
| 1863 | knowledge of exponent assumption. |
| 1864 | |
| 1865 | We suspect that a key-exchange protocol like ours can be constructed using |
| 1866 | Stinson-Wu rather than the Wrestlers identification scheme. We haven't, |
| 1867 | however, gone through all the details, since we believe our protocol is just |
| 1868 | as efficient and is based on much more conservative assumptions. |
| 1869 | |
| 1870 | %%%-------------------------------------------------------------------------- |
| 1871 | |
| 1872 | \section{A simple key-exchange protocol} |
| 1873 | \label{sec:kx} |
| 1874 | |
| 1875 | In this section, we describe a simple key-exchange protocol built out of the |
| 1876 | identification protocol shown previously. |
| 1877 | |
| 1878 | The key-exchange protocol arises from the following observation. If Bob |
| 1879 | sends a challenge, presumably to Alice, and gets a correct response, then not |
| 1880 | only did he really send the challenge to Alice but he knows that she received |
| 1881 | it correctly. |
| 1882 | |
| 1883 | So, if Alice and Bob authenticate each other, by the end of it, they should |
| 1884 | each have chosen a random private value, sent the corresponding public value |
| 1885 | to the other, and been convinced that it arrived safely. |
| 1886 | |
| 1887 | Unfortunately, life isn't quite this kind, and we have to do some more work |
| 1888 | to make this scheme secure. |
| 1889 | |
| 1890 | |
| 1891 | Our key exchange protocol essentially consists of two parallel instances of |
| 1892 | the identification protocol. If Alice receives a correct response to her |
| 1893 | challenge, she will know that Bob received her challenge correctly, and |
| 1894 | \emph{vice versa}. If we let Alice's challenge be $R_0 = r_0 P$ and Bob's |
| 1895 | challenge be $R_1 = r_1 P$ then each can compute a shared secret $Z = r_0 R_1 |
| 1896 | = r_0 r_1 P = r_1 R_0$ unknown to an adversary. There are, unfortunately, a |
| 1897 | few subtleties involved in turning this intuition into a secure key-exchange |
| 1898 | protocol, which we now describe. |
| 1899 | |
| 1900 | |
| 1901 | \subsection{Overview} |
| 1902 | \label{sec:kx-overview} |
| 1903 | |
| 1904 | We present a quick, informal description of our basic key-exchange protocol. |
| 1905 | In addition to our group $G$, we shall also need a secure symmetric |
| 1906 | encryption scheme $\E = (\kappa, E, D)$, and two secure hash functions |
| 1907 | $H_I\colon \Bin^{2\ell_G} \to \Bin^{\ell_I}$ and $H_K\colon \Bin^{\ell_G+1} |
| 1908 | \to \Bin^\kappa$. |
| 1909 | |
| 1910 | Suppose that Alice's and Bob's private keys are $a$ and $b$ respectively, and |
| 1911 | their public keys are $A = a P$ and $B = b P$. |
| 1912 | \begin{enumerate} |
| 1913 | \item Alice chooses a random index $r \inr \Nupto{|G|}$. She computes $R = r |
| 1914 | P$ and $c = r \xor H_I(R, r B)$. She sends the pair $(R, c)$ to Bob. |
| 1915 | \item Similarly, Bob chooses a random $s \inr \Nupto{|G|}$. He computes $S |
| 1916 | = s P$ and $d = s \xor H_I(S, s A)$. He sends $(S, d)$ to Alice. |
| 1917 | \item Alice receives $(S', d')$ from Bob. She computes $s' = d' \xor H_I(S', |
| 1918 | a S')$, and verifies that $S' = s' P$. If so, she computes $K_A = H_K(0 |
| 1919 | \cat r S')$, and sends $R, E_{K_A}(a S')$ to Bob. |
| 1920 | \item Similarly, Bob receives $(R', c')$ from Alice. He verifies that $R' = |
| 1921 | \bigl( c' \xor H_I(R', b R') \bigr) P$. If so, he computes $K_B = H_K(0 |
| 1922 | \cat s R')$ and sends S, $E_{K_B}(b R')$ to Alice. |
| 1923 | \item Alice receives a ciphertext $(S'', \chi_B)$ from Bob. She checks that |
| 1924 | $S'' = S'$, decrypts $\chi_B$, and checks that $D_{K_A}(\chi_B) = r B$. If |
| 1925 | so, she uses $H_K(1 \cat r S')$ as her shared secret. |
| 1926 | \item Similarly, Bob receives $(R'', \chi_A)$ from Alice, and checks $R'' = |
| 1927 | R'$ and $D_{K_B}(\chi_A) = s A$. If so, he uses $H_K(1 \cat s R')$ as his |
| 1928 | shared secret. |
| 1929 | \end{enumerate} |
| 1930 | This is the Wrestlers Key Exchange protocol, $\Wkx^{G, \E}$ (again, we omit |
| 1931 | the superscripts when referring to the general protocol, or when confusion is |
| 1932 | unlikely). A diagrammatic summary of the protocol is shown in |
| 1933 | figure~\ref{fig:wkx}. |
| 1934 | |
| 1935 | \begin{figure} |
| 1936 | \begin{description} |
| 1937 | \item[Setup] Group $G = \langle P \rangle$; $|G| = q$ is prime. |
| 1938 | $H_I(\cdot, \cdot)$ and $H_K(\cdot)$ are secure hashes. $\E = (\kappa, |
| 1939 | E, D)$ is an IND-CCA2 symmetric encryption scheme. |
| 1940 | \item[Parties] $U_i$ for $0 \le i < n$. |
| 1941 | \item[Private keys] $x_i \inr \Nupto{q}$. |
| 1942 | \item[Public keys] $X_i = x_i P$. |
| 1943 | \end{description} |
| 1944 | \[ \begin{graph} |
| 1945 | !{0; <8cm, 0cm>: <0cm, 3cm>::} |
| 1946 | *+[F]\dbox{$r_i \getsr I$; $R_i \gets r_i P$ \\ |
| 1947 | $c_i \gets r_i \xor H_I(R_i, r_i X_j)$} ="i0" |
| 1948 | [d] |
| 1949 | *+[F]\dbox{Check $R_j = \bigl(c_j \xor H_I(x_i R_j)\bigr) P$ \\ |
| 1950 | $Z \gets r_i R_j$; $K \gets H_K(0, Z)$ \\ |
| 1951 | $\chi_i \gets E_K(x_i R_j)$} ="i1" |
| 1952 | [d] |
| 1953 | *+[F]\dbox{Check $D_K(\chi_j) = r_i X_j$ \\ |
| 1954 | Shared key is $H_K(1, Z)$} ="i2" |
| 1955 | "i0" [r] |
| 1956 | *+[F]\dbox{$r_j \getsr I$; $R_j \gets r_j P$ \\ |
| 1957 | $c_j \gets r_j \xor H_I(R_j, r_j X_i)$} ="j0" |
| 1958 | [d] |
| 1959 | *+[F]\dbox{Check $R_i = \bigl(c_i \xor H_I(x_j R_i)\bigr) P$ \\ |
| 1960 | $Z \gets r_j R_i$; $K \gets H_K(0, Z)$ \\ |
| 1961 | $\chi_j \gets E_K(x_j R_i)$} ="j1" |
| 1962 | [d] |
| 1963 | *+[F]\dbox{Check $D_K(\chi_i) = r_j X_i$ \\ |
| 1964 | Shared key is $H_K(1, Z)$} ="j2" |
| 1965 | % |
| 1966 | "i0" : |(0)/3.25cm/*+{(R_i, c_i)} "j1" |
| 1967 | "i1" : |(0)/3.25cm/*+{(R_i, \chi_i)} "j2" |
| 1968 | "j0" : |(0)/3.25cm/*+{(R_j, c_j)} "i1" |
| 1969 | "j1" : |(0)/3.25cm/*+{(R_j, \chi_j)} "i2" |
| 1970 | \end{graph} \] |
| 1971 | |
| 1972 | \caption{Summary of the Wrestlers Key Exchange protocol, $\Wkx$} |
| 1973 | \label{fig:wkx} |
| 1974 | \end{figure} |
| 1975 | |
| 1976 | Assume, for the moment, that Alice and Bob's messages are relayed honestly. |
| 1977 | Then: |
| 1978 | \begin{itemize} |
| 1979 | \item $a S' = a S = a (s P) = s (a P) = s A$, so $s' = d' \xor H_I(S' a S') = |
| 1980 | d \xor H_I(S, s A) = s$, and $S' = S = s P = s' P$, and therefore Alice |
| 1981 | responds to Bob's message; |
| 1982 | \item similarly $b R' = r B$, so $r' = r$ and $R' = r' P$, and therefore Bob |
| 1983 | responds to Alice's message; |
| 1984 | \item $b R' = b R = b (r P) = r (b P) = r B$, and $a S' = a S = a (s P) = s |
| 1985 | (a P) = s A$, and therefore both parties compute their responses correctly; |
| 1986 | and |
| 1987 | \item $r S' = r S = r (s P) = s (r P) = s R = s R'$, so $K_A = K_B$, and |
| 1988 | therefore they can decrypt each others' responses, and agree the same |
| 1989 | shared secret. |
| 1990 | \end{itemize} |
| 1991 | This shows that the protocol is basically valid, but says little about its |
| 1992 | security. The remainder of this section will describe our protocol in more |
| 1993 | formal detail, and prove its security in a model with multiple parties and an |
| 1994 | adversary who controls the network. |
| 1995 | |
| 1996 | Observe that the protocol as we've presented here is \emph{symmetrical}. |
| 1997 | There's no notion of `initiator' or `responder'. There are a total of four |
| 1998 | messages which must be sent before both parties accept. However, this can be |
| 1999 | reduced to three by breaking the symmetry of the protocol and combining one |
| 2000 | or other party's challenge and response messages. We choose to analyse the |
| 2001 | symmetrical version, since to do so, it suffices to consider only the two |
| 2002 | different kinds of messages. Since our security model allows the messages to |
| 2003 | be adversarially delayed and reordered, it is easy to show that the security |
| 2004 | of an optimized, asymmetrical protocol is no worse than the symmetrical |
| 2005 | version we present here. |
| 2006 | |
| 2007 | |
| 2008 | \subsection{Security model and security definition} |
| 2009 | \label{sec:um} |
| 2010 | |
| 2011 | Our model is very similar to that of Canetti and Krawczyk |
| 2012 | \cite{Canetti:2001:AKE}, though we have modified it in two ways. |
| 2013 | \begin{enumerate} |
| 2014 | \item We allow all the participants (including the adversary) in the protocol |
| 2015 | access to the various random oracles required to implement it. |
| 2016 | \item Since we want to analyse a specific, practical scheme, asymptotic |
| 2017 | results are useless. We measure the adversary's resource usage carefully, |
| 2018 | and produce a quantitative bound on the adversary's advantage in the |
| 2019 | SK-security game. |
| 2020 | \end{enumerate} |
| 2021 | |
| 2022 | \subsubsection{Overview} |
| 2023 | We briefly describe our modified model, pointing out the changes we have |
| 2024 | made, and how they apply to our protocol. Much of Canetti and Krawczyk's |
| 2025 | model (for example, the local and global outputs) is useful for proving more |
| 2026 | general security properties such as demonstrating that SK-security suffices |
| 2027 | for constructing secure channels, and we shall not concern ourselves with |
| 2028 | such details. Other parts deal with issues such as security parameters and |
| 2029 | ensuring that all the computation is polynomially bounded, which are |
| 2030 | irrelevant since we are dealing with a single concrete protocol rather than a |
| 2031 | family of them. |
| 2032 | |
| 2033 | The entities in the model are the \emph{adversary}~$A$, and a (fixed) number |
| 2034 | of \emph{parties}~$P_i$, for $0 \le i < n$. If the protocol under |
| 2035 | consideration makes use of random oracles, then all the participants -- the |
| 2036 | adversary and the parties -- are all allowed access to the random oracles. |
| 2037 | |
| 2038 | The parties and the adversary play a `game'. At the beginning of the game, |
| 2039 | the participants are given some inputs computed by a randomized |
| 2040 | \emph{initialization procedure}~$\id{init}$. This produces as output a pair |
| 2041 | $(i_U, \mathbf{i})$; the value $i_U$ is the \emph{global input}, and is given |
| 2042 | to all the participants including the adversary. The vector $\mathbf{i}$ has |
| 2043 | $n$ components, and party $P_i$ is given $(i_U, \mathbf{i}[i])$ as input. |
| 2044 | |
| 2045 | \subsubsection{Sessions} |
| 2046 | Parties don't act directly. Instead, each party runs a number of |
| 2047 | \emph{sessions}. A session a triple $S = (P_i, P_j, s)$, where $i, j \in |
| 2048 | \Nupto{n}$ identify the owning party and a \emph{partner}, and $s \in |
| 2049 | \Bin^{\ell_S}$ is a \emph{session-id}. (The original model includes a |
| 2050 | r\^ole, for distinguishing between initiators and responders. Our protocol |
| 2051 | is symmetrical, so this distinction isn't useful.) If $P_i$ runs a session |
| 2052 | $S = (P_i, P_j, s)$ and $P_j$ runs a session $S' = (P_j, P_i, s)$ then we say |
| 2053 | that $S$ and $S'$ are \emph{matching}, and that $P_j$ is $P_i$'s |
| 2054 | \emph{partner} for the session. |
| 2055 | |
| 2056 | At most one participant in the game is \emph{active} at any given time. |
| 2057 | Initially the adversary is active. The adversary may \emph{activate} a |
| 2058 | session in one of two ways. |
| 2059 | \begin{enumerate} |
| 2060 | \item It may \emph{create a session} of a party~$P_i$, by selecting a |
| 2061 | session-id~$s \in \Bin^{\ell_S}$ and a partner $j$. There is no |
| 2062 | requirement that $P_j$ ever have a matching session. However, all sessions |
| 2063 | of a party must be distinct, i.e., sessions with the same partner must have |
| 2064 | different session-ids. |
| 2065 | \item It may \emph{deliver a message}~$\mu \in \Bin^*$, from party~$P_j$, to |
| 2066 | an existing session~$S = (P_i, P_j, s)$. There is no requirement that any |
| 2067 | party previously sent $\mu$: the adversary is free to make up messages as |
| 2068 | it sees fit. |
| 2069 | \end{enumerate} |
| 2070 | The adversary becomes inactive, and the session becomes active. The session |
| 2071 | performs some computation, according to its protocol, and may request a |
| 2072 | message~$\mu$ be delivered to the matching session running in its partner |
