| 1 | %%% -*-latex-*- |
| 2 | %%% |
| 3 | %%% $Id: wrestlers.tex,v 1.7 2004/04/08 01:36:17 mdw Exp $ |
| 4 | %%% |
| 5 | %%% Description of the Wrestlers Protocol |
| 6 | %%% |
| 7 | %%% (c) 2001 Mark Wooding |
| 8 | %%% |
| 9 | |
| 10 | \newif\iffancystyle\fancystyletrue |
| 11 | |
| 12 | \iffancystyle |
| 13 | \documentclass |
| 14 | [a4paper, article, 10pt, numbering, noherefloats, notitlepage] |
| 15 | {strayman} |
| 16 | \usepackage[palatino, helvetica, courier, maths=cmr]{mdwfonts} |
| 17 | \usepackage[mdwmargin]{mdwthm} |
| 18 | \PassOptionsToPackage{dvips}{xy} |
| 19 | \else |
| 20 | \documentclass{llncs} |
| 21 | \fi |
| 22 | |
| 23 | \usepackage{mdwtab, mathenv, mdwlist, mdwmath, crypto} |
| 24 | \usepackage{amssymb, amstext} |
| 25 | \usepackage{tabularx} |
| 26 | \usepackage{url} |
| 27 | \usepackage[all]{xy} |
| 28 | |
| 29 | \errorcontextlines=999 |
| 30 | \showboxbreadth=999 |
| 31 | \showboxdepth=999 |
| 32 | \makeatletter |
| 33 | |
| 34 | \title{The Wrestlers Protocol: proof-of-receipt and secure key exchange} |
| 35 | \author{Mark Wooding \and Clive Jones} |
| 36 | |
| 37 | \bibliographystyle{mdwalpha} |
| 38 | |
| 39 | \newcolumntype{G}{p{0pt}} |
| 40 | \def\Nupto#1{\N_{<{#1}}} |
| 41 | \let\Bin\Sigma |
| 42 | \let\epsilon\varepsilon |
| 43 | \let\emptystring\lambda |
| 44 | \def\bitsto{\mathbin{..}} |
| 45 | \turnradius{4pt} |
| 46 | \def\fixme{\marginpar{FIXME}} |
| 47 | \def\messages{% |
| 48 | \basedescript{% |
| 49 | \desclabelwidth{2.5cm}% |
| 50 | \desclabelstyle\pushlabel% |
| 51 | \let\makelabel\cookie% |
| 52 | }% |
| 53 | } |
| 54 | \let\endmessages\endbasedescript |
| 55 | |
| 56 | \begin{document} |
| 57 | |
| 58 | \maketitle |
| 59 | \begin{abstract} |
| 60 | The Wrestlers Protocol\footnote{% |
| 61 | `The Wrestlers' is a pub in Cambridge which serves good beer and |
| 62 | excellent Thai food. It's where the authors made their first attempts at |
| 63 | a secure key-exchange protocol which doesn't use signatures.} % |
| 64 | is a key-exchange protocol with the interesting property that it leaves no |
| 65 | evidence which could be used to convince a third party that any of the |
| 66 | participants are involved. We describe the protocol and prove its security |
| 67 | in the random oracle model. |
| 68 | |
| 69 | Almost incidentally, we provide a new security proof for the CBC encryption |
| 70 | mode. Our proof is much simpler than that of \cite{Bellare:2000:CST}, and |
| 71 | gives a slightly better security bound. |
| 72 | |
| 73 | % I've not yet decided whose key-exchange model to use, but this ought to |
| 74 | % be mentioned. |
| 75 | \end{abstract} |
| 76 | \tableofcontents |
| 77 | \newpage |
| 78 | |
| 79 | %%%-------------------------------------------------------------------------- |
| 80 | |
| 81 | \section{Introduction} |
| 82 | \label{sec:intro} |
| 83 | % Some waffle here about the desirability of a key-exchange protocol that |
| 84 | % doesn't leave signatures lying around, followed by an extended report of |
| 85 | % the various results. |
| 86 | |
| 87 | %%%-------------------------------------------------------------------------- |
| 88 | |
| 89 | \section{Preliminaries} |
| 90 | \label{sec:prelim} |
| 91 | % Here we provide definitions of the various kinds of things we use and make, |
| 92 | % and describe some of the notation we use. |
| 93 | |
| 94 | \subsection{Bit strings} |
| 95 | |
| 96 | Most of our notation for bit strings is standard. The main thing to note is |
| 97 | that everything is zero-indexed. |
| 98 | |
| 99 | \begin{itemize} |
| 100 | \item We write $\Bin = \{0, 1\}$ for the set of binary digits. Then $\Bin^n$ |
| 101 | is the set of $n$-bit strings, and $\Bin^*$ is the set of all bit strings. |
| 102 | \item If $x$ is a bit string then $|x|$ is the length of $x$. If $x \in |
| 103 | \Bin^n$ then $|x| = n$. |
| 104 | \item If $x, y \in \Bin^n$ are strings of bits of the same length then $x |
| 105 | \xor y \in \Bin^n$ is their bitwise XOR. |
| 106 | \item If $x$ and $y$ are bit strings then $x \cat y$ is the result of |
| 107 | concatenating $y$ to $x$. If $z = x \cat y$ then we have $|z| = |x| + |
| 108 | |y|$. |
| 109 | \item The empty string is denoted $\emptystring$. We have $|\emptystring| = |
| 110 | 0$, and $x = x \cat \emptystring = \emptystring \cat x$ for all strings $x |
| 111 | \in \Bin^*$. |
| 112 | \item If $x$ is a bit string and $i$ is an integer satisfying $0 \le i < |x|$ |
| 113 | then $x[i]$ is the $i$th bit of $x$. If $a$ and $b$ are integers |
| 114 | satisfying $0 \le a \le b \le |x|$ then $x[a \bitsto b]$ is the substring |
| 115 | of $x$ beginning with bit $a$ and ending just \emph{before} bit $b$. We |
| 116 | have $|x[i]| = 1$ and $|x[a \bitsto b]| = b - a$; if $y = x[a \bitsto b]$ |
| 117 | then $y[i] = x[a + i]$. |
| 118 | \item If $x$ is a bit string and $n$ is a natural number then $x^n$ is the |
| 119 | result of concatenating $x$ to itself $n$ times. We have $x^0 = |
| 120 | \emptystring$ and if $n > 0$ then $x^n = x^{n-1} \cat x = x \cat x^{n-1}$. |
| 121 | \end{itemize} |
| 122 | |
| 123 | \subsection{Other notation} |
| 124 | |
| 125 | \begin{itemize} |
| 126 | \item If $n$ is any natural number, then $\Nupto{n}$ is the set $\{\, i \in |
| 127 | \Z \mid 0 \le i < n \,\} = \{ 0, 1, \ldots, n \}$. |
| 128 | \item The symbol $\bot$ (`bottom') is different from every bit string and |
| 129 | group element. |
| 130 | \item We write $\Func{l}{L}$ as the set of all functions from $\Bin^l$ to |
| 131 | $\Bin^L$, and $\Perm{l}$ as the set of all permutations on $\Bin^l$. |
| 132 | \end{itemize} |
| 133 | |
| 134 | \subsection{Algorithm descriptions} |
| 135 | |
| 136 | Most of the notation used in the algorithm descriptions should be obvious. |
| 137 | We briefly note a few features which may be unfamiliar. |
| 138 | \begin{itemize} |
| 139 | \item The notation $a \gets x$ denotes the action of assigning the value $x$ |
| 140 | to the variable $a$. |
| 141 | \item The notation $a \getsr X$, where $X$ is a finite set, denotes the |
| 142 | action of assigning to $a$ a random value $x \in X$ according to the |
| 143 | uniform probability distribution on $X$; i.e., following $a \getsr X$, |
| 144 | $\Pr[a = x] = 1/|X|$ for any $x \in X$. |
| 145 | \end{itemize} |
| 146 | The notation is generally quite sloppy about types and scopes. In |
| 147 | particular, there are implicit coercions between bit strings, integers and |
| 148 | group elements. Any simple injective mapping will do for handling the |
| 149 | conversions. We don't think these informalities cause much confusion, and |
| 150 | they greatly simplify the presentation of the algorithms. |
| 151 | |
| 152 | \subsection{Random oracles} |
| 153 | |
| 154 | We shall analyse the Wrestlers Protocol in the random oracle model |
| 155 | \cite{Bellare:1993:ROP}. That is, each participant including the adversary |
| 156 | is given oracle access (only) to a uniformly-distributed random function |
| 157 | $H\colon \Bin^* \to \Bin^\infty$ chosen at the beginning of the game: for any |
| 158 | input string $x$, the oracle can produce, on demand, any prefix of an |
| 159 | infinitely long random answer $y = H(x)$. Repeating a query yields a prefix |
| 160 | of the same random result string; asking a new query yields a prefix of a new |
| 161 | randomly-chosen string. |
| 162 | |
| 163 | We shan't need either to query the oracle on very long input strings nor |
| 164 | shall we need outputs much longer than a representation of a group index. |
| 165 | Indeed, since all the programs we shall be dealing with run in finite time, |
| 166 | and can therefore make only a finite number of oracle queries, each with a |
| 167 | finitely long result, we can safely think about the random oracle as a finite |
| 168 | object. |
| 169 | |
| 170 | Finally, we shall treat the oracle as a function of multiple inputs and |
| 171 | expect it to operate on some unambiguous encoding of all of the arguments in |
| 172 | order. |
| 173 | |
| 174 | \subsection{Symmetric encryption} |
| 175 | |
| 176 | \begin{definition}[Symmetric encryption] |
| 177 | \label{def:sym-enc} |
| 178 | A \emph{symmetric encryption scheme} $\mathcal{E} = (E, D)$ is a pair of |
| 179 | algorithms: |
| 180 | \begin{itemize} |
| 181 | \item a randomized \emph{encryption algorithm} $E\colon \keys \mathcal{E} |