| 2073 | (which may not exist). The session may also \emph{terminate}. In the case |
| 2074 | we are interested in, of key-exchange protocols, a session~$S = (P_i, P_j, |
| 2075 | s)$ may terminate in one of two ways: |
| 2076 | \begin{enumerate} |
| 2077 | \item it may \emph{complete}, outputting $(i, j, s, K)$, for some |
| 2078 | \emph{session key}~$K$, or |
| 2079 | \item it may \emph{abort}, outputting $(i, j, s, \bot)$. |
| 2080 | \end{enumerate} |
| 2081 | Once it has performed these actions, the session deactivates and the |
| 2082 | adversary becomes active again. The adversary is given the message~$\mu$, if |
| 2083 | any, and informed of whether the session completed or aborted, but, in the |
| 2084 | case of completion, not of the value of the key~$K$. A session is |
| 2085 | \emph{running} if it has been created and has not yet terminated. |
| 2086 | |
| 2087 | \subsubsection{Other adversarial actions} |
| 2088 | As well as activating sessions, the adversary has other capabilities, as |
| 2089 | follows. |
| 2090 | \begin{itemize} |
| 2091 | \item It may \emph{expire} any session~$S$, causing the owning party to |
| 2092 | `forget' the session key output by that session. |
| 2093 | \item It may \emph{corrupt} any party~$P_i$, at will: the adversary learns |
| 2094 | the entire state of the corrupted party, including its initial |
| 2095 | input~$\mathbf{i}[i]$, the state of any sessions it was running at the |
| 2096 | time, and the session keys of any completed but unexpired sessions. Once |
| 2097 | corrupted, a party can no longer be activated. Of course, the adversary |
| 2098 | can continue to send messages allegedly from the corrupted party. |
| 2099 | \item It may \emph{reveal the state} of a running session~$S$, learning any |
| 2100 | interesting values specific to that session, but \emph{not} the owning |
| 2101 | party's long-term secrets. |
| 2102 | \item It may \emph{reveal the session-key} of a completed session~$S$. |
| 2103 | \item It may elect to be \emph{challenged} with a completed session~$S$, |
| 2104 | provided. Challenge sessions form part of the security notion for |
| 2105 | key-exchange protocols. See below for more details. |
| 2106 | \end{itemize} |
| 2107 | We say that a session $S = (P_i, P_j, s)$ is \emph{locally exposed} if |
| 2108 | \begin{itemize} |
| 2109 | \item it has had its state revealed, |
| 2110 | \item it has had its session-key revealed, or |
| 2111 | \item $P_i$ has been corrupted, and $S$ had not been expired when this |
| 2112 | happened. |
| 2113 | \end{itemize} |
| 2114 | A session is \emph{exposed} if it is locally exposed, or if its matching |
| 2115 | session exists and has been locally exposed. |
| 2116 | |
| 2117 | At the beginning of the game, a bit $b^*$ is chosen at random. The adversary |
| 2118 | may choose to be \emph{challenged} with any completed, unexposed |
| 2119 | session;\footnote{% |
| 2120 | The original Canetti-Krawczyk definition restricts the adversary to a |
| 2121 | single challenge session, but our proof works independent of the number of |
| 2122 | challenge sessions, so we get a stronger result by relaxing the requirement |
| 2123 | here.)} |
| 2124 | the adversary is then given either the session's key -- if $b^* = 1$ -- or a |
| 2125 | string chosen at random and independently of the game so far from a |
| 2126 | protocol-specific distribution -- if $b^* = 0$. At the end of the game, the |
| 2127 | adversary outputs a single bit~$b$. |
| 2128 | |
| 2129 | \subsubsection{SK-security} |
| 2130 | We've now described the game; it is time to explain the adversary's goal in |
| 2131 | it. The adversary \emph{wins} the game if either |
| 2132 | \begin{enumerate} |
| 2133 | \item two unexposed, matching sessions complete, but output different |
| 2134 | keys,\footnote{% |
| 2135 | The original Canetti-Krawczyk definition differs slightly here. It |
| 2136 | requires that `if two \emph{uncorrupted} parties complete matching |
| 2137 | sessions then they both output the same key' [original emphasis]. This |
| 2138 | can't be taken at face value, since none of the protocols they claim to |
| 2139 | be secure actually meet this requirement: they meet only the weaker |
| 2140 | requirement that parties completing matching sessions output different |
| 2141 | keys with negligible probability. We assume here that this is what they |
| 2142 | meant.} |
| 2143 | or |
| 2144 | \item the adversary correctly guesses the hidden bit~$b^*$. |
| 2145 | \end{enumerate} |
| 2146 | More formally, we make the following definition. |
| 2147 | \begin{definition}[SK-security] |
| 2148 | \label{def:sk} |
| 2149 | Let $\Pi^{H_0(\cdot), H_1(\cdot), \ldots}$ be a key-exchange protocol |
| 2150 | which makes use of random oracles $H_0(\cdot)$, $H_1(\cdot)$, \dots, and |
| 2151 | let $A$ be an adversary playing the game described previously, where $n$ |
| 2152 | parties run the protocol~$\Pi$. Let $V$ be the event that any pair of |
| 2153 | matching, unexposed sessions completed, but output different session keys. |
| 2154 | Let $W$ be the event that the adversary's output bit matches the game's |
| 2155 | hidden bit~$b^*$. We define the adversary's \emph{advantage against the |
| 2156 | SK-security of the protocol~$\Pi$} to be |
| 2157 | \[ \Adv{sk}{\Pi}(A, n) = \max(\Pr[V], 2\Pr[W] - 1). \] |
| 2158 | Furthermore, we define the \emph{SK insecurity function of the |
| 2159 | protocol~$\Pi$} to be |
| 2160 | \[ \InSec{sk}(\Pi; t, n, q_S, q_M, q_{H_0}, q_{H_1}, \dots) = |
| 2161 | \max_A \Adv{sk}{\Pi}(A, n) |
| 2162 | \] |
| 2163 | where the maximum is taken over all adversaries~$A$ with total running |
| 2164 | time~$t$ (not including time taken by the parties), create at most $q_S$ |
| 2165 | sessions, deliver at most $q_M$~messages, and (if applicable) make at most |
| 2166 | $q_{H_i}$ random-oracle queries to each random oracle $H_i(\cdot)$. |
| 2167 | \end{definition} |
| 2168 | |
| 2169 | |
| 2170 | \subsection{Security} |
| 2171 | |
| 2172 | In order to analyse our protocol $\Wkx^{G, \E}$ properly, we must describe |
| 2173 | exactly how it fits into the formal model described in our formal model. |
| 2174 | |
| 2175 | \subsubsection{Sessions and session-ids} |
| 2176 | Our formal model introduced the concept of sessions, which the informal |
| 2177 | description of section~\ref{sec:kx-overview} neglected to do. (One could |
| 2178 | argue that we described a single session only.) As we shall show in |
| 2179 | section~\ref{sec:kx-insecure}, our protocol is \emph{insecure} unless we |
| 2180 | carefully modify it to distinguish between multiple sessions. |
| 2181 | |
| 2182 | In the model, distinct key-exchange sessions are given distinct partners and |
| 2183 | session-ids. In order to prevent sessions interfering with each other, we |
| 2184 | shall make explicit use of the session-ids. |
| 2185 | |
| 2186 | Suppose the session-ids are $\ell_S$-bit strings. We expand the domain of |
| 2187 | the random oracle $H_I$ so that it's now |
| 2188 | \[ H_I\colon G \times \Bin^{\ell_S} \times G \times G \to \Bin_{\ell_I}. \] |
| 2189 | |
| 2190 | \subsubsection{Messages} |
| 2191 | We split the messages our protocols into two parts: a \emph{type}~$\tau$ and |
| 2192 | a \emph{body}~$\mu$. We assume some convenient, unambiguous encoding of |
| 2193 | pairs $(\tau, \mu)$ as bit-strings. For readability, we present message |
| 2194 | types as text strings, e.g., `\cookie{challenge}', though in practice one |
| 2195 | could use numerical codes instead. |
| 2196 | |
| 2197 | The message body itself may be a tuple of values, which, again, we assume are |
| 2198 | encoded as bit-strings in some convenient and unambiguous fashion. We shall |
| 2199 | abuse the notation for the sake of readability by dropping a layer of nesting |
| 2200 | in this case: for example, we write $(\cookie{hello}, x, y, z)$ rather than |
| 2201 | $\bigl(\cookie{hello}, (x, y, z)\bigr)$. |
| 2202 | |
| 2203 | \subsubsection{The protocol} |
| 2204 | Our protocol is represented by three functions, shown in |
| 2205 | figure~\ref{fig:wkx-formal}. |
| 2206 | \begin{itemize} |
| 2207 | \item $\id{init}(n)$ is the initialization function, as described in |
| 2208 | section~\ref{sec:um}. It outputs a pair $(\mathbf{p}, \mathbf{i})$, where |
| 2209 | $\mathbf{i}[i]$ is the private key of party~$P_i$ and $\mathbf{p}[i]$ is |
| 2210 | the corresponding public key. Only $P_i$ is given $\mathbf{i}[i]$, whereas |
| 2211 | all parties and the adversary are given $\mathbf{p}$. |
| 2212 | \item $\id{new-session}^{H_I(\cdot, \cdot, \cdot, \cdot), H_K(\cdot)} |
| 2213 | (\mathbf{p}, x, i, j, s)$ is the new-session function. This is executed by |
| 2214 | party~$P_i$ when the adversary decides to create a new session~$S = (P_i, |
| 2215 | P_j, s)$. It is also given the relevant outputs of $\id{init}$, and |
| 2216 | allowed access to the random oracles $H_I$ and $H_K$. |
| 2217 | \item $\id{message}^{H_I(\cdot, \cdot, \cdot, \cdot), H_K(\cdot)} (\tau, |
| 2218 | \mu)$ is the incoming-message function. This is executed by a session when |
| 2219 | the adversary decides to deliver a message $(\tau, \mu)$ to it. It makes |
| 2220 | use of the subsidiary functions $\id{msg-challenge}$ and |
| 2221 | $\id{msg-response}$ to handle the messages. |
| 2222 | \end{itemize} |
| 2223 | We observe that the protocol never aborts. We could make it do so if it |
| 2224 | receives an invalid message, but we prefer to ignore invalid messages for the |
| 2225 | sake of robustness.\footnote{% |
| 2226 | Note that this protocol would therefore require modification before it was |
| 2227 | acceptable in the simulation-based model of \cite{Shoup:1999:OFM}. There |
| 2228 | it is required that a key-exchange protocol terminate after a |
| 2229 | polynomially-bounded number of messages are delivered to it.} |
| 2230 | |
| 2231 | \begin{figure} |
| 2232 | \begin{program} |
| 2233 | Function $\id{init}(n)$: \+ \\ |
| 2234 | \FOR $i \in \Nupto{n}$ \DO \\ \ind |
| 2235 | $x \getsr \Nupto{|G|}$; \\ |
| 2236 | $\mathbf{i}[i] \gets x$; \\ |
| 2237 | $\mathbf{p}[i] \gets x P$; \- \\ |
| 2238 | \RETURN $(\mathbf{p}, \mathbf{i})$; |
| 2239 | \- \\[\medskipamount] |
| 2240 | Function $\id{new-session}^{H_I(\cdot, \cdot, \cdot, \cdot), H_K(\cdot)} |
| 2241 | (\mathbf{p}, x, i, j, s)$: \+ \\ |
| 2242 | $X \gets \mathbf{p}[i]$; |
| 2243 | $X' \gets \mathbf{p}[j]$; |
| 2244 | $C \gets \emptyset$; \\ |
| 2245 | $r \getsr \Nupto{|G|}$; |
| 2246 | $R \gets r P$; |
| 2247 | $Y \gets r X'$; \\ |
| 2248 | $h \gets H_I(X, s, R, Y)$; |
| 2249 | $c \gets r \xor h$; \\ |
| 2250 | \SEND $(\cookie{challenge}, R, c)$; |
| 2251 | \- \\[\medskipamount] |
| 2252 | Function $\id{message}^{H_I(\cdot, \cdot, \cdot, \cdot), H_K(\cdot)} |
| 2253 | (\tau, \mu)$: \+ \\ |
| 2254 | \IF $\tau = \cookie{challenge}$ \THEN $\id{msg-challenge}(\mu)$; \\ |
| 2255 | \ELSE \IF $\tau = \cookie{response}$ \THEN $\id{msg-response}(\mu)$; |
| 2256 | \next |
| 2257 | Function $\id{msg-challenge}(\mu)$: \+ \\ |
| 2258 | $(R', c') \gets \mu$; \\ |
| 2259 | $Y' \gets x R'$; \\ |
| 2260 | $h' \gets H_I(X', s, R', Y')$; \\ |
| 2261 | $r' \gets c' \xor h'$; \\ |
| 2262 | \IF $R' \ne r' P$ \THEN \RETURN; \\ |
| 2263 | $C \gets C \cup \{R\}$; \\ |
| 2264 | $Z \gets r R'$; \\ |
| 2265 | $(K_0, K_1) \gets H_K(Z)$; \\ |
| 2266 | $\chi \gets E_{K_0}(Y')$; \\ |
| 2267 | \SEND $(\cookie{response}, R, \chi)$; |
| 2268 | \- \\[\medskipamount] |
| 2269 | Function $\id{msg-response}(\mu)$: \+ \\ |
| 2270 | $(R', \chi') \gets \mu$; \\ |
| 2271 | \IF $R' \notin C$ \THEN \RETURN; \\ |
| 2272 | $Z \gets r R'$; \\ |
| 2273 | $(K_0, K_1) \gets H_K(Z)$; \\ |
| 2274 | $Y' \gets D_{K_0}(\chi')$; \\ |
| 2275 | \IF $Y' \ne Y$ \THEN \RETURN; \\ |
| 2276 | \OUTPUT $K_1$; |
| 2277 | \STOP; |
| 2278 | \end{program} |
| 2279 | |
| 2280 | \caption{Formalization of $\Wkx$} |
| 2281 | \label{fig:wkx-formal} |
| 2282 | \end{figure} |
| 2283 | |
| 2284 | \subsubsection{Session states} |
| 2285 | We must specify what the adversary obtains when it chooses to reveal a |
| 2286 | session's state. Given the program in figure~\ref{fig:wkx-formal}, we can |
| 2287 | see that the session state consists of the variables $(x, X, X', r, R, Y, |
| 2288 | C)$. |
| 2289 | |
| 2290 | However, $x$ is the owning party's long-term secret, and it seems |
| 2291 | unreasonable to disclose this to an adversary who stops short of total |
| 2292 | corruption. |
| 2293 | |
| 2294 | The public keys $X$ and $X'$ are part of the adversary's input anyway, so |
| 2295 | revealing them doesn't help. Similarly, the set $C$ of valid challenges |
| 2296 | could have been computed easily by the adversary, since a group element $R' |
| 2297 | \in C$ if and only if the session $S$ responded to some message |
| 2298 | $(\cookie{challenge}, R', c')$. |
| 2299 | |
| 2300 | The value $R = r P$ is easily computed given $r$, and besides is sent in the |
| 2301 | clear in the session's first message. The expected response $Y = r X'$ is |
| 2302 | also easily computed from $r$. The converse is also true, since $r$ can be |
| 2303 | recovered from $R$ and $c$ in the session's challenge message and the value |
| 2304 | $Y$. Besides, $r$ is necessary for computing $Z$ in response to incoming |
| 2305 | challenges. |
| 2306 | |
| 2307 | We conclude that the only `interesting' session state is $r$. |
| 2308 | |
| 2309 | \subsubsection{Security} |
| 2310 | Having formally presented the protocol, we can now state our main theorem |
| 2311 | about its security. The proof is given in appendix~\ref{sec:sk-proof}. |
| 2312 | \begin{theorem}[SK-security of $\Wkx$] |
| 2313 | \label{thm:sk} |
| 2314 | Let $G$ be a cyclic group. Let $\E = (\kappa, E, D)$ be a symmetric |
| 2315 | encryption scheme. Then |
| 2316 | \begin{spliteqn*} |
| 2317 | \InSec{sk}(\Wkx^{G, \E}; t, n, q_S, q_M, q_I, q_K) \le |
| 2318 | 2 q_S \bigl( \InSec{ind-cca}(\E; t', q_M, q_M) + {} \\ |
| 2319 | \InSec{mcdh}(G; t', q_K) + |
| 2320 | n \,\InSec{mcdh}(G; t', q_M + q_I) \bigr) + |
| 2321 | \frac{n (n - 1)}{|G|} + |
| 2322 | \frac{2 q_M}{2^{\ell_I}}. |
| 2323 | \end{spliteqn*} |
| 2324 | where $t' = t + O(n) + O(q_S) + O(q_M q_I) + O(q_K)$. |
| 2325 | \end{theorem} |
| 2326 | |
| 2327 | |
| 2328 | \subsection{Insecure protocol variants} |
| 2329 | \label{sec:kx-insecure} |
| 2330 | |
| 2331 | It's important to feed the session-id and verifier's public key to the random |
| 2332 | oracle $H_I$ when constructing the check-value~$c$. Without these, the |
| 2333 | protocol is vulnerable to attack. In this section, we consider some variants |
| 2334 | of our protocol which hash less information, and show attacks against them. |
| 2335 | |
| 2336 | To simplify the presentation, we shall consider Alice and Bob, and a third |
| 2337 | character Carol. We shall be attacking a pair of matching sessions $A$ |
| 2338 | and~$B$, run by Alice and Bob respectively. Let Alice and Bob's private keys |
| 2339 | be $x_A$ and~$x_B$, and let their public keys be $X_A = x_A P$ and $X_B = x_B |
| 2340 | P$ respectively. |
| 2341 | |
| 2342 | \subsubsection{Protocol diagram notation} |
| 2343 | In order to keep the presentation as clear as possible, we use a simple |
| 2344 | diagrammatic notation for describing protocol runs. A line of the form |
| 2345 | \protocolrun{\textit{action} \ar[r] & \PRsession{S} & \textit{result}} |
| 2346 | states that the adversary performs the given \textit{action} on session~$S$, |
| 2347 | with the stated \textit{result}. The \textit{action} may be |
| 2348 | \begin{itemize} |
| 2349 | \item \textsf{Create session $(P_i, P_j, s)$}: the session is created, |
| 2350 | running in party~$P_i$, with partner~$P_j$ and session-id~$s$. |
| 2351 | \item \textsf{Receive $\mu$}: the session is activated with an incoming message~$\mu$. |
| 2352 | \item \textsf{Session-state reveal}: The adversary requests the session's |
| 2353 | internal state. |
| 2354 | \end{itemize} |
| 2355 | The \textit{result} may be |
| 2356 | \begin{itemize} |
| 2357 | \item \textsf{Send $\mu'$}: the session requests the delivery of the |
| 2358 | message~$\mu'$. |
| 2359 | \item \textsf{Complete: $K$}: the session completes, outputting the key~$K$. |
| 2360 | \item \textsf{State: $\sigma$}: the session's state is revealed to |
| 2361 | be~$\sigma$. |
| 2362 | \item \textsf{(Ignore)}: the result of the action is unimportant. |
| 2363 | \end{itemize} |
| 2364 | |
| 2365 | \subsubsection{Omitting the session-id} |
| 2366 | Consider a protocol variant where session $S$ sets $h_S = H_I(X_N, R_S, |
| 2367 | Y_S)$, where $N$ is the session's partner. That is, we've omitted the |
| 2368 | session-id from the hash. An adversary can cross over two sessions between |
| 2369 | Alice and Bob. Here's how. |
| 2370 | |
| 2371 | The attack assumes that Alice and Bob set up two pairs of matching sessions |
| 2372 | with each other, thus. |
| 2373 | \protocolrun{ |
| 2374 | \PRcreate{Alice}{Bob}{s} & \PRsession{A} & |
| 2375 | \PRsend{challenge}{R_A, c_A} \\ |
| 2376 | \PRcreate{Bob}{Alice}{s} & \PRsession{B} & |
| 2377 | \PRsend{challenge}{R_B, c_B} \\ |
| 2378 | \PRcreate{Alice}{Bob}{s'} & \PRsession{A'} & |
| 2379 | \PRsend{challenge}{R_{A'}, c_{A'}} \\ |
| 2380 | \PRcreate{Bob}{Alice}{s'} & \PRsession{B'} & |
| 2381 | \PRsend{challenge}{R_{B'}, c_{B'}} |
| 2382 | } |
| 2383 | Observe that the session pairs use distinct session-ids $s \ne s'$, so this |
| 2384 | is allowed. Now the adversary crosses over the challenges, using the second |
| 2385 | pair of sessions to provide responses to the challenges issued by the first |
| 2386 | pair. By revealing the state in the second pair of sessions, the adversary |
| 2387 | can work out the (probably different) session keys accepted by the first |
| 2388 | pair. |
| 2389 | \protocolrun{ |
| 2390 | \PRreceive{challenge}{R_B, c_B} & \PRsession{A'} & |
| 2391 | \PRsend{response}{R_{A'}, E_{K_{A'B,0}}(x_A R_B)} \\ |
| 2392 | \PRreceive{challenge}{R_A, c_A} & \PRsession{B'} & |
| 2393 | \PRsend{response}{R_{B'}, E_{K_{B'A,0}}(x_B R_A)} \\ |
| 2394 | \PRreceive{challenge}{R_{A'}, c_{A'}} & \PRsession{A} & \PRignore \\ |
| 2395 | \PRreceive{challenge}{R_{B'}, c_{B'}} & \PRsession{B} & \PRignore \\ |
| 2396 | \PRreveal & \PRsession{A'} & r_{A'} \\ |
| 2397 | \PRreveal & \PRsession{B'} & r_{B'} \\ |
| 2398 | \PRreceive{response}{R_{B'}, E_{K_{B'A,0}}(x_B R_A)} & \PRsession{A} & |
| 2399 | \PRcomplete{K_{AB',1}} \\ |
| 2400 | \PRreceive{response}{R_{A'}, E_{K_{A'B,0}}(x_A R_B)} & \PRsession{B} & |
| 2401 | \PRcomplete{K_{BA',1}} \\ |
| 2402 | } |
| 2403 | The adversary can now compute $K_{AB'} = H_K(r_{B'} R_A)$ and $K_{B'A} = |
| 2404 | H_K(r_{A'} R_B)$. Safely in possession of both keys, the adversary can now |
| 2405 | read and impersonate freely. |
| 2406 | |
| 2407 | \subsubsection{Omitting the partner's public key} |
| 2408 | Now consider a protocol variant where session $S$ sets $h_S = H_I(s, R_S, |
| 2409 | Y_S)$, where $s$ is the session-id. An adversary can use a sessions with |
| 2410 | Carol to attack a session between Alice and Bob. Here's how the sessions are |
| 2411 | set up. |
| 2412 | \protocolrun{ |
| 2413 | \PRcreate{Alice}{Bob}{s} & \PRsession{A} & |
| 2414 | \PRsend{challenge}{R_A, c_A} \\ |
| 2415 | \PRcreate{Bob}{Alice}{s} & \PRsession{B} & |
| 2416 | \PRsend{challenge}{R_B, c_B} \\ |
| 2417 | \PRcreate{Alice}{Carol}{s} & \PRsession{A'} & |
| 2418 | \PRsend{challenge}{R_{A'}, c_{A'}} \\ |
| 2419 | \PRcreate{Bob}{Carol}{s} & \PRsession{B'} & |
| 2420 | \PRsend{challenge}{R_{B'}, c_{B'}} |
| 2421 | } |
| 2422 | Although each of Alice and Bob have two sessions with session-id~$s$, this is |
| 2423 | allowed, since they are with different partners. The rest of the attack in |
| 2424 | fact proceeds identically to the previous case. |
| 2425 | |
| 2426 | |
| 2427 | \subsection{Deniability} |
| 2428 | \label{sec:denial} |
| 2429 | |
| 2430 | We have claimed that the Wrestlers key-exchange protocol is \emph{deniable}. |
| 2431 | In this section, we define what we mean, explain the limits of the |
| 2432 | deniablility of the protocol as currently presented, fix the protocol with an |
| 2433 | additional pass (which can be collapsed to a single message flow by breaking |
| 2434 | the protocol's symmetry), and prove the deniability of the resulting |
| 2435 | protocol. |
| 2436 | |
| 2437 | Our notion of deniability is taken from Di~Raimondo, Gennaro and Krawczyk |
| 2438 | \cite{DiRaimondo:2006:DAK}, except that, as usual, we opt for a concrete |
| 2439 | security approach. |
| 2440 | |
| 2441 | \subsubsection{Discussion} |
| 2442 | Our definition for deniability is that, for any adversary interacting with |
| 2443 | participants in the protocol, a simulator exists which can compute the same |
| 2444 | things as the adversary. In particular, since an adversary could output a |
| 2445 | transcript of the interactions between itself and the parties, it would |
| 2446 | follow that a simulator could do this too. If the simulator is effective, |
| 2447 | its output is indistinguishable from that of the real adversary, and hence no |
| 2448 | `judge' (distinguisher) should be persuaded by evidence presented by someone |
| 2449 | who claims to have witnessed or participated in an interaction. |
| 2450 | |
| 2451 | We work again the model described in~\ref{sec:um}. That is, our adversary |
| 2452 | has complete control over the ordering and delivery of all messages. The |
| 2453 | adversary is also able, at any time, to reveal the state of any session. |
| 2454 | However, deniability is obviously impossible against an adversary who can |
| 2455 | \emph{corrupt} other parties, since simulating such an adversary's actions |
| 2456 | would necessarily require the simulator to compute private keys corresponding |
| 2457 | to the known public keys, and this is (we believe) difficult, because an |
| 2458 | efficient algorithm for doing this could easily attack our protocol, which we |
| 2459 | already proved secure. Therefore, we forbid the adversary from corrupting |
| 2460 | parties. |
| 2461 | |
| 2462 | In order to allow the adversary to participate in the protocol, rather than |
| 2463 | merely observing it, we need to give it one or more private keys. We could |
| 2464 | modify the initialization function \id{init} from figure~\ref{fig:wkx-formal} |
| 2465 | to give the adversary a private key, but there's an easier way: we can just |
| 2466 | give the adversary a number of private keys in its auxiliary input. |
| 2467 | |
| 2468 | \subsubsection{Definitions} |
| 2469 | Let $\Pi$ be a key-exchange protocol, in the model described in |
| 2470 | section~\ref{sec:um}. We use the simulation framework of |
| 2471 | section~\ref{sec:sim}. We define the initialization function $I_\Pi$ to be |
| 2472 | the initialization function of $\Pi$, as before, and the corresponding |
| 2473 | world~$W_\Pi(\iota, \sigma, \tau, \mu)$ is a fairly straightforward mapping |
| 2474 | of the adversary's possible actions to the simulation model: |
| 2475 | \begin{itemize} |
| 2476 | \item The invocation $\cookie{new-session}$ with $\mu = (i, j, s)$ creates a |
| 2477 | new session on party~$P_i$, with partner~$P_j$ and session-id~$s$. The |
| 2478 | reply $\rho = (\delta, m)$ is a \emph{decision} $\delta \in |
| 2479 | \{\cookie{continue}, \cookie{abort}, \cookie{complete}\}$ and an output |
| 2480 | message $m \in \Bin^* \cup \{\bot\}$. If $m \ne \bot$ then $m$ is a |
| 2481 | message to be sent to the matching session (if any). |
| 2482 | \item The invocation $\cookie{deliver}$ with $\mu = (i, j, s, m)$ delivers |
| 2483 | message $m$ to the session $S = (P_i, P_j, s)$. The response $\rho$ is as |
| 2484 | for $\cookie{new-session}$ invocations. |
| 2485 | \item The invocation $\cookie{reveal-session-state}$ with $\mu = (i, j, s)$ |
| 2486 | reveals to the adversary the state of the running session $S = (P_i, P_j, |
| 2487 | s)$. The response $\rho$ is the session's state if $S$ is indeed a running |
| 2488 | session, or $\bot$ otherwise. |
| 2489 | \item The invocation $\cookie{reveal-session-key}$ with $\mu = (i, j, s)$ |
| 2490 | reveals to the adversary the session-key of the completed session~$S = |
| 2491 | (P_i, P_j, s)$. The response $\rho$ is the session key~$K$ if the session |
| 2492 | is indeed complete, or $\bot$ otherwise. |
| 2493 | \end{itemize} |
| 2494 | There are no invocations corresponding to the adversary actions of corrupting |
| 2495 | parties (since deniability against an corrupting adversary is impossible, as |
| 2496 | discussed earlier), or session expiry or challenging (since they are useless |
| 2497 | in this context). |