| 182 | \times \Bin^* \to \Bin^*$; and |
| 183 | \item a deterministic \emph{decryption algorithm} $E\colon \keys |
| 184 | \mathcal{E} \times \Bin^* \to \Bin^* \cup \{ \bot \}$ |
| 185 | \end{itemize} |
| 186 | with the property that, for any $K \in \keys \mathcal{E}$, any plaintext |
| 187 | message $x$, and any ciphertext $y$ returned as a result of $E_K(x)$, we |
| 188 | have $D_K(y) = x$. |
| 189 | \end{definition} |
| 190 | |
| 191 | \begin{definition}[Chosen plaintext security for symmetric encryption] |
| 192 | \label{def:sym-cpa} |
| 193 | Let $\mathcal{E} = (E, D)$ be a symmetric encryption scheme. Let $A$ be |
| 194 | any algorithm. Define |
| 195 | \begin{program} |
| 196 | Experiment $\Expt{lor-cpa-$b$}{\mathcal{E}}(A)$: \+ \\ |
| 197 | $K \getsr \keys \mathcal{E}$; \\ |
| 198 | $b' \getsr A^{E_K(\id{lr}(b, \cdot, \cdot))}$; \\ |
| 199 | \RETURN $b'$; |
| 200 | \next |
| 201 | Function $\id{lr}(b, x_0, x_1)$: \+ \\ |
| 202 | \RETURN $x_b$; |
| 203 | \end{program} |
| 204 | An adversary $A$ is forbidden from querying its encryption oracle |
| 205 | $E_K(\id{lr}(b, \cdot, \cdot))$ on a pair of strings with differing |
| 206 | lengths. We define the adversary's \emph{advantage} in this game by |
| 207 | \begin{equation} |
| 208 | \Adv{lor-cpa}{\mathcal{E}}(A) = |
| 209 | \Pr[\Expt{lor-cpa-$1$}{\mathcal{E}}(A) = 1] - |
| 210 | \Pr[\Expt{lor-cpa-$0$}{\mathcal{E}}(A) = 1] |
| 211 | \end{equation} |
| 212 | and the \emph{left-or-right insecurity of $\mathcal{E}$ under |
| 213 | chosen-plaintext attack} is given by |
| 214 | \begin{equation} |
| 215 | \InSec{lor-cpa}(\mathcal{E}; t, q_E, \mu_E) = |
| 216 | \max_A \Adv{lor-cpa}{\mathcal{E}}(A) |
| 217 | \end{equation} |
| 218 | where the maximum is taken over all adversaries $A$ running in time $t$ and |
| 219 | making at most $q_E$ encryption queries, totalling most $\mu_E$ bits of |
| 220 | plaintext. |
| 221 | \end{definition} |
| 222 | |
| 223 | \subsection{The decision Diffie-Hellman problem} |
| 224 | |
| 225 | Let $G$ be some cyclic group. The standard \emph{Diffie-Hellman problem} |
| 226 | \cite{Diffie:1976:NDC} is to compute $g^{\alpha\beta}$ given $g^\alpha$ and |
| 227 | $g^\beta$. We need a slightly stronger assumption: that, given $g^\alpha$ |
| 228 | and $g^\beta$, it's hard to tell the difference between the correct |
| 229 | Diffie-Hellman value $g^{\alpha\beta}$ and a randomly-chosen group element |
| 230 | $g^\gamma$. This is the \emph{decision Diffie-Hellman problem} |
| 231 | \cite{Boneh:1998:DDP}. |
| 232 | |
| 233 | \begin{definition} |
| 234 | \label{def:ddh} |
| 235 | Let $G$ be a cyclic group of order $q$, and let $g$ be a generator of $G$. |
| 236 | Let $A$ be any algorithm. Then $A$'s \emph{advantage in solving the |
| 237 | decision Diffie-Hellman problem in $G$} is |
| 238 | \begin{equation} |
| 239 | \begin{eqnalign}[rl] |
| 240 | \Adv{ddh}{G}(A) = |
| 241 | & \Pr[\alpha \getsr \Nupto{q}; \beta \getsr \Nupto{q} : |
| 242 | A(g^\alpha, g^\beta, g^{\alpha\beta}) = 1] - {} \\ |
| 243 | & \Pr[\alpha \getsr \Nupto{q}; \beta \getsr \Nupto{q}; |
| 244 | \gamma \getsr \Nupto{q} : |
| 245 | A(g^\alpha, g^\beta, g^\gamma) = 1]. |
| 246 | \end{eqnalign} |
| 247 | \end{equation} |
| 248 | The \emph{insecurity function of the decision Diffie-Hellman problem in |
| 249 | $G$} is |
| 250 | \begin{equation} |
| 251 | \InSec{ddh}(G; t) = \max_A \Adv{ddh}{G}(A) |
| 252 | \end{equation} |
| 253 | where the maximum is taken over all algorithms $A$ which run in time $t$. |
| 254 | \end{definition} |
| 255 | |
| 256 | %%%-------------------------------------------------------------------------- |
| 257 | |
| 258 | \section{The protocol} |
| 259 | \label{sec:protocol} |
| 260 | |
| 261 | The Wrestlers Protocol is parameterized. We need the following things: |
| 262 | \begin{itemize} |
| 263 | \item A cyclic group $G$ whose order~$q$ is prime. Let $g$ be a generator |
| 264 | of~$G$. We require that the (decision?\fixme) Diffie-Hellman problem be |
| 265 | hard in~$G$. The group operation is written multiplicatively. |
| 266 | \item A symmetric encryption scheme $\mathcal{E} = (E, D)$. We require that |
| 267 | $\mathcal{E}$ be secure against adaptive chosen-plaintext attacks. Our |
| 268 | implementation uses Blowfish \cite{Schneier:1994:BEA} in CBC mode with |
| 269 | ciphertext stealing. See section~\ref{sec:cbc} for a description of |
| 270 | ciphertext stealing and an analysis of its security. |
| 271 | \item A message authentication scheme $\mathcal{M} = (T, V)$. We require |
| 272 | that $\mathcal{M}$ be (strongly) existentially unforgeable under |
| 273 | chosen-message attacks. Our implementation uses RIPEMD-160 |
| 274 | \cite{Dobbertin:1996:RSV} in the HMAC \cite{Bellare:1996:HC} construction. |
| 275 | \item An instantiation for the random oracle. We use RIPEMD-160 again, |
| 276 | either on its own, if the output is long enough, or in the MGF-1 |
| 277 | \cite{RFC2437} construction, if we need a larger output.\footnote{% |
| 278 | The use of the same hash function in the MAC as for instantiating the |
| 279 | random oracle is deliberate, with the aim of reducing the number of |
| 280 | primitives whose security we must assume. In an application of HMAC, the |
| 281 | message to be hashed is prefixed by a secret key padded out to the hash |
| 282 | function's block size. In a `random oracle' query, the message is |
| 283 | prefixed by a fixed identification string and not padded. Interference |
| 284 | between the two is then limited to the case where one of the HMAC keys |
| 285 | matches a random oracle prefix, which happens only with very tiny |
| 286 | probability.}% |
| 287 | \end{itemize} |
| 288 | |
| 289 | An authenticated encryption scheme with associated data (AEAD) |
| 290 | \cite{Rogaway:2002:AEAD, Rogaway:2001:OCB, Kohno:2003:CWC} could be used |
| 291 | instead of a separate symmetric encryption scheme and MAC. |
| 292 | |
| 293 | \subsection{Symmetric encryption} |
| 294 | |
| 295 | The same symmetric encryption subprotocol is used both within the key |
| 296 | exchange, to ensure secrecy and binding, and afterwards for message |
| 297 | transfer. It provides a secure channel between two players, assuming that |
| 298 | the key was chosen properly. |
| 299 | |
| 300 | A \id{keyset} contains the state required for communication between the two |
| 301 | players. In particular it maintains: |
| 302 | \begin{itemize} |
| 303 | \item separate encryption and MAC keys in each direction (four keys in |
| 304 | total), chosen using the random oracle based on an input key assumed to be |
| 305 | unpredictable by the adversary and a pair of nonces chosen by the two |
| 306 | players; and |
| 307 | \item incoming and outgoing sequence numbers, to detect and prevent replay |
| 308 | attacks. |
| 309 | \end{itemize} |
| 310 | |
| 311 | The operations involved in the symmetric encryption protocol are shown in |
| 312 | figure~\ref{fig:keyset}. |
| 313 | |
| 314 | The \id{keygen} procedure initializes a \id{keyset}, resetting the sequence |
| 315 | numbers, and selecting keys for the encryption scheme and MAC using the |
| 316 | random oracle. It uses the nonces $r_A$ and $r_B$ to ensure that with high |
| 317 | probability the keys are different for the two directions: assuming that |
| 318 | Alice chose her nonce $r_A$ at random, and that the keys and nonce are |
| 319 | $\kappa$~bits long, the probability that the keys in the two directions are |
| 320 | the same is at most $2^{\kappa - 2}$. |
| 321 | |
| 322 | The \id{encrypt} procedure constructs a ciphertext from a message $m$ and a |
| 323 | \emph{message type} $\id{ty}$. It encrypts the message giving a ciphertext |
| 324 | $y$, and computes a MAC tag $\tau$ for the triple $(\id{ty}, i, y)$, where |
| 325 | $i$ is the next available outgoing sequence number. The ciphertext message |
| 326 | to send is then $(i, y, \tau)$. The message type codes are used to |
| 327 | separate ciphertexts used by the key-exchange protocol itself from those sent |
| 328 | by the players later. |
| 329 | |
| 330 | The \id{decrypt} procedure recovers the plaintext from a ciphertext triple |
| 331 | $(i, y, \tau)$, given its expected type code $\id{ty}$. It verifies that the |
| 332 | tag $\tau$ is valid for the message $(\id{ty}, i, y)$, checks that the |