| 2498 | |
| 2499 | We measure the deniability of a protocol~$\Pi$, using a given simulator~$S$, |
| 2500 | by the insecurity function $\InSec{sim}(W_\Pi, I_\Pi, S; t_D, t_A, |
| 2501 | \mathcal{Q}_D, \mathcal{Q}_A, \mathcal{R}, \mathcal{U})$ of |
| 2502 | definition~\ref{def:sim}. The interaction bounds $\mathcal{R} = (q_S, q_M)$ |
| 2503 | we place on the adversary are on the number ($q_S$) of \cookie{new-session} |
| 2504 | and ($q_M$) \cookie{deliver} invocations it makes. |
| 2505 | |
| 2506 | We shall (informally) say that a protocol~$\Pi$ is deniable if there is a |
| 2507 | simulator~$S_\Pi$ for which the insecurity function is small for appropriate |
| 2508 | resource bounds on the adversary and distinguisher. |
| 2509 | |
| 2510 | \subsubsection{The current protocol} |
| 2511 | As it stands, $\Wkx$ isn't deniable, according to our definition, for |
| 2512 | arbitrary auxiliary inputs. Let's see why. |
| 2513 | |
| 2514 | Suppose that Bob is an informant for the secret police, and wants to convince |
| 2515 | a judge that Alice is involved in subversive activities in cyberspace. |
| 2516 | Unfortunately, Bob's job is difficult, because of the strong zero-knowledge |
| 2517 | nature of the Wrestlers identification protocol. However, Bob can work with |
| 2518 | the judge to bring him the evidence necessary to convict Alice. Here's how. |
| 2519 | |
| 2520 | Alice's public key is $A$, and Bob's public key is $B$. The judge chooses |
| 2521 | some session-id $s$, and $r \inr \Nupto{q}$. He computes $R = r P$ and $c = |
| 2522 | r \xor H_I(B, s, R, r A)$, and gives Bob the triple $(s, R, c)$, keeping $r$ |
| 2523 | secret. Bob can now persuade Alice to enter into a key-exchange with him, |
| 2524 | with session-id $s$. He uses $(R, c)$ as his challenge message. When Alice |
| 2525 | sends back her response $(R', \chi)$ (because Bob's challenge is correctly |
| 2526 | formed), Bob aborts and shows $(R', \chi)$ to the judge. The judge knows $r$ |
| 2527 | and can therefore decrypt $\chi$ and check the response. If it's wrong, then |
| 2528 | Bob was cheating and gets demoted -- he can't get the right answer by himself |
| 2529 | because that would require him to impersonate Alice. If it's right, Alice is |
| 2530 | really a subversive element and `disappears' in the night. |
| 2531 | |
| 2532 | We shall show in theorem~\ref{thm:denial} below that this is basically the |
| 2533 | only attack against the deniability of the protocol. However, we can do |
| 2534 | better. |
| 2535 | |
| 2536 | \subsubsection{Fixing deniability} |
| 2537 | We can fix the protocol to remove even the attack discussed above. The |
| 2538 | change is simple: we feed \emph{both} parties' challenges to the hash |
| 2539 | function~$H_I$ rather than just the sender's. We use a five-argument hash |
| 2540 | function (random oracle) $H_I\colon G^2 \times \Bin^{\ell_S} \times G^2 \to |
| 2541 | \Bin^{\ell_I}$. We introduce a new message pass at the beginning of the |
| 2542 | protocol: each session simply sends its challenge point $R = r P$ in the |
| 2543 | clear as a `pre-challenge'. The actual challenge is $R$ and $c = r \xor |
| 2544 | H_I(X, R', s, R, c)$, where $R'$ is the challenge of the matching session. |
| 2545 | |
| 2546 | By breaking symmetry, we can reduce the communication complexity of this |
| 2547 | variant to four messages. As before, we analyse the symmetrical version. |
| 2548 | The extra flow might seem a high price to pay, but we shall see that it has |
| 2549 | additional benefits beyond deniability. |
| 2550 | |
| 2551 | A summary of the new protocol is shown in figure~\ref{fig:wdkx}, and the |
| 2552 | formal description is shown in figure~\ref{fig:wdkx-formal}. |
| 2553 | |
| 2554 | \begin{figure} |
| 2555 | \begin{description} |
| 2556 | \item[Setup] Group $G = \langle P \rangle$; $|G| = q$ is prime. |
| 2557 | $H_I(\cdot, \cdot, \cdot, \cdot, \cdot)$ and $H_K(cdot)$ are secure |
| 2558 | hashes. $\E = (\kappa, E, D)$ is an IND-CCA2 symmetric encryption |
| 2559 | scheme. |
| 2560 | \item[Parties] $U_i$ for $0 \le i < n$. |
| 2561 | \item[Private keys] $x_i \inr \Nupto{q}$. |
| 2562 | \item[Public keys] $X_i = x_i P$. |
| 2563 | \end{description} |
| 2564 | \[ \begin{graph} |
| 2565 | !{0; <8cm, 0cm>: <0cm, 3cm>::} |
| 2566 | *+[F]\dbox{$r_i \getsr I$; $R_i \gets r_i P$} ="i-1" |
| 2567 | [d] |
| 2568 | *+[F]\dbox{$c_i \gets r_i \xor H_I(R_j, X_i, s, R_i, r_i X_j)$} ="i0" |
| 2569 | [d] |
| 2570 | *+[F]\dbox{Check $R_j = \bigl(c_j \xor H_I(x_i R_j)\bigr) P$ \\ |
| 2571 | $Z \gets r_i R_j$; $K \gets H_K(0, Z)$ \\ |
| 2572 | $\chi_i \gets E_K(x_i R_j)$} ="i1" |
| 2573 | [d] |
| 2574 | *+[F]\dbox{Check $D_K(\chi_j) = r_i X_j$ \\ |
| 2575 | Shared key is $H_K(1, Z)$} ="i2" |
| 2576 | "i-1" [r] |
| 2577 | *+[F]\dbox{$r_j \getsr I$; $R_j \gets r_j P$} ="j-1" |
| 2578 | [d] |
| 2579 | *+[F]\dbox{$c_j \gets r_j \xor H_I(R_i, X_j, s, R_j, r_j X_i)$} ="j0" |
| 2580 | [d] |
| 2581 | *+[F]\dbox{Check $R_i = \bigl(c_i \xor H_I(x_j R_i)\bigr) P$ \\ |
| 2582 | $Z \gets r_j R_i$; $K \gets H_K(0, Z)$ \\ |
| 2583 | $\chi_j \gets E_K(x_j R_i)$} ="j1" |
| 2584 | [d] |
| 2585 | *+[F]\dbox{Check $D_K(\chi_i) = r_j X_i$ \\ |
| 2586 | Shared key is $H_K(1, Z)$} ="j2" |
| 2587 | % |
| 2588 | "i-1" : |(0)/3.25cm/*+{R_i} "j0" |
| 2589 | "i0" : |(0)/3.25cm/*+{(R_i, c_i)} "j1" |
| 2590 | "i1" : |(0)/3.25cm/*+{(R_i, \chi_i)} "j2" |
| 2591 | "j-1" : |(0)/3.25cm/*+{R_j} "i0" |
| 2592 | "j0" : |(0)/3.25cm/*+{(R_j, c_j)} "i1" |
| 2593 | "j1" : |(0)/3.25cm/*+{(R_j, \chi_j)} "i2" |
| 2594 | \end{graph} \] |
| 2595 | |
| 2596 | \caption{Summary of the Deniable Wrestlers Key Exchange protocol, $\Wdkx$} |
| 2597 | \label{fig:wdkx} |
| 2598 | \end{figure} |
| 2599 | |
| 2600 | \begin{figure} |
| 2601 | \begin{program} |
| 2602 | Function $\id{init}(n)$: \+ \\ |
| 2603 | \FOR $i \in \Nupto{n}$ \DO \\ \ind |
| 2604 | $x \getsr \Nupto{|G|}$; \\ |
| 2605 | $\mathbf{i}[i] \gets x$; \\ |
| 2606 | $\mathbf{p}[i] \gets x P$; \- \\ |
| 2607 | \RETURN $(\mathbf{p}, \mathbf{i})$; |
| 2608 | \- \\[\medskipamount] |
| 2609 | Function $\id{new-session}^{H_I(\cdot, \cdot, \cdot, \cdot, \cdot), H_K(\cdot)} |
| 2610 | (\mathbf{p}, x, i, j, s)$: \+ \\ |
| 2611 | $X \gets \mathbf{p}[i]$; |
| 2612 | $X' \gets \mathbf{p}[j]$; |
| 2613 | $C \gets \emptyset$; \\ |
| 2614 | $r \getsr \Nupto{|G|}$; |
| 2615 | $R \gets r P$; |
| 2616 | $Y \gets r X'$; \\ |
| 2617 | \SEND $(\cookie{pre-challange}, R)$; |
| 2618 | \- \\[\medskipamount] |
| 2619 | Function $\id{message}^{H_I(\cdot, \cdot, \cdot, \cdot, \cdot), H_K(\cdot)} |
| 2620 | (\tau, \mu)$: \+ \\ |
| 2621 | \IF $\tau = \cookie{pre-challenge}$ \THEN |
| 2622 | $\id{msg-pre-challenge}(\mu)$; \\ |
| 2623 | \ELSE \IF $\tau = \cookie{challenge}$ \THEN |
| 2624 | $\id{msg-challenge}(\mu)$; \\ |
| 2625 | \ELSE \IF $\tau = \cookie{response}$ \THEN $\id{msg-response}(\mu)$; |
| 2626 | \next |
| 2627 | Function $\id{msg-pre-challenge}(\mu)$: \+ \\ |
| 2628 | $R' \gets \mu$; \\ |
| 2629 | $h \gets H_I(R', X, s, R, c)$; \\ |
| 2630 | $c \gets r \xor h$; \\ |
| 2631 | \SEND $(\id{msg-challenge}, R, c)$; |
| 2632 | \- \\[\medskipamount] |
| 2633 | Function $\id{msg-challenge}(\mu)$: \+ \\ |
| 2634 | $(R', c') \gets \mu$; \\ |
| 2635 | $Y' \gets x R'$; \\ |
| 2636 | $h' \gets H_I(R, X', s, R', Y')$; \\ |
| 2637 | $r' \gets c' \xor h'$; \\ |
| 2638 | \IF $R' \ne r' P$ \THEN \RETURN; \\ |
| 2639 | $C \gets C \cup \{R\}$; \\ |
| 2640 | $Z \gets r R'$; \\ |
| 2641 | $(K_0, K_1) \gets H_K(Z)$; \\ |
| 2642 | $\chi \gets E_{K_0}(Y')$; \\ |
| 2643 | \SEND $(\cookie{response}, R, \chi)$; |
| 2644 | \- \\[\medskipamount] |
| 2645 | Function $\id{msg-response}(\mu)$: \+ \\ |
| 2646 | $(R', \chi') \gets \mu$; \\ |
| 2647 | \IF $R' \notin C$ \THEN \RETURN; \\ |
| 2648 | $Z \gets r R'$; \\ |
| 2649 | $(K_0, K_1) \gets H_K(Z)$; \\ |
| 2650 | $Y' \gets D_{K_0}(\chi')$; \\ |
| 2651 | \IF $Y' \ne Y$ \THEN \RETURN; \\ |
| 2652 | \OUTPUT $K_1$; |
| 2653 | \STOP; |
| 2654 | \end{program} |
| 2655 | |
| 2656 | \caption{Deniable key-exchange: formalization of $\Wdkx$} |
| 2657 | \label{fig:wdkx-formal} |
| 2658 | \end{figure} |
| 2659 | |
| 2660 | The security of this variant is given by the following theorem, whose proof |
| 2661 | is in appendix~\ref{sec:sk2-proof}. |
| 2662 | \begin{theorem}[SK-security of $\Wdkx$] |
| 2663 | \label{thm:sk2} |
| 2664 | Let $G$ be a cyclic group. Let $\E = (\kappa, E, D)$ be a symmetric |
| 2665 | encryption scheme. Then |
| 2666 | \[ \InSec{sk}(\Wdkx^{G, \E}; t, n, q_S, q_M, q_I, q_K) = |
| 2667 | \InSec{sk}(\Wkx^{G, \E}; t, n, q_S, q_M, q_I, q_K) |
| 2668 | \] |
| 2669 | \end{theorem} |
| 2670 | |
| 2671 | \subsubsection{Deniability of the Wrestlers protocols} |
| 2672 | In order to quantify the level of deniability our protocols provide, we shall |
| 2673 | impose a limit on the auxiliary input to the adversary. In particular, we |
| 2674 | shall use $\mathcal{U}$ of definition~\ref{def:sim} to count the number of |
| 2675 | \emph{challenges} in the auxiliary input. That is, $\mathcal{U} = n_C$ is |
| 2676 | the number of tuples $(i, j, s, R', R, c)$ for which there is an $r$ such |
| 2677 | that $R = r P$ and $c = r \xor H_I(R', X_j, s, R, r X_i)$ (or without the |
| 2678 | $R'$ for $\Wkx$). |
| 2679 | |
| 2680 | With this restriction in place, we can state the following theorem about the |
| 2681 | deniability of our protocols. |
| 2682 | \begin{theorem}[Deniability of $\Wkx$ and $\Wdkx$] |
| 2683 | \label{thm:denial} |
| 2684 | There exist simulators $S_\Wkx$ and $\Wdkx$ such that |
| 2685 | \[ \InSec{sim}(W_{\Wkx^{G, \E}}, I_{\Wkx^{G, \E}}, S_{\Wkx^{G, \E}}; |
| 2686 | t_D, t_A, \mathcal{Q}_D, \mathcal{Q}_A, (q_S, q_M), 0) \le |
| 2687 | \frac{q_M}{2^{\ell_I}} |
| 2688 | \] |
| 2689 | and |
| 2690 | \[ \InSec{sim}(W_{\Wdkx^{G, \E}}, I_{\Wdkx^{G, \E}}, S_{\Wdkx^{G, \E}}; |
| 2691 | t_D, t_A, \mathcal{Q}_D, \mathcal{Q}_A, (q_S, q_M), n_C) \le |
| 2692 | \frac{n_C q_S}{|G|} + |
| 2693 | \frac{q_M}{2^{\ell_I}}. |
| 2694 | \] |
| 2695 | The running time of the simulators is $O(t_A) + O(\mathcal{Q}_A q_M)$. |
| 2696 | \end{theorem} |
| 2697 | \begin{proof} |
| 2698 | The simulators $S_\Wkx$ and $S_\Wdkx$ are very similar. We describe both |
| 2699 | here. Both are fake-world simulators, working as follows. |
| 2700 | \begin{enumerate} |
| 2701 | \item Initially, it constructs simulated parties $P_i$, for $0 \le i < n$, |
| 2702 | giving each the public key $X_i$ from the common input. |
| 2703 | \item Suppose the adversary requests creation of a new session $S = (P_i, |
| 2704 | P_j, s)$. Then the simulator creates a new session, including a random |
| 2705 | value $r_S \inr \Nupto{|G|}$, and computes $R_S = r_S P$, and $Y_S = r_S |
| 2706 | X_j$. For $\Wdkx$, it sends the message $(\cookie{pre-challenge}, R_S)$; |
| 2707 | for $\Wkx$, it additionally computes $h = H_I(X_i, s, R_S, Y_S)$ and |
| 2708 | sends $(\cookie{challenge}, R_S, r_S \xor h)$. |
| 2709 | \item Suppose, for $\Wdkx$, the adversary sends a message |
| 2710 | $(\cookie{pre-challenge}, R')$ to a session~$S = (P_i, P_j, s)$. The |
| 2711 | simulator computes $h = H_I(R', X_i, s, R_S, Y_S)$, and sends |
| 2712 | $(\cookie{challenge}, R_S, r_S \xor h)$. |
| 2713 | \item Suppose the adversary sends a message $(\cookie{challenge}, R', c')$ |
| 2714 | to session $S = (P_i, P_j, s)$. The simulator doesn't know $x_i$. |
| 2715 | \begin{enumerate} |
| 2716 | \item If $R' = R_T$ for some other simulated session $T$, then the |
| 2717 | simulator knows $r_T$ such that $R_T = r_T P$. Let $Y' = r_T X_i$. |
| 2718 | The simulator computes $h = H_I(X_j, s, R', Y')$ (resp.\ $h = H_I(R_S, |
| 2719 | X_j, s, R', Y')$) for $\Wkx$ (resp.\ $\Wdkx$) and checks that $r_T = c' |
| 2720 | \xor h$. If not, the simulator discards the message. Otherwise, it |
| 2721 | computes $(K_0, K_1) = H_K(r_S R')$, and sends the message |
| 2722 | $(\cookie{response}, R, E_{K_0}(Y'))$. |
| 2723 | \item \label{en:simextract} Otherwise the simulator runs the extractor |
| 2724 | $T_\Wident$ on the adversary's history of queries $H_I(X_j, s, R', |
| 2725 | \cdot)$ (resp.\ $H_I(R_S, X_j, s, R', \cdot)$) for $\Wkx$ (resp.\ |
| 2726 | $\Wdkx$). The extractor returns $(r', Y')$. If $Y' = \bot$ then the |
| 2727 | simulator ignores the message. Otherwise, the simulator computes |