| 333 | sequence number $i$ hasn't been seen before,\footnote{% |
| 334 | The sequence number checking shown in the figure is simple but obviously |
| 335 | secure. The actual implementation maintains a window of 32 previous |
| 336 | sequence numbers, to allow out-of-order reception of messages while still |
| 337 | preventing replay attacks. This doesn't affect our analysis.}% |
| 338 | and then decrypts the ciphertext $y$. |
| 339 | |
| 340 | \begin{figure} |
| 341 | \begin{program} |
| 342 | Structure $\id{keyset}$: \+ \\ |
| 343 | $\Xid{K}{enc-in}$; $\Xid{K}{enc-out}$; \\ |
| 344 | $\Xid{K}{mac-in}$; $\Xid{K}{mac-out}$; \\ |
| 345 | $\id{seq-in}$; $\id{seq-out}$; \- \\[\medskipamount] |
| 346 | Function $\id{gen-keys}(r_A, r_B, K)$: \+ \\ |
| 347 | $k \gets \NEW \id{keyset}$; \\ |
| 348 | $k.\Xid{K}{enc-in} \gets H(\cookie{encryption}, r_A, r_B, K)$; \\ |
| 349 | $k.\Xid{K}{enc-out} \gets H(\cookie{encryption}, r_B, r_A, K)$; \\ |
| 350 | $k.\Xid{K}{mac-in} \gets H(\cookie{integrity}, r_A, r_B, K)$; \\ |
| 351 | $k.\Xid{K}{mac-out} \gets H(\cookie{integrity}, r_B, r_A, K)$; \\ |
| 352 | $k.\id{seq-in} \gets 0$; \\ |
| 353 | $k.\id{seq-out} \gets 0$; \\ |
| 354 | \RETURN $k$; |
| 355 | \next |
| 356 | Function $\id{encrypt}(k, \id{ty}, m)$: \+ \\ |
| 357 | $y \gets (E_{k.\Xid{K}{enc-out}}(m))$; \\ |
| 358 | $i \gets k.\id{seq-out}$; \\ |
| 359 | $\tau \gets T_{k.\Xid{K}{mac-out}}(\id{ty}, i, y)$; \\ |
| 360 | $k.\id{seq-out} \gets i + 1$; \\ |
| 361 | \RETURN $(i, y, \tau)$; \- \\[\medskipamount] |
| 362 | Function $\id{decrypt}(k, \id{ty}, c)$: \+ \\ |
| 363 | $(i, y, \tau) \gets c$; \\ |
| 364 | \IF $V_{k.\Xid{K}{mac-in}}((\id{ty}, i, y), \tau) = 0$ \THEN \\ \ind |
| 365 | \RETURN $\bot$; \- \\ |
| 366 | \IF $i < k.\id{seq-in}$ \THEN \RETURN $\bot$; \\ |
| 367 | $m \gets D_{k.\Xid{K}{enc-in}}(y)$; \\ |
| 368 | $k.\id{seq-in} \gets i + 1$; \\ |
| 369 | \RETURN $m$; |
| 370 | \end{program} |
| 371 | |
| 372 | \caption{Symmetric-key encryption functions} |
| 373 | \label{fig:keyset} |
| 374 | \end{figure} |
| 375 | |
| 376 | \subsection{The key-exchange} |
| 377 | |
| 378 | The key-exchange protocol is completely symmetrical. Either party may |
| 379 | initiate, or both may attempt to converse at the same time. We shall |
| 380 | describe the protocol from the point of view of Alice attempting to exchange |
| 381 | a key with Bob. |
| 382 | |
| 383 | Alice's private key is a random index $\alpha \inr \Nupto{q}$. Her public |
| 384 | key is $a = g^\alpha$. Bob's public key is $b \in G$. We'll subscript the |
| 385 | variables Alice computes with an~$A$, and the values Bob has sent with a~$B$. |
| 386 | Of course, if Bob is following the protocol correctly, he will have computed |
| 387 | his $B$ values in a completely symmetrical way. |
| 388 | |
| 389 | There are six messages in the protocol, and we shall briefly discuss the |
| 390 | purpose of each before embarking on the detailed descriptions. At the |
| 391 | beginning of the protocol, Alice chooses a new random index $\rho_A$ and |
| 392 | computes her \emph{challenge} $r_A = g^{\rho_A}$. Eventually, the shared |
| 393 | secret key will be computed as $K = r_B^{\rho_A} = r_A^{\rho_B} = |
| 394 | g^{\rho_A\rho_B}$, as for standard Diffie-Hellman key agreement. |
| 395 | |
| 396 | Throughout, we shall assume that messages are implicitly labelled with the |
| 397 | sender's identity. If Alice is actually trying to talk to several other |
| 398 | people she'll need to run multiple instances of the protocol, each with its |
| 399 | own state, and she can use the sender label to decide which instance a |
| 400 | message should be processed by. There's no need for the implicit labels to |
| 401 | be attached securely. |
| 402 | |
| 403 | We'll summarize the messages and their part in the scheme of things before we |
| 404 | start on the serious detail. For a summary of the names and symbols used in |
| 405 | these descriptions, see table~\ref{tab:kx-names}. The actual message |
| 406 | contents are summarized in table~\ref{tab:kx-messages}. A state-transition |
| 407 | diagram of the protocol is shown in figure~\ref{fig:kx-states}. If reading |
| 408 | pesudocode algorithms is your thing then you'll find message-processing |
| 409 | procedures in figure~\ref{fig:kx-messages} with the necessary support procedures |
| 410 | in figure~\ref{fig:kx-support}. |
| 411 | |
| 412 | \begin{table} |
| 413 | \begin{tabularx}{\textwidth}{Mr X} |
| 414 | G & A cyclic group known by all participants \\ |
| 415 | q = |G| & The prime order of $G$ \\ |
| 416 | g & A generator of $G$ \\ |
| 417 | E_K(\cdot) & Encryption under key $K$, here used to denote |
| 418 | application of the $\id{encrypt}(K, \cdot)$ |
| 419 | operation \\ |
| 420 | \alpha \inr \Nupto{q} & Alice's private key \\ |
| 421 | a = g^{\alpha} & Alice's public key \\ |
| 422 | \rho_A \inr \Nupto{q} & Alice's secret Diffie-Hellman value \\ |
| 423 | r_A = g^{\rho_A} & Alice's public \emph{challenge} \\ |
| 424 | c_A = H(\cookie{cookie}, r_A) |
| 425 | & Alice's \emph{cookie} \\ |
| 426 | v_A = \rho_A \xor H(\cookie{expected-reply}, r_A, r_B, b^{\rho_A}) |
| 427 | & Alice's challenge \emph{check value} \\ |
| 428 | r_B^\alpha = a^{\rho_B} |
| 429 | & Alice's reply \\ |
| 430 | K = r_B^{\rho_A} = r_B^{\rho_A} = g^{\rho_A\rho_B} |
| 431 | & Alice and Bob's shared secret key \\ |
| 432 | w_A = H(\cookie{switch-request}, c_A, c_B) |
| 433 | & Alice's \emph{switch request} value \\ |
| 434 | u_A = H(\cookie{switch-confirm}, c_A, c_B) |
| 435 | & Alice's \emph{switch confirm} value \\ |
| 436 | \end{tabularx} |
| 437 | |
| 438 | \caption{Names used during key-exchange} |
| 439 | \label{tab:kx-names} |
| 440 | \end{table} |
| 441 | |
| 442 | \begin{table} |
| 443 | \begin{tabular}[C]{Ml} |
| 444 | \cookie{kx-pre-challenge}, r_A \\ |
| 445 | \cookie{kx-cookie}, r_A, c_B \\ |
| 446 | \cookie{kx-challenge}, r_A, c_B, v_A \\ |
| 447 | \cookie{kx-reply}, c_A, c_B, v_A, E_K(r_B^\alpha)) \\ |
| 448 | \cookie{kx-switch}, c_A, c_B, E_K(r_B^\alpha, w_A)) \\ |
| 449 | \cookie{kx-switch-ok}, E_K(u_A)) |
| 450 | \end{tabular} |
| 451 | |
| 452 | \caption{Message contents, as sent by Alice} |
| 453 | \label{tab:kx-messages} |
| 454 | \end{table} |
| 455 | |
| 456 | \begin{messages} |
| 457 | \item[kx-pre-challenge] Contains a plain statement of Alice's challenge. |
| 458 | This is Alice's first message of a session. |
| 459 | \item[kx-cookie] A bare acknowledgement of a received challenge: it restates |
| 460 | Alice's challenge, and contains a hash of Bob's challenge. This is an |
| 461 | engineering measure (rather than a cryptographic one) which prevents |
| 462 | trivial denial-of-service attacks from working. |
| 463 | \item[kx-challenge] A full challenge, with a `check value' which proves the |
| 464 | challenge's honesty. Bob's correct reply to this challenge informs Alice |
| 465 | that she's received his challenge correctly. |
| 466 | \item[kx-reply] A reply. This contains a `check value', like the |
| 467 | \cookie{kx-challenge} message above, and an encrypted reply which confirms |
| 468 | to Bob Alice's successful receipt of his challenge and lets Bob know he |
| 469 | received Alice's challenge correctly. |
| 470 | \item[kx-switch] Acknowledges Alice's receipt of Bob's \cookie{kx-reply} |
| 471 | message, including Alice's own reply to Bob's challenge. Tells Bob that |
| 472 | she can start using the key they've agreed. |
| 473 | \item[kx-switch-ok] Acknowlegement to Bob's \cookie{kx-switch} message. |
| 474 | \end{messages} |
| 475 | |
| 476 | \begin{figure} |
| 477 | \small |
| 478 | \let\ns\normalsize |
| 479 | \let\c\cookie |
| 480 | \[ \begin{graph} |
| 481 | []!{0; <4.5cm, 0cm>: <0cm, 1.5cm>::} |
| 482 | *++[F:<4pt>]\txt{\ns Start \\ Choose $\rho_A$} ="start" |
| 483 | :[dd] |
| 484 | *++[F:<4pt>]\txt{ |
| 485 | \ns State \c{challenge} \\ |
| 486 | Send $(\c{pre-challenge}, r_A)$} |
| 487 | ="chal" |
| 488 | [] "chal" !{!L(0.5)} ="chal-cookie" |
| 489 | :@(d, d)[l] |
| 490 | *+\txt{Send $(\c{cookie}, r_A, c_B)$} |
| 491 | ="cookie" |
| 492 | |*+\txt{Receive \\ $(\c{pre-challenge}, r_B)$ \\ (no spare slot)} |
| 493 | :@(u, u)"chal-cookie" |
| 494 | "chal" :@/_0.8cm/ [ddddl] |
| 495 | *+\txt{Send \\ $(\c{challenge}, $\\$ r_A, c_B, v_A)$} |
| 496 | ="send-chal" |