| 2728 | $(K_0, K_1) = H_K(r R')$ and sends back $(\cookie{response}, R, |
| 2729 | E_{K_0}(Y'))$. |
| 2730 | \end{enumerate} |
| 2731 | \item Suppose the adversary sends a message $(\cookie{response}, R', \chi)$ |
| 2732 | to session $S = (P_i, P_j, s)$. The simulator computes $(K_0, K_1) = |
| 2733 | H_K(r_S R')$, and decrypts $Y' = D_{K_0}(\chi)$. If $Y' \ne Y_S$ then |
| 2734 | the simulator discards the message. Otherwise, it makes the simulated |
| 2735 | session complete, and outputs key $K_1$. |
| 2736 | \item Finally, if the adversary reveals a session's state, the simulator |
| 2737 | reveals $r_S$ as required; if the adversary reveals a session-key, the |
| 2738 | simulator reveals the $K_1$ it output. |
| 2739 | \end{enumerate} |
| 2740 | The only point where the simulation fails to be perfect is in |
| 2741 | \ref{en:simextract}. Let $R'$ and $c'$ be the values from an incoming |
| 2742 | challenge message to session $S = (P_i, P_j, s)$. Let $r'$ be such that |
| 2743 | $R' = r' P$ and let $Y' = r' X_i$. If a random-oracle query $H_I(X_j, s, |
| 2744 | R', Y')$ (or $H_I(R_S, X_j, s, R', Y')$ for $\Wdkx$) has been issued, then |
| 2745 | there are a number of possibilities. Let $h'$ be the result of this query. |
| 2746 | \begin{itemize} |
| 2747 | \item The adversary made this query. Then the extractor will find it and |
| 2748 | return $Y'$ if $c' = h' \xor r'$, or $\bot$ otherwise. |
| 2749 | \item Some simulated session $U = (P_{i'}, P_{j'}, s')$ made this query. |
| 2750 | But simulated sessions only make $H_I$-queries when constructing |
| 2751 | challenges, so $R' = R_U$ for some session~$U$. But the simulator does |
| 2752 | something different in that case. |
| 2753 | \item In $\Wdkx$, the quadruple $(s, R_S, R', c')$ came from the |
| 2754 | adversary's auxiliary input. In this case the simulator must fail. But |
| 2755 | $R_S = r_S P$, and $r_S$ was chosen uniformly at random. If there are at |
| 2756 | most $n_C$ challenge sets in the auxiliary input then this happens with |
| 2757 | probability at most $n_C/|G|$ for any given session. |
| 2758 | \end{itemize} |
| 2759 | We conclude that the simulator fails with probability |
| 2760 | \[ \frac{q_M}{2^{\ell_I}} + \frac{q_S n_C}{|G|}. \] |
| 2761 | (Note that we only consider $n_C = 0$ for $\Wkx$.) No adversary can |
| 2762 | distinguish the simulator from a real interaction unless the simulator |
| 2763 | fails, and the simulator is a fake-world simulator. We therefore apply |
| 2764 | proposition~\ref{prop:fakesim}; the theorem follows. |
| 2765 | \end{proof} |
| 2766 | |
| 2767 | |
| 2768 | \subsection{Practical issues} |
| 2769 | \label{sec:practice} |
| 2770 | |
| 2771 | \subsubsection{Denial of service from spoofers} |
| 2772 | The adversary we considered in~\ref{sec:um} is very powerful. Proving |
| 2773 | security against such a powerful adversary is good and useful. However, |
| 2774 | there are other useful security properties we might like against weaker |
| 2775 | adversaries. |
| 2776 | |
| 2777 | Eavesdropping on the Internet is actually nontrivial. One needs control of |
| 2778 | one of the intermediate routers between two communicating parties. (There |
| 2779 | are tricks one can play involving subversion of DNS servers, but this is also |
| 2780 | nontrivial.) However, injecting packets with bogus source addresses is very |
| 2781 | easy. |
| 2782 | |
| 2783 | Layering the protocol over TCP \cite{RFC0793} ameliorates this problem because |
| 2784 | an adversary needs to guess or eavesdrop in order to obtain the correct |
| 2785 | sequence numbers for a spoofed packet; but the Wrestlers protocol is |
| 2786 | sufficiently simple that we'd like to be able to run it over UDP |
| 2787 | \cite{RFC0768}, for which spoofing is trivial. |
| 2788 | |
| 2789 | Therefore, it's possible for anyone on the 'net to send Alice a spurious |
| 2790 | challenge message $(R, c)$. She will then compute $Y = a R$, recover $r' = c |
| 2791 | \xor H_I(\ldots, R, Y)$ check that $R = r' P$ and so on. That's at least two |
| 2792 | scalar multiplications to respond to a spoofed packet, and even with very |
| 2793 | efficient group operations, coping with this kind of simple denial-of-service |
| 2794 | attack might be difficult. |
| 2795 | |
| 2796 | A straightforward solution is to use the Deniable variant of the protocol, |
| 2797 | and require a challenge to quote its matching session's challenge $R'$ in its |
| 2798 | challenge. That is, upon receiving a $(\cookie{pre-challenge}, R')$, the |
| 2799 | session sends $(\cookie{challenge}, R', R, c)$. Alice then rejects any |
| 2800 | \emph{incoming} challenge message which doesn't quote her current challenge |
| 2801 | value. Now only eavesdroppers can force her to perform expensive |
| 2802 | computations. |
| 2803 | |
| 2804 | Indeed, one need not quote the entire challenge $R'$: it suffices to send |
| 2805 | some short hard-to-guess hash of it, maybe just the bottom 128 bits or so. |
| 2806 | |
| 2807 | This can't reduce security. Consider any adversary attacking this protocol |
| 2808 | variant. We can construct an adversary which attacks the original protocol |
| 2809 | just as efficiently. The new adversary attaches fake $R'$ values to |
| 2810 | challenges output by other parties, and strips them off on delivery, |
| 2811 | discarding messages with incorrect $R'$ values. |
| 2812 | |
| 2813 | \subsubsection{Key confirmation} |
| 2814 | Consider an application which uses the Wrestlers protocol to re-exchange keys |
| 2815 | periodically. The application can be willing to \emph{receive} incoming |
| 2816 | messages using the new key as soon as the key exchange completes |
| 2817 | successfully; however, it should refrain from \emph{sending} messages under |
| 2818 | the new key until it knows that its partner has also completed. The partner |
| 2819 | may not have received the final response message, and therefore be unwilling |
| 2820 | to accept a new key; it will therefore (presumably) reject incoming messages |
| 2821 | under this new key. |
| 2822 | |
| 2823 | While key confirmation is unnecessary for \emph{security}, it has |
| 2824 | \emph{practical} value, since it solves the above problem. If the |
| 2825 | application sends a \cookie{switch} message when it `completes', it can |
| 2826 | signal its partner that it is indeed ready to accept messages under the new |
| 2827 | key. Our implementation sends $(\cookie{switch-rq}, E_{K_0}(H_S(0, R, R')))$ |
| 2828 | as its switch message; the exact contents aren't important. Our |
| 2829 | retransmission policy (below) makes use of an additional message |
| 2830 | \cookie{switch-ok}, which can be defined similarly. |
| 2831 | |
| 2832 | It's not hard to show that this doesn't adversely affect the security of the |
| 2833 | protocol, since the encrypted message is computed only from public values. |
| 2834 | In the security proof, we modify the generation of \cookie{response} |
| 2835 | messages, so that the plaintexts are a constant string rather than the true |
| 2836 | responses, guaranteeing that the messages give no information about the |
| 2837 | actual response. To show this is unlikely to matter, we present an adversary |
| 2838 | attacking the encryption scheme by encrypting either genuine responses or |
| 2839 | fake constant strings. Since the adversary can't distinguish which is being |
| 2840 | encrypted (by the definition of IND-CCA security, |
| 2841 | definition~\ref{def:ind-cca}), the change goes unnoticed. In order to allow |
| 2842 | incorporate our switch messages, we need only modify this adversary, to |
| 2843 | implement the modified protocol. This is certainly possible, since the |
| 2844 | messages contain (hashes of) public values. We omit the details. |
| 2845 | |
| 2846 | However, while the extra message doesn't affect the security of our protocol, |
| 2847 | it would be annoying if an adversary could forge the switch request message, |
| 2848 | since this would be a denial of service. In the strong adversarial model, |
| 2849 | this doesn't matter, since the adversary can deny service anyway, but it's a |
| 2850 | concern against less powerful adversaries. Most IND-CCA symmetric encryption |
| 2851 | schemes also provide integrity of plaintexts \cite{Bellare:2000:AER} (e.g., |
| 2852 | the encrypt-then-MAC generic composition approach \cite{Bellare:2000:AER,% |
| 2853 | Krawczyk:2001:OEA}, and the authenticated-encryption modes of |
| 2854 | \cite{Rogaway:2003:OCB,Bellare:2004:EAX,McGrew:2004:SPG}), so this isn't a |
| 2855 | great imposition. |
| 2856 | |
| 2857 | \subsubsection{Optimization and piggybacking} |
| 2858 | We can optimize the number of messages sent by combining them. Here's one |
| 2859 | possible collection of combined messages: |
| 2860 | \begin{description} |
| 2861 | \item [\cookie{pre-challenge}] $R$ |
| 2862 | \item [\cookie{challenge}] $R'$, $R$, $c = H_I(R', X, s, R, c) \xor r$ |
| 2863 | \item [\cookie{response}] $R'$, $R$, $c$, $E_{K_0}(x R')$ |
| 2864 | \item [\cookie{switch}] $R'$, $E_{K_0}(x R', H_S(0, R, R'))$ |
| 2865 | \item [\cookie{switch-ok}] $R'$, $E_{K_0}(H_S(1, R, R'))$ |
| 2866 | \end{description} |
| 2867 | The combination is safe: |
| 2868 | \begin{itemize} |
| 2869 | \item the \cookie{switch} and \cookie{switch-ok} messages are safe by the |
| 2870 | argument above; and |
| 2871 | \item the other recombinations can be performed and undone in a `black box' |
| 2872 | way, by an appropriately defined SK-security adversary. |
| 2873 | \end{itemize} |
| 2874 | |
| 2875 | \subsubsection{Unreliable transports} |
| 2876 | The Internet UDP \cite{RFC0768} is a simple, unreliable protocol for |
| 2877 | transmitting datagrams. However, it's very efficient, and rather attractive |
| 2878 | as a transport for datagram-based applications such as virtual private |
| 2879 | networks (VPNs). Since UDP is a best-effort rather than a reliable |
| 2880 | transport, it can occasionally drop packets. Therefore it is necessary for a |
| 2881 | UDP application to be able to retransmit messages which appear to have been |
| 2882 | lost. |
| 2883 | |
| 2884 | We recommend the following simple retransmission policy for running the |
| 2885 | Wrestlers protocol over UDP. |
| 2886 | \begin{itemize} |
| 2887 | \item Initially, send out the \cookie{pre-challenge} message every minute. |
| 2888 | \item On receipt of a \cookie{pre-challenge} message, send the corresponding |
| 2889 | full \cookie{challenge}, but don't retain any state. |
| 2890 | \item On receipt of a (valid) \cookie{challenge}, record the challenge value |
| 2891 | $R'$ in a table, together with $K = (K_0, K_1)$ and the response $Y' = x |
| 2892 | R'$. If the table is full, overwrite an existing entry at random. Send |
| 2893 | the corresponding \cookie{response} message, and retransmit it every ten |
| 2894 | seconds or so. |
| 2895 | \item On receipt of a (valid) \cookie{response}, discard any other |
| 2896 | challenges, and stop sending \cookie{pre-challenge} and \cookie{response} |
| 2897 | retransmits. At this point, the basic protocol described above would |
| 2898 | \emph{accept}, so the key $K_1$ is known to be good. Send the |
| 2899 | \cookie{switch} message, including its response to the (now known-good) |
| 2900 | sender's challenge. |
| 2901 | \item On receipt of a (valid) \cookie{switch}, send back a \cookie{switch-ok} |
| 2902 | message and stop retransmissions. It is now safe to start sending messages |
| 2903 | under $K_1$. |
| 2904 | \item On receipt of a (valid) \cookie{switch-ok}, stop retransmissions. It |
| 2905 | is now safe to start sending messages under $K_1$. |
| 2906 | \end{itemize} |
| 2907 | |
| 2908 | \subsubsection{Key reuse} |
| 2909 | Suppose our symmetric encryption scheme $\E$ is not only IND-CCA secure |
| 2910 | (definition~\ref{def:ind-cca}) but also provides integrity of plaintexts |
| 2911 | \cite{Bellare:2000:AER} (or, alternatively, is an AEAD scheme |
| 2912 | \cite{Rogaway:2002:AEA}. Then we can use it to construct a secure channel, |
| 2913 | by including message type and sequence number fields in the plaintexts, along |
| 2914 | with the message body. If we do this, we can actually get away with just the |
| 2915 | one key $K = H_K(Z)$ rather than both $K_0$ and $K_1$. |
| 2916 | |
| 2917 | To do this, it is essential that the plaintext messages (or additional data) |
| 2918 | clearly distinguish between the messages sent as part of the key-exchange |
| 2919 | protocol and messages sent over the `secure channel'. Otherwise, there is a |
| 2920 | protocol-interference attack: an adversary can replay key-exchange |
| 2921 | ciphertexts to insert the corresponding plaintexts into the channel. |
| 2922 | |
| 2923 | We offer a sketch proof of this claim in appendix~\ref{sec:sc-proof}. |
| 2924 | |
| 2925 | %%%-------------------------------------------------------------------------- |
| 2926 | |
| 2927 | \section{Conclusions} |
| 2928 | \label{sec:conc} |
| 2929 | |
| 2930 | We have presented new protocols for identification and authenticated |
| 2931 | key-exchange, and proven their security. We have shown them to be efficient |
| 2932 | and simple. We have also shown that our key-exchange protocol is deniable. |
| 2933 | Finally, we have shown how to modify the key-exchange protocol for practical |
| 2934 | use, and proven that this modification is still secure. |
| 2935 | |
| 2936 | %%%-------------------------------------------------------------------------- |
| 2937 | |
| 2938 | \section{Acknowledgements} |
| 2939 | |
| 2940 | The Wrestlers Protocol is named after the Wrestlers pub in Cambridge where |
| 2941 | Clive Jones and I worked out the initial design. |
| 2942 | |
| 2943 | %%%-------------------------------------------------------------------------- |
| 2944 | |
| 2945 | \bibliography{mdw-crypto,cryptography2000,cryptography,rfc,std} |
| 2946 | |
| 2947 | %%%-------------------------------------------------------------------------- |
| 2948 | |
| 2949 | \appendix |
| 2950 | \section{Proofs} |
| 2951 | |
| 2952 | \subsection{Proof of theorem~\ref{thm:sk}} |
| 2953 | \label{sec:sk-proof} |
| 2954 | |
| 2955 | Before we embark on the proof proper, let us settle on some notation. Let |
| 2956 | $P_i$ be a party. Then we write $x_i$ for $P_i$'s private key and $X_i = x_i |
| 2957 | P$ is $P_i$'s public key. Let $S = (P_i, P_j, s)$ be a session. We write |
| 2958 | $r_S$ for the random value chosen at the start of the session, and $R_S$, |
| 2959 | $c_S$ etc.\ are the corresponding derived values in that session. |
| 2960 | |
| 2961 | The proof uses a sequence of games. For each game~$\G{i}$, let $V_i$ be the |
| 2962 | event that some pair of unexposed, matching sessions both complete but output |
| 2963 | different keys, and let $W_i$ be the event that the adversary's final output |
| 2964 | equals the game's hidden bit~$b^*$. To save on repetition, let us write |
| 2965 | \[ \diff{i}{j} = \max(|\Pr[V_i] - \Pr[V_j]|, |\Pr[W_i] - \Pr[W_j]|). \] |
| 2966 | Obviously, |
| 2967 | \[ \diff{i}{j} \le \sum_{i\le k<j} \diff{k}{k + 1}. \] |
| 2968 | |
| 2969 | Here's a quick overview of the games we use. |
| 2970 | \begin{itemize} |
| 2971 | \item $\G0$ is the original SK-security game. |
| 2972 | \item In $\G1$, we abort the game unless all parties' public keys are |
| 2973 | distinct. Since keys are generated at random, parties are unlikely to be |
| 2974 | given the same key by accident. |
| 2975 | \item In $\G2$, we change the way sessions respond to challenge messages, by |
| 2976 | using the extractor to fake up answers to most challenges. Since the |
| 2977 | extractor is good, the adversary is unlikely to notice. |
| 2978 | \item In $\G3$, we abort the game if the adversary ever queries $H_K(\cdot)$ |
| 2979 | on the Diffie-Hellman secret $r_S r_T P$ shared between two unexposed |
| 2980 | matching sessions. We show that this is unlikely to happen if the |
| 2981 | Diffie-Hellman problem is hard. |
| 2982 | \item In $\G4$, we abort the game if any unexposed session \emph{accepts} a |
| 2983 | response message which wasn't sent by a matching session. |
| 2984 | \end{itemize} |
| 2985 | Finally, we show that the adversary has no advantage in $\G4$. The theorem |
| 2986 | follows. |
| 2987 | |
| 2988 | For ease of comprehension, we postpone the detailed proofs of some of the |
| 2989 | steps until after we conclude the main proof. |
| 2990 | |
| 2991 | Let $A$ be a given adversary which runs in time~$t$, creates at most~$q_S$ |
| 2992 | sessions, delivers at most~$q_M$ messages, and makes at most~$q_I$ queries to |
| 2993 | its $H_I(\cdot, \cdot, \cdot, \cdot)$ oracle and at most~$q_K$ queries to its |
| 2994 | $H_K(\cdot)$ oracle. Let $\G0$ be the original SK-security game of |
| 2995 | definition~\ref{def:sk}, played with adversary~$A$. |
| 2996 | |
| 2997 | Game~$\G1$ is the same as game~$\G0$ except, if the initialization function |
| 2998 | reports two parties as having the same public key (i.e., we have $X_i \ne |
| 2999 | X_j$ where $0 \le i < j < n$), we stop the game immediately and without |
| 3000 | crediting the adversary with a win. This only happens when the corresponding |
| 3001 | private keys are equal, i.e., $x_i = x_j$, and since the initialization |
| 3002 | function chooses private keys uniformly at random, this happens with |
| 3003 | probability at most $\binom{n}{2}/|G|$. Since if this doesn't happen, the |
| 3004 | game is identical to $\G0$, we can apply lemma~\ref{lem:shoup}, and see that |
| 3005 | \begin{equation} |
| 3006 | \label{eq:sk-g0-g1} |
| 3007 | \diff{0}{1} \le \frac{1}{|G|} \binom{n}{2} = \frac{n (n - 1)}{2 |G|}. |
| 3008 | \end{equation} |
| 3009 | In game~$\G1$ and onwards, we can assume that public keys for distinct |
| 3010 | parties are themselves distinct. Note that the game now takes at most |
| 3011 | $O(q_I)$ times longer to process each message delivered by the adversary. |
| 3012 | This is where the $O(q_I q_M)$ term comes from in the theorem statement. |
| 3013 | |
| 3014 | Game~$\G2$ is the same as game~$\G1$, except that we change the way that we |
| 3015 | make parties respond to \cookie{challenge} messages $(\cookie{challenge}, R, |
| 3016 | c)$. Specifically, suppose that $S = (P_i, P_j, s)$ is a session. |
| 3017 | \begin{itemize} |
| 3018 | \item Suppose $T = (P_j, P_i, s)$ is the matching session of $S$. The game |
| 3019 | proceeds as before if $(R, c) = (R_T, c_T)$ is the challenge issued by $T$. |
| 3020 | \item Otherwise, we run the extractor $T_\Wident$ on the adversary's history |
| 3021 | so far of oracle queries $H_I(X_i, s, R, \cdot)$ to determine a pair $(r, |
| 3022 | Y)$. If $r = \bot$ then we discard the message. Otherwise, we add $R$ to |
| 3023 | the list~$C$, and return a fake response to the adversary by computing $K = |
| 3024 | H_K(r R_S)$ and handing the adversary $(\cookie{response}, R_S, E_K(Y))$. |
| 3025 | \end{itemize} |
| 3026 | The following lemma shows how this affects the adversary's probabilities of |
| 3027 | winning. |
| 3028 | \begin{lemma} |
| 3029 | \label{lem:sk-g1-g2} |
| 3030 | \begin{equation} |
| 3031 | \label{eq:sk-g1-g2} |
| 3032 | \diff{1}{2} \le \frac{q_M}{2^{\ell_I}}. |
| 3033 | \end{equation} |
| 3034 | \end{lemma} |
| 3035 | |
| 3036 | Let us say that a session $S = (P_i, P_j, s)$ is \emph{ripe} if |
| 3037 | \begin{itemize} |
| 3038 | \item there is a matching session $T = (P_j, P_i, s)$, and |
| 3039 | \item $S$ is unexposed. |
| 3040 | \end{itemize} |
| 3041 | Suppose that $S$ is a ripe session, and that it has a matching session~$T$: |
| 3042 | let $Z_S = Z_T = r_S r_T P$. |
| 3043 | |
| 3044 | Game~$\G3$ is the same as $\G2$, except that the game is immediately aborted |
| 3045 | if ever the adversary queries its random oracle $H_K(\cdot)$ at a value $Z_S$ |
| 3046 | for any ripe session~$S$. The following lemma shows how this affects the |
| 3047 | adversary's probabilities of winning. |
| 3048 | \begin{lemma} |
| 3049 | \label{lem:sk-g2-g3} |
| 3050 | For some $t'$ within the bounds given in the theorem statement we have |
| 3051 | \begin{equation} |
| 3052 | \label{eq:sk-g2-g3} |
| 3053 | \diff{2}{3} \le q_S \InSec{mcdh}(G; t', q_K). |
| 3054 | \end{equation} |
| 3055 | \end{lemma} |
| 3056 | |
| 3057 | Game~$\G4$ is the same as $\G3$ except that the game is immediately aborted |
| 3058 | if ever the adversary sends a response message to a ripe session~$S$ which |
| 3059 | wasn't output by its matching session as a response to $S$'s challenge, with |
| 3060 | the result that $S$ completes. |
| 3061 | |
| 3062 | Let's make this more precise. Let $U$ and $V$ be a pair of matching |
| 3063 | sessions. Let $C_U = (\cookie{challenge}, R_U, c_U$ be the challenge message |
| 3064 | sent by $U$. Let $M_T$ be the set of messages which $T$ has sent upon |
| 3065 | delivery of $C_U$. Then, in $\G4$, we abort the game if, for any pair $S$ |
| 3066 | and~$T$ of matching, unexposed sessions, $S$ has completed as a result of |
| 3067 | being sent a message $\mu \notin M_T$. We have the following lemma. |
| 3068 | \begin{lemma} |
| 3069 | \label{lem:sk-g3-g4} |
| 3070 | For a $t'$ within the stated bounds, we have |
| 3071 | \begin{equation} |
| 3072 | \label{eq:sk-g3-g4} |
| 3073 | \diff{3}{4} \le q_S \bigl( \InSec{ind-cca}(\E; t', q_M, q_M) + |
| 3074 | n \cdot \InSec{mcdh}(G; t', q_M + q_I) \bigr) |
| 3075 | \end{equation} |
| 3076 | \end{lemma} |
| 3077 | |
| 3078 | Finally, let us consider the state we're in with $\G4$. |
| 3079 | \begin{itemize} |
| 3080 | \item No ripe session completes except as a result the adversary faithfully |
| 3081 | delivering messages between it and its matching session. |
| 3082 | \item The adversary never queries $Z_S$ for any ripe session~$S$. If we set |
| 3083 | $K_S = (K_{S, 0}, K_{S, 1}) = H_K(Z_S)$, then $K_{S, 1}$ is the key output |
| 3084 | by $S$ when it completes. |
| 3085 | \item If $S$ and $T$ are matching ripe sessions, then $K_S = K_T$, since $Z_S |
| 3086 | = r_S R_T = r_T R_S = Z_T$. |
| 3087 | \item For any ripe session~$S$, $K_{S, 1}$ is uniformly distributed in |
| 3088 | $\Bin^\kappa$ and independent of the adversary's view. |
| 3089 | \item If $S = (P_i, P_j, s)$ and $T = (P_j, P_i, s)$ are matching ripe |
| 3090 | sessions, then $Z_S$ depends only $r_S$ and $r_T$. Hence, once $S$ and~$T$ |
| 3091 | complete, and erase their states, $Z_S$ is independent of everything except |
| 3092 | the messages sent between the two sessions. In particular, $Z_S$ is |
| 3093 | independent of the long-term secrets $x_i$ and~$x_j$, so if either player |
| 3094 | is later corrupted, the key $K_{S, 1}$ remains independent of the |
| 3095 | adversary's view. |
| 3096 | \item Hence, the keys output by unexposed sessions are indistinguishable from |
| 3097 | freshly-generated random strings, and remain so indefinitely. |
| 3098 | \end{itemize} |
| 3099 | We conclude that, for any adversary $A$, |
| 3100 | \begin{equation} |
| 3101 | \label{eq:sk-g4} |
| 3102 | \Pr[V_4] = 0 \qquad \text{and} \qquad \Pr[W_4] = \frac{1}{2}. |
| 3103 | \end{equation} |
| 3104 | Putting equations~\ref{eq:sk-g0-g1}--\ref{eq:sk-g4} together, we find |
| 3105 | \begin{spliteqn} |
| 3106 | \Adv{sk}{\Wident^{G, \E}}(A) \le |
| 3107 | 2 q_S \bigl(\InSec{ind-cca}(\E; t', q_M, q_M) + {} \\ |
| 3108 | \InSec{mcdh}(G; t', q_K) + |
| 3109 | n \,\InSec{mcdh}(G; t', q_M + q_I) \bigr) + {} |
| 3110 | \frac{n (n - 1)}{|G|} + |
| 3111 | \frac{2 q_M}{2^{\ell_I}}. |
| 3112 | \end{spliteqn} |
| 3113 | The theorem follows, since $A$ was chosen arbitrarily. |
| 3114 | |
| 3115 | |
| 3116 | \begin{proof}[Proof of lemma~\ref{lem:sk-g1-g2}] |
| 3117 | The two games $\G1$ and~$\G2$ differ only in whether they accept or reject |
| 3118 | particular challenge messages $(\cookie{challenge}, R, c)$. |
| 3119 | |
| 3120 | We claim firstly that no message is \emph{accepted} by $\G2$ which would |
| 3121 | have been rejected by $\G1$. To prove the claim, it is sufficient to note |
| 3122 | that the extractor's output, if not $\bot$, is always correct, and hence if |
| 3123 | $\G2$ accepts a message then $\G1$ would have done so too. |
| 3124 | |
| 3125 | Since $\G2$ also behaves identically when the adversary submits to~$S$ the |
| 3126 | challenge from the matching session~$T$, we have nothing to prove in this |
| 3127 | case. Let $F$ be the event that the adversary submits a message |
| 3128 | $(\cookie{challenge}, R, c)$ to a session~$S$ which $S$ would have accepted |
| 3129 | in $\G1$ but would be rejected by the new rule in~$\G2$. By |
| 3130 | lemma~\ref{lem:shoup} we have $\diff{1}{2} \le \Pr[F]$. To prove the |
| 3131 | current lemma, therefore, we must show that $\Pr[F] \le q_M/2^{\ell_I}$. |