| 497 | |<>(0.67) *+\txt\small{ |
| 498 | Receive \\ $(\c{pre-challenge}, r_B)$ \\ (spare slot)} |
| 499 | "chal" :@/^0.8cm/ "send-chal" |<>(0.33) |
| 500 | *+\txt{Receive \\ $(\c{cookie}, r_B, c_A)$} |
| 501 | :[rr] |
| 502 | *+\txt{Send \\ $(\c{reply}, c_A, c_B, $\\$ v_A, E_K(r_B^\alpha))$} |
| 503 | ="send-reply" |
| 504 | |*+\txt{Receive \\ $(\c{challenge}, $\\$ r_B, c_A, v_B)$} |
| 505 | "chal" :"send-reply" |
| 506 | |*+\txt{Receive \\ $(\c{challenge}, $\\$ r_B, c_A, v_B)$} |
| 507 | "send-chal" :[ddd] |
| 508 | *++[F:<4pt>]\txt{ |
| 509 | \ns State \c{commit} \\ |
| 510 | Send \\ $(\c{switch}, c_A, c_B, $\\$ E_K(r_B^\alpha, w_A))$} |
| 511 | ="commit" |
| 512 | |*+\txt{Receive \\ $(\c{reply}, c_B, c_A, $\\$ v_B, E_K(b^{\rho_A}))$} |
| 513 | :[rr] |
| 514 | *+\txt{Send \\ $(\c{switch-ok}, E_K(u_A))$} |
| 515 | ="send-switch-ok" |
| 516 | |*+\txt{Receive \\ $(\c{switch}, c_B, c_A, $\\$ E_K(b^{\rho_A}, w_B))$} |
| 517 | "send-reply" :"commit" |
| 518 | |*+\txt{Receive \\ $(\c{reply}, c_B, c_A, $\\$ v_B, E_K(b^{\rho_A}))$} |
| 519 | "send-reply" :"send-switch-ok" |
| 520 | |*+\txt{Receive \\ $(\c{switch}, c_B, c_A, $\\$ E_K(b^{\rho_A}, w_B))$} |
| 521 | :[dddl] |
| 522 | *++[F:<4pt>]\txt{\ns Done} |
| 523 | ="done" |
| 524 | "commit" :"done" |
| 525 | |*+\txt{Receive \\ $(\c{switch-ok}, E_K(u_B))$} |
| 526 | "send-chal" [r] !{+<0cm, 0.75cm>} |
| 527 | *\txt\itshape{For each outstanding challenge} |
| 528 | ="for-each" |
| 529 | !{"send-chal"+DL-<8pt, 8pt> ="p0", |
| 530 | "for-each"+U+<8pt> ="p1", |
| 531 | "send-reply"+UR+<8pt, 8pt> ="p2", |
| 532 | "send-reply"+DR+<8pt, 8pt> ="p3", |
| 533 | "p0" !{"p1"-"p0"} !{"p2"-"p1"} !{"p3"-"p2"} |
| 534 | *\frm<8pt>{--}} |
| 535 | \end{graph} \] |
| 536 | |
| 537 | \caption{State-transition diagram for key-exchange protocol} |
| 538 | \label{fig:kx-states} |
| 539 | \end{figure} |
| 540 | |
| 541 | We now describe the protocol message by message, and Alice's actions when she |
| 542 | receives each. Since the protocol is completely symmetrical, Bob should do |
| 543 | the same, only swapping round $A$ and $B$ subscripts, the public keys $a$ and |
| 544 | $b$, and using his private key $\beta$ instead of $\alpha$. |
| 545 | |
| 546 | \subsubsection{Starting the protocol} |
| 547 | |
| 548 | As described above, at the beginning of the protocol Alice chooses a random |
| 549 | $\rho_A \inr \Nupto q$, and computes her \emph{challenge} $r_A = g^{\rho_A}$ |
| 550 | and her \emph{cookie} $c_A = H(\cookie{cookie}, r_A)$. She sends her |
| 551 | announcement of her challenge as |
| 552 | \begin{equation} |
| 553 | \label{eq:kx-pre-challenge} |
| 554 | \cookie{kx-pre-challenge}, r_A |
| 555 | \end{equation} |
| 556 | and enters the \cookie{challenge} state. |
| 557 | |
| 558 | \subsubsection{The \cookie{kx-pre-challenge} message} |
| 559 | |
| 560 | If Alice receieves a \cookie{kx-pre-challenge}, she ensures that she's in the |
| 561 | \cookie{challenge} state: if not, she rejects the message. |
| 562 | |
| 563 | She must first calculate Bob's cookie $c_B = H(\cookie{cookie}, r_B)$. Then |
| 564 | she has a choice: either she can send a full challenge, or she can send the |
| 565 | cookie back. |
| 566 | |
| 567 | Suppose she decides to send a full challenge. She must compute a \emph{check |
| 568 | value} |
| 569 | \begin{equation} |
| 570 | \label{eq:v_A} |
| 571 | v_A = \rho_A \xor H(\cookie{expected-reply}, r_A, r_B, b^{\rho_A}) |
| 572 | \end{equation} |
| 573 | and sends |
| 574 | \begin{equation} |
| 575 | \label{eq:kx-challenge} |
| 576 | \cookie{kx-challenge}, r_A, c_B, v_A |
| 577 | \end{equation} |
| 578 | to Bob. Then she remembers Bob's challenge for later use, and awaits his |
| 579 | reply. |
| 580 | |
| 581 | If she decides to send only a cookie, she just transmits |
| 582 | \begin{equation} |
| 583 | \label{eq:kx-cookie} |
| 584 | \cookie{kx-cookie}, r_A, c_B |
| 585 | \end{equation} |
| 586 | to Bob and forgets all about it. |
| 587 | |
| 588 | Why's this useful? Well, if Alice sends off a full \cookie{kx-challenge} |
| 589 | message, she must remember Bob's $r_B$ so she can check his reply and that |