| 3132 | |
| 3133 | Rather than consider individual challenge messages, we consider |
| 3134 | \emph{classes} of messages. We shall refer to a quadruple~$\Cid = (i, j, |
| 3135 | s, R)$ as a \emph{class-id}, and define some useful functions: |
| 3136 | \begin{itemize} |
| 3137 | \item the class's \emph{session} $\Csession(\Cid) = (P_i, P_j, s)$; |
| 3138 | \item the class's \emph{index} $\Cindex(\Cid)$ is $r \in I$ where $R = r |
| 3139 | P$, which is well-defined by lemma~\ref{lem:unique-dl}; |
| 3140 | \item the class's \emph{query} $\Cquery(\Cid) = (X_j, s, R, x_i R)$; |
| 3141 | \item the class's \emph{hash} $\Chash(\Cid) = H_I(\Cquery(\Cid)) = H_I(X_j, |
| 3142 | s, R, x_i R)$; |
| 3143 | \item the class's \emph{check-value} $\Ccheck(\Cid) = \Chash(\Cid) \xor |
| 3144 | \Cindex(\Cid)$; |
| 3145 | \item the class's \emph{check-set} $\Ccheckset(\Cid)$ is the set of |
| 3146 | check-values $c$ such that a message $(\cookie{challenge}, R, c)$ was |
| 3147 | sent to session $S = (P_i, P_j, s)$; and |
| 3148 | \item the class's \emph{count} $\Ccount(\Cid) = |\Ccheckset(\Cid)|$. |
| 3149 | \end{itemize} |
| 3150 | |
| 3151 | Consider any class-id~$\Cid = (i, j, s, R)$. A natural question which |
| 3152 | arises is: which participants have issued $\Cid$'s query, i.e., queried |
| 3153 | $H_I$ at $\Cquery(\Cid)$? |
| 3154 | |
| 3155 | We can characterise the $H_I(\cdot, \cdot, \cdot, \cdot)$ queries of a |
| 3156 | session $U = (P_{i'}, P_{j'}, s')$ as follows: |
| 3157 | \begin{itemize} |
| 3158 | \item computing the check-value for the challenge $R_U$ by querying |
| 3159 | $H_I(X_{i'}, s', R_U, r_U X_{j'})$, and |
| 3160 | \item checking an incoming challenge~$R'$ by querying $H_I(X_{j'}, s', R', |
| 3161 | x_{i'} R')$. |
| 3162 | \end{itemize} |
| 3163 | The class~$\Cid$'s query $\Cquery(\Cid)$ is $U$'s check-value query if |
| 3164 | \[ (j, i, s, R) = (i', j', s', R_U) \] |
| 3165 | i.e., $U$ is the matching session of $\Csession(\Cid)$, and moreover $R = |
| 3166 | R_U$ is the challenge value issued by $U$. For any $c \in |
| 3167 | \Ccheckset(\Cid)$, if $c = \Ccheck(\Cid)$ then $(\cookie{challenge}, R, c)$ |
| 3168 | is precisely the challenge message issued by~$U$ to $\Csession(\Cid)$; the |
| 3169 | rules for handling this message didn't change. However, if $c \ne |
| 3170 | \Ccheck(\Cid)$ then the message would have been rejected in $\G1$, and we |
| 3171 | have already shown that $\G2$ continues to reject all messages rejected by |
| 3172 | $\G1$. |
| 3173 | |
| 3174 | Let us say that a class-id~$\Cid = (i, j, s, R)$ is \emph{bad} if |
| 3175 | \begin{enumerate} |
| 3176 | \item the value $R$ is not the challenge issued by $\Csession(\Cid)$'s |
| 3177 | matching session, and |
| 3178 | \item the adversary has not issued $\Cid$'s query $\Cquery(\Cid)$, |
| 3179 | \emph{but} |
| 3180 | \item $\Ccheck(\Cid) \in \Ccheckset(\Cid)$, so one of the check-values |
| 3181 | submitted to $S$ was actually correct. |
| 3182 | \end{enumerate} |
| 3183 | We claim that our extractor will work perfectly unless some class-id is |
| 3184 | bad. Certainly, if $R$ was issued by the matching session, there is |
| 3185 | nothing to prove; if the adversary has issued the relevant query then the |
| 3186 | extractor will recover $\Cindex(\Cid)$ just fine; and if $\Ccheck(\Cid) |
| 3187 | \notin \Ccheckset(\Cid)$ then all messages in the class would have been |
| 3188 | rejected by $\G1$ anyway. |
| 3189 | |
| 3190 | Let $B(\Cid)$ be the event that the class~$\Cid$ is bad. We claim that |
| 3191 | \[ \Pr[B(\Cid)] \le \frac{\Ccount(\Cid)}{2^{\ell_I}}. \] |
| 3192 | The present lemma follows, since |
| 3193 | \[ \diff{1}{2} |
| 3194 | \le \Pr[F] |
| 3195 | \le \sum_\Cid \Pr[B(\Cid)] |
| 3196 | \le \sum_\Cid \frac{\Ccount(\Cid)}{2^{\ell_I}} |
| 3197 | = \frac{1}{2^{\ell_I}} \sum_\Cid \Ccount(\Cid) |
| 3198 | \le \frac{q_M}{2^{\ell_I}} |
| 3199 | \] |
| 3200 | as required. |
| 3201 | |
| 3202 | Now observe that, in $\G2$, sessions don't actually check incoming |
| 3203 | challenges in this way any more -- instead we run the extractor. So, to |
| 3204 | prove the claim, we consider a class~$\Cid$ where properties~1 and~2 above |
| 3205 | hold. The correct hash $\Chash(\Cid)$ is then independent of the rest of |
| 3206 | the game, so the probability that $\Ccheck(\Cid) \in \Ccheckset(\Cid)$ is |
| 3207 | precisely $\Ccount(\Cid)/2^{\ell_I}$ as required. |
| 3208 | |
| 3209 | This completes the proof the lemma. |
| 3210 | \end{proof} |
| 3211 | |
| 3212 | \begin{proof}[Proof of lemma~\ref{lem:sk-g2-g3}] |
| 3213 | Let $F$ be the event that the adversary makes a query $H_K(Z_S)$ for some |
| 3214 | ripe session~$S$. Since $\G3$ proceeds exactly as $\G2$ did unless $F_2$ |
| 3215 | occurs, we apply lemma~\ref{lem:shoup}, which tells us that $\diff{2}{3} |
| 3216 | \le \Pr[F_2]$. We must therefore bound this probability. |
| 3217 | |
| 3218 | To do this, we consider a new game~$\G3'$, which is the same as $\G3$, |
| 3219 | except that, at the start of the game, we choose a random number $k \inr |
| 3220 | \Nupto{q_S}$. For $0 \le i < q_S$, let $S_i$ be the $i$th session created |
| 3221 | by the adversary. We define $F'$ to be the event that the adversary |
| 3222 | queries $H_K(Z_{S_k})$ when $S_k$ is ripe. |
| 3223 | |
| 3224 | The lemma now follows from these two claims. |
| 3225 | |
| 3226 | \begin{claim} |
| 3227 | $\Pr[F] \le q_S \Pr[F']$. |
| 3228 | \end{claim} |
| 3229 | To see this, for any session~$S$, let $F_S$ be the event that the adversary |
| 3230 | queries~$H_K(Z_S)$ when $S$ is ripe. Then |
| 3231 | \[ \Pr[F] \le \sum_{0\le i<q_S} \Pr[F_{S_i}]. \] |
| 3232 | Hence, |
| 3233 | \[ \Pr[F'] = \Pr[F_{S_k}] = \sum_{0\le i<q_S} \Pr[F_{S_i}] \Pr[k = i] |
| 3234 | = \frac{1}{q_S} \sum_{0\le i<q_S} \Pr[F_{S_i}] |
| 3235 | \ge \frac{\Pr[F]}{q_S} |
| 3236 | \] |
| 3237 | proving the claim. |
| 3238 | |
| 3239 | \begin{claim} |
| 3240 | For some $t' = t + O(n) + O(q_S q_M) + O(q_I) + O(q_K)$, we have |
| 3241 | $\Pr[F'] \le \InSec{mcdh}(G; t', q_K).$ |
| 3242 | \end{claim} |
| 3243 | To prove this claim, we construct an adversary~$B$ which solves the MCDH |
| 3244 | problem in~$G$. The adversary works as follows. |
| 3245 | \begin{enumerate} |
| 3246 | \item It is given a pair $(R^*, S^*) = (r^* P, s^* P)$ of group elements; |
| 3247 | its objective is to make a verification-oracle query $V(Z^*)$ where $Z^* |
| 3248 | = r^* s^* P$. |
| 3249 | \item It sets up a simulation of the game~$\G3'$, by running the |
| 3250 | $\id{init}$ function, and simulating all of the parties. In particular, |
| 3251 | it chooses a random~$k \in \Nupto{q_S}$. |
| 3252 | \item It sets up accurate simulations of the random oracles $H_K(\cdot)$ |
| 3253 | and $H_I(\cdot, \cdot, \cdot, \cdot)$, which choose random outputs for |
| 3254 | new, fresh inputs. However, whenever $A$ queries $H_K(\cdot)$ on a group |
| 3255 | element $Z$, $B$ also queries $V(Z)$. |
| 3256 | \item It runs $A$ in its simulated game. It responds to all of $A$'s |
| 3257 | instructions faithfully, until the $k$th session-creation. |
| 3258 | \item When creating the $k$th session $S = S_k = (P_i, P_j, s)$, $B$ has |
| 3259 | party~$P_i$ choose $R^*$ as its challenge, rather than choosing $r_S$ and |
| 3260 | setting $R_S = r_S P$. Because it simulates all the parties, $B$ can |
| 3261 | compute $Y_S = x_j R$, which is still correct. |
| 3262 | \item If $A$ requests the creation of a matching session $T = (P_j, P_i, |
| 3263 | s)$ then $B$ has party~$P_j$ choose $S^*$ as its challenge. Again, $B$ |
| 3264 | computes $Y_T = x_i S^*$. |
| 3265 | \item If $A$ ever corrupts the parties $P_i$ or $P_j$, or reveals the |
| 3266 | session state of $S$ or $T$ then $B$ stops the simulation abruptly and |
| 3267 | halts. |
| 3268 | \end{enumerate} |
| 3269 | Adversary $B$'s running time is within the bounds required of $t'$, and $B$ |
| 3270 | makes at most $q_K$ queries to $V(\cdot)$; we therefore have |
| 3271 | \[ \Pr[F'] \le \Succ{mcdh}{G}(B) \le \InSec{mcdh}(G; t', q_K) \] |
| 3272 | as required. |
| 3273 | \end{proof} |
| 3274 | |
| 3275 | \begin{proof}[Proof of lemma~\ref{lem:sk-g3-g4}] |
| 3276 | Let $F_4$ be the event under which we abort the game~$\G4$. Clearly, if |
| 3277 | $F$ doesn't occur, games $\G3$ and $\G4$ proceed identically, so we can |
| 3278 | apply lemma~\ref{lem:shoup} to see that $\diff{3}{4} \le \Pr[F_4]$. |
| 3279 | Bounding $\Pr[F_4]$, however, is somewhat complicated. We use a further |
| 3280 | sequence of games. |
| 3281 | |
| 3282 | Firstly, let $\G5$ be like $\G4$ with the exception that we choose, at |
| 3283 | random, an integer~$k \inr \Nupto{q_S}$. As we did in the proof for |
| 3284 | lemma~\ref{lem:sk-g3-g4}, let $S_i$ be the $i$th session created by the |
| 3285 | adversary. For each session~$S_i$, let $T_i$ be its matching session, if |
| 3286 | there is one. We define $F_5$ to be the event that |
| 3287 | \begin{itemize} |
| 3288 | \item $S_k$ completes immediately following delivery of a message $\mu |
| 3289 | \notin M_{T_k}$, and |
| 3290 | \item $S_k$ was ripe at this point. |
| 3291 | \end{itemize} |
| 3292 | For games~$\G{i}$, for $i > 5$, we define the event $F_i$ to be the event |
| 3293 | corresponding to $F_5$ in $\G{i}$. Note that if $S_k$ \emph{is} sent a |
| 3294 | message in $M_{T_k}$ then $S_k$ immediately completes. |
| 3295 | |
| 3296 | \begin{claim} |
| 3297 | $\Pr[F_4] \le \Pr[F_5]/q_S$. |
| 3298 | \end{claim} |
| 3299 | This claim is proven exactly as we did for claim~1 of |
| 3300 | lemma~\ref{lem:sk-g2-g3}. |
| 3301 | |
| 3302 | Let~$\G6$ be the same as $\G5$ except that we change the encrypted |
| 3303 | responses of session~$S_k$ and its matching session~$T_k$. Let $K^* = |
| 3304 | (K_0^*, K_1^*) = H_K(Z_S)$. Then, rather than sending $(\cookie{response}, |
| 3305 | R_S, E_{K_0^*}(Y_T))$, session~$S$ sends $(\cookie{response}, R_S, |
| 3306 | E_{K_0^*}(1^{\ell_G}))$. |
| 3307 | \begin{claim} |
| 3308 | $|\Pr[F_6] - \Pr[F_5]| \le \InSec{ind-cca}(\E; t', q_M, q_M).$ |
| 3309 | \end{claim} |
| 3310 | To prove this claim, we construct an adversary~$B$ which attacks the |
| 3311 | IND-CCA security of our encryption scheme $\E$. The adversary~$B$ works as |
| 3312 | follows. |
| 3313 | \begin{enumerate} |
| 3314 | \item It is given no input, but a pair of oracles $E(\cdot, \cdot)$ and |
| 3315 | $D(\cdot)$; the former encrypts either the left or right input, according |
| 3316 | to a hidden bit, and the latter decrypts ciphertexts other than those |
| 3317 | returned by $E(\cdot, \cdot)$. Its goal is to guess the hidden bit. |
| 3318 | \item It sets up a simulation of the game~$\G5$, by running the $\id{init}$ |
| 3319 | function, and simulating all of the parties. In particular, it chooses a |
| 3320 | random $k \in \Nupto{q_S}$. |
| 3321 | \item It sets up accurate simulations of the random oracles $H_K(\cdot)$ |
| 3322 | and $H_I(\cdot, \cdot, \cdot, \cdot)$. |
| 3323 | \item It runs $A$ in its simulated game. It responds to all of $A$'s |
| 3324 | instructions faithfully, except for the matching sessions $S_k$ and |
| 3325 | $T_k$. Let $S = S_k = (P_i, P_j, s)$, and $T = T_k = (P_j, P_i, s)$. |
| 3326 | \item Suppose $T$ is sent the message $C_S = (\cookie{challenge}, R_S, |
| 3327 | c_S)$. Rather than computing $K^* = H_K(r_T R_S)$ and performing the |
| 3328 | encryption properly, $B$ queries its left-or-right encryption oracle |
| 3329 | $E(\cdot, \cdot)$ on $E(1^{\ell_G}, x_j R_S)$, and sends the resulting |
| 3330 | ciphertext~$\chi$ back to~$S$ as part of a message $(\cookie{response}, |
| 3331 | R_T, \chi)$. The set $M_T$ is precisely the set of messages constructed |
| 3332 | in this fashion. (Recall that challenge messages other than $C_S$ aren't |
| 3333 | actually delivered to $T$, since we simulate the responses using the |
| 3334 | extractor, as of $\G2$.) |
| 3335 | \item Suppose $S$ is sent a message $M = (\cookie{response}, R_T, \chi) \in |
| 3336 | M_T$. We immediately stop the simulation, and $B$ returns $0$. |
| 3337 | \item Suppose, instead, that $S$ is sent some message $M' = |
| 3338 | (\cookie{response}, R, \chi) \notin M_T$. There are two cases to |
| 3339 | consider. If $R = R_T$ then we must have $\chi$ distinct from the |
| 3340 | ciphertexts returned by the $E(\cdot, \cdot)$ oracle, so we can invoke |
| 3341 | the decryption oracle $D(\cdot)$ on $\chi$ to obtain a response $Y$. |
| 3342 | Alternatively, if $R \ne R_T$, we can compute the key $K = (K_0, K_1) = |
| 3343 | H_K(r_S R)$, and recover $Y = D_{K_0}(\chi)$. In either case, if $Y = |
| 3344 | r_S X_j)$ then $S$ would complete at this point: $B$ stops the simulation |
| 3345 | and returns $1$. |
| 3346 | \item If $A$ exposes $S$ (by corrupting $P_i$ or~$P_j$, or revealing $S$ or |
| 3347 | $T$) then we stop the simulation and $B$ returns $0$. |
| 3348 | \item Finally, if the game stops, either because $A$ halts, or because of |
| 3349 | one of the special rules introduced in earlier games, $B$ returns $0$. |
| 3350 | \end{enumerate} |
| 3351 | It is clear that $B$'s oracle queries are acceptable, since $|x_j R_S| = |
| 3352 | \ell_G$ by definition, and $B$ never queries $D(\cdot)$ on a ciphertext |
| 3353 | returned by its encryption oracle. By the rules of~$\G3$, we know that the |
| 3354 | game stops immediately if $A$ ever queries $Z_S$, so the key $K^*$ is |
| 3355 | independent of everything in $A$'s view except the ciphertexts $\chi$ |
| 3356 | output by $S$ and $T$. Therefore, if the hidden bit of the IND-CCA game is |
| 3357 | $1$, $B$ accurately simulates $\G5$, whereas if the bit is $0$ then $B$ |
| 3358 | accurately simulates $\G6$. We issue no more that $q_M$ encryption or |
| 3359 | decryption queries. Finally, $B$'s running time is within the bounds |
| 3360 | allowed for $t'$. Therefore, |
| 3361 | \[ \Adv{ind-cca}{\E}(B) = \Pr[F_5] - \Pr[F_6] |
| 3362 | \le \InSec{ind-cca}(\E; t', q_M, q_M). \] |
| 3363 | We construct the adversary~$\bar{B}$ which is the same as $B$ above, except |
| 3364 | that $\bar{B}$ returns $0$ whenever $B$ returns~$1$, and \emph{vice versa}. |
| 3365 | Clearly |
| 3366 | \[ \Adv{ind-cca}{\E}(\bar{B}) |
| 3367 | = (1 - \Pr[F_5]) - (1 - \Pr[F_6]) |
| 3368 | = \Pr[F_6] - \Pr[F_5] |
| 3369 | \le \InSec{ind-cca}(\E; t', q_M, q_M). |
| 3370 | \] |
| 3371 | This proves the claim. |
| 3372 | |
| 3373 | Let $\G7$ be the same as $\G6$, except that at the start of the game we |
| 3374 | choose a random $m \in \Nupto{n}$, and when the adversary creates the |
| 3375 | session $S = S_k = (P_i, P_j, s)$, we abort the game unless $j = m$. |
| 3376 | Clearly we have $\Pr[F_6] = n \Pr[F_7]$. |
| 3377 | |
| 3378 | Finally, we can explicitly bound $F_6$. In $\G6$, the adversary's view is |
| 3379 | independent of the correct response $Y_S = r_S X_S = x_j R_S$ to $S$'s |
| 3380 | challenge. Therefore, if $A$ manages to send any message $\mu \notin M_T$ |
| 3381 | which causes $S$ to complete, then it has impersonated~$P_j$. |
| 3382 | \begin{claim} |
| 3383 | $\Pr[F_7] \le \InSec{mcdh}(G; t', q_M + q_I)$. |
| 3384 | \end{claim} |
| 3385 | The present lemma follows from this and the previous claims. |
| 3386 | |
| 3387 | To prove the claim formally, we construct an adversary~$B'$, which behaves |
| 3388 | as follows. |
| 3389 | \begin{enumerate} |
| 3390 | \item It is given as input a public key $X^*$ and a single challenge $(R^*, |
| 3391 | c^*)$, a random oracle $H^*_I(\cdot, \cdot)$, and an oracle $V(\cdot, |
| 3392 | \cdot)$, which verifies responses $(R, Y)$. Its goal is to invoke |
| 3393 | $V(\cdot, \cdot)$ with a correct response to the challenge. |
| 3394 | \item It chooses a random $k \in \Nupto{q_S}$ and $m \in \Nupto{n}$. It |
| 3395 | sets up a simulation of the game~$\G7$, by running the $\id{init}$ |
| 3396 | function, and simulating all of the parties, except that it gives party |
| 3397 | $P_m$ the public key $X^*$. This makes no difference, since $P_m$ |
| 3398 | doesn't actually need to give any `honest' responses because of the |
| 3399 | change we made in $\G6$. |
| 3400 | \item It sets up accurate simulations of the random oracles $H_K(\cdot)$ |
| 3401 | and $H_I(\cdot, \cdot, \cdot, \cdot)$, with one exception -- see below. |
| 3402 | \item It runs $A$ in its simulated game. It responds to all of $A$'s |
| 3403 | instructions faithfully, except for the session $S_k$. Let $S = S_k = |
| 3404 | (P_i, P_j, s)$, and let $T = T_k = (P_j, P_i, s)$ be its matching |
| 3405 | session. |
| 3406 | \item When session~$S$ is created, $B'$ checks that $j = m$, and if not |
| 3407 | stops the simulation and halts. Otherwise, $B'$ invokes its oracle~$C()$ |
| 3408 | to obtain a pair $(R, c)$. Session~$S$ sends $C_S = (\cookie{challenge}, |
| 3409 | R, c)$ as its challenge to~$T$. |
| 3410 | \item When $A$ makes a query $H_I(X^*, s, R, Y)$, $B$ answers it by |
| 3411 | querying its \emph{own} random oracle $H^*_I(R, Y)$. |
| 3412 | \item When $S$ receives a message $(\cookie{response}, R, \chi)$, we |
| 3413 | compute $(K_0, K_1) = r_S R$, and $Y = D_{K_0}(\chi)$. If $Y \ne \bot$ |
| 3414 | then $B'$ calls $V(R, Y)$. |
| 3415 | \item If $A$ reveals $S$ or corrupts $P_i$ or $P_j$ then $B'$ stops the |
| 3416 | simulation immediately and halts. |
| 3417 | \end{enumerate} |
| 3418 | The running time of $B'$ is within the bounds required of $t'$; it makes at |
| 3419 | most $q_I$ random-oracle and at most $q_M$ verification queries. Clearly |
| 3420 | $B'$ succeeds whenever $F_7$ occurs. The claim follows from |
| 3421 | theorem~\ref{thm:wident-sound}. |
| 3422 | \end{proof} |
| 3423 | |
| 3424 | |
| 3425 | \subsection{Proof of theorem~\ref{thm:sk2}} |
| 3426 | \label{sec:sk2-proof} |
| 3427 | |
| 3428 | The proof is almost identical to the proof of theorem~\ref{thm:sk}, in |
| 3429 | appendix~\ref{sec:sk-proof}. Unfortunately a black-box reduction doesn't |
| 3430 | seem possible. |
| 3431 | |
| 3432 | We use the games and notation of section~\ref{sec:sk-proof}. |
| 3433 | |
| 3434 | The change to the check-value calculation doesn't affect key-generation at |
| 3435 | all, so the transition to $\G1$ goes through as before. |
| 3436 | |
| 3437 | The transition from $\G1$ to $\G2$ -- answering challenges using the |
| 3438 | extractor -- needs a little care. Let $S = (P_i, P_j, s)$ be a session, and |
| 3439 | consider an incoming message $(\cookie{challenge}, R, c)$. |
| 3440 | \begin{itemize} |
| 3441 | \item If $T = (P_j, P_i, s)$ is the matching session to $S$, and $R = R_T$ is |
| 3442 | the public challenge value of $T$, and $c = r_T \xor H_I(R_S, X_j, s, R_T, |
| 3443 | r_T X_i)$ is the check-value output by $T$ when it received |
| 3444 | $(\cookie{pre-challenge}, R_S)$ as input, then $S$ replies as it did in |
| 3445 | $\G1$. |
| 3446 | \item If the challenge message is any other message, then we use the |
| 3447 | extractor. |
| 3448 | \end{itemize} |
| 3449 | As in lemma~\ref{lem:sk-g1-g2}, we examine which sessions could have queried |
| 3450 | $H_I(R_S, X_j, s, R, x_i R)$, and for the same reasons conclude that only the |
| 3451 | matching session would have done this, and only in response to the |
| 3452 | pre-challenge $R_S$. It follows that $\diff{1}{2} \le q_M/2^{\ell_I}$ as |
| 3453 | before. |
| 3454 | |
| 3455 | The remaining game steps go through unchanged. In particular, we conclude |
| 3456 | that a ripe session will only complete if the adversary has transmitted |
| 3457 | messages from its matching session correctly, and the session key is |
| 3458 | independent of the adversary's view. The theorem follows. |
| 3459 | |
| 3460 | |
| 3461 | \subsection{Sketch proof of single-key protocol for secure channels} |
| 3462 | \label{sec:sc-proof} |
| 3463 | |
| 3464 | We want to show that the Wrestlers Key-Exchange protocol, followed by use of |
| 3465 | the encryption scheme $\E$, with the \emph{same} key $K = K_0$, still |
| 3466 | provides secure channels. |
| 3467 | |
| 3468 | \subsubsection{Secure channels definition} |
| 3469 | We (very briefly!) recall the \cite{Canetti:2001:AKE} definition of a secure |
| 3470 | channels protocol. We again play a game with the adversary. At the |
| 3471 | beginning, we choose a bit $b^* \inr \{0, 1\}$ at random. We allow the |
| 3472 | adversary the ability to establish \emph{secure channels} sessions within the |
| 3473 | parties. Furthermore, for any running session $S = (P_i, P_j, s)$, we allow |
| 3474 | the adversary to request $S$ to send a message~$\mu$ through its secure |
| 3475 | channel. Finally, the adversary is allowed to choose one ripe |
| 3476 | \emph{challenge} session, and ask for it to send of one of a \emph{pair} of |
| 3477 | messages $(\mu_0, \mu_1)$, subject to the restriction that $|\mu_0| = |
| 3478 | |\mu_1|$; the session sends message $\mu_{b^*}$. The adversary may not |
| 3479 | expose the challenge session. |
| 3480 | |
| 3481 | The adversary wins if (a)~it can guess the bit $b^*$, or (b)~it can cause a |
| 3482 | ripe session $S$ (i.e., an unexposed, running session), with a matching |
| 3483 | session~$T$ to output a message other than one that it requested that $T$ |
| 3484 | send. |
| 3485 | |
| 3486 | \subsubsection{Protocol definition} |
| 3487 | The protocol begins with Wrestlers key-exchange. The encryption in the |
| 3488 | key-exchange protocol is performed as $E_K(\cookie{kx}, \cdot)$; encryption |
| 3489 | for secure channels is performed as $E_K(\cookie{sc}, i, o, \cdot)$, where |
| 3490 | $i$ is a sequence number to prevent replays and $o \in \{S, T\}$ identifies |
| 3491 | the sender. |
| 3492 | |
| 3493 | \subsubsection{Proof sketch} |
| 3494 | We use the games and notation of appendix~\ref{sec:sk-proof}. |
| 3495 | |
| 3496 | The proof goes through as far as the step between $\G5$ and $\G6$ in the |
| 3497 | proof of lemma~\ref{lem:sk-g3-g4}. Here we make the obvious adjustments to |
| 3498 | our adversary against the IND-CCA security of $\E$. (Our adversary will need |
| 3499 | to use the left-or-right oracle for messages sent using the secure channel |
| 3500 | built on $K^*$. That's OK.) |
| 3501 | |
| 3502 | In $\G4$, we know that ripe sessions agree the correct key, and the adversary |
| 3503 | never queries the random oracle, so the key is independent of the adversary's |
| 3504 | view. |
| 3505 | |
| 3506 | We define a new game $\G8$, which is like $\G4$, except that we stop the game |
| 3507 | if the adversary ever forges a message sent over the secure channel. That |
| 3508 | is, if a ripe session $S$ ever announces receipt of a message not sent (at |
| 3509 | the adversary's request) by its matching session $T$. Let $F_8$ be the event |
| 3510 | that a forgery occurs. We apply lemma~\ref{lem:shoup}, which tells us that |
| 3511 | $\diff{4}{8} \le \Pr[F_8]$. To bound $F_8$, we isolate a session at random |
| 3512 | (as in lemmata \ref{lem:sk-g2-g3} and~\ref{lem:sk-g3-g4}), which tells us |
| 3513 | that |
| 3514 | \begin{equation} |
| 3515 | \label{eq:sc-g4-g8} |
| 3516 | \diff{4}{8} \le q_S \cdot \InSec{int-ptxt}(\E; t', q_M, q_M) |
| 3517 | \end{equation} |
| 3518 | Finally, we can bound the adversary's advantage at guessing the hidden bit |
| 3519 | $b^*$. We isolate (we hope) the challenge session $S$ by choosing a target |
| 3520 | session at random, as before. Let $K_* = H_K(Z_S)$ be the key agreed by the |
| 3521 | session (if it becomes ripe). We define an adversary $B$ against the IND-CCA |
| 3522 | security of $\E$. The adversary $B$ simulates the game. If the adversary |
| 3523 | exposes the target session, or doesn't choose it as the challenge session, |
| 3524 | $B$ fails (and exits 0); otherwise it uses the left-or-right encryption |
| 3525 | oracle to encrypt both of the adversary's message choices, and outputs the |
| 3526 | adversary's choice. Let $b$ be the adversary's output, and let $\epsilon$ be |
| 3527 | the advantage of our IND-CCA distinguisher. Then |
| 3528 | \begin{eqnarray*}[rl] |
| 3529 | \Pr[b = b^*] |
| 3530 | & = \Pr[b = b^* \wedge b^* = 1] + \Pr[b = b^* \wedge b^* = 0] \\ |
| 3531 | & = \frac{1}{2}\bigl( \Pr[b = b^* \mid b^* = 1] + |
| 3532 | \Pr[b = b^* \mid b^* = 0] \bigr) \\ |
| 3533 | & = \frac{1}{2}\bigl( \Pr[b = b^* \mid b^* = 1] + |
| 3534 | (1 - \Pr[b \ne b^* \mid b^* = 0]) \bigr) \\ |
| 3535 | & = \frac{1}{2}\bigl( \Pr[b = 1 \mid b^* = 1] - |
| 3536 | \Pr[b = 1 \mid b^* = 0] + 1 \bigr) \\ |
| 3537 | & = \frac{1}{2}\bigl(1 + q_S\,\Adv{ind-cca}{\E}(B) \bigr) \\ |
| 3538 | & \le \frac{1}{2} \bigl( 1 + q_S\,\InSec{ind-cca}(\E; t', q_M, q_M) \bigr). |
| 3539 | \eqnumber |
| 3540 | \end{eqnarray*} |
| 3541 | |
| 3542 | \end{document} |
| 3543 | |
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