| 590 | involves using up a table slot. That means that someone can send Alice |
| 591 | messages purporting to come from Bob which will chew up Alice's memory, and |
| 592 | they don't even need to be able to read Alice's messages to Bob to do that. |
| 593 | If this protocol were used over the open Internet, script kiddies from all |
| 594 | over the world might be flooding Alice with bogus \cookie{kx-pre-challenge} |
| 595 | messages and she'd never get around to talking to Bob. |
| 596 | |
| 597 | By sending a cookie intead, she avoids committing a table slot until Bob (or |
| 598 | someone) sends either a cookie or a full challenge, thus proving, at least, |
| 599 | that he can read her messages. This is the best we can do at this stage in |
| 600 | the protocol. Against an adversary as powerful as the one we present in |
| 601 | section~\fixme\ref{sec:formal} this measure provides no benefit (but we have |
| 602 | to analyse it anyway); but it raises the bar too sufficiently high to |
| 603 | eliminate a large class of `nuisance' attacks in the real world. |
| 604 | |
| 605 | Our definition of the Wrestlers Protocol doesn't stipulate when Alice should |
| 606 | send a full challenge or just a cookie: we leave this up to individual |
| 607 | implementations, because it makes no difference to the security of the |
| 608 | protocol against powerful adversaries. But we recommend that Alice proceed |
| 609 | `optimistically' at first, sending full challenges until her challenge table |
| 610 | looks like it's running out, and then generating cookies only if it actually |
| 611 | looks like she's under attack. This is what our pseudocode in |
| 612 | figure~\ref{fig:kx-messages} does. |
| 613 | |
| 614 | \subsubsection{The \cookie{kx-cookie} message} |
| 615 | |
| 616 | When Alice receives a \cookie{kx-cookie} message, she must ensure that she's |
| 617 | in the \cookie{challenge} state: if not, she rejects the message. She checks |
| 618 | the cookie in the message against the value of $c_A$ she computed earlier. |
| 619 | If all is well, Alice sends a \cookie{kx-challenge} message, as in |
| 620 | equation~\ref{eq:kx-challenge} above. |
| 621 | |
| 622 | This time, she doesn't have a choice about using up a table slot to remember |
| 623 | Bob's $r_B$. If her table size is fixed, she must choose a slot to recycle. |
| 624 | We suggest simply recycling slots at random: this means there's no clever |
| 625 | pattern of \cookie{kx-cookie} messages an attacker might be able to send to |
| 626 | clog up all of Alice's slots. |
| 627 | |
| 628 | \subsubsection{The \cookie{kx-challenge} message} |
| 629 | |
| 630 | |
| 631 | |
| 632 | \begin{figure} |
| 633 | \begin{program} |
| 634 | Procedure $\id{kx-initialize}$: \+ \\ |
| 635 | $\rho_A \getsr [q]$; \\ |
| 636 | $r_a \gets g^{\rho_A}$; \\ |
| 637 | $\id{state} \gets \cookie{challenge}$; \\ |
| 638 | $\Xid{n}{chal} \gets 0$; \\ |
| 639 | $k \gets \bot$; \\ |
| 640 | $\id{chal-commit} \gets \bot$; \\ |
| 641 | $\id{send}(\cookie{kx-pre-challenge}, r_A)$; \- \\[\medskipamount] |
| 642 | Procedure $\id{kx-receive}(\id{type}, \id{data})$: \\ \ind |
| 643 | \IF $\id{type} = \cookie{kx-pre-challenge}$ \THEN \\ \ind |
| 644 | \id{msg-pre-challenge}(\id{data}); \- \\ |
| 645 | \ELSE \IF $\id{type} = \cookie{kx-cookie}$ \THEN \\ \ind |
| 646 | \id{msg-cookie}(\id{data}); \- \\ |
| 647 | \ELSE \IF $\id{type} = \cookie{kx-challenge}$ \THEN \\ \ind |
| 648 | \id{msg-challenge}(\id{data}); \- \\ |
| 649 | \ELSE \IF $\id{type} = \cookie{kx-reply}$ \THEN \\ \ind |
| 650 | \id{msg-reply}(\id{data}); \- \\ |
| 651 | \ELSE \IF $\id{type} = \cookie{kx-switch}$ \THEN \\ \ind |
| 652 | \id{msg-switch}(\id{data}); \- \\ |
| 653 | \ELSE \IF $\id{type} = \cookie{kx-switch-ok}$ \THEN \\ \ind |
| 654 | \id{msg-switch-ok}(\id{data}); \-\- \\[\medskipamount] |
| 655 | Procedure $\id{msg-pre-challenge}(\id{data})$: \+ \\ |
| 656 | \IF $\id{state} \ne \cookie{challenge}$ \THEN \RETURN; \\ |
| 657 | $r \gets \id{data}$; \\ |
| 658 | \IF $\Xid{n}{chal} \ge \Xid{n}{chal-thresh}$ \THEN \\ \ind |
| 659 | $\id{send}(\cookie{kx-cookie}, r_A, \id{cookie}(r_A)))$; \- \\ |
| 660 | \ELSE \+ \\ |
| 661 | $\id{new-chal}(r)$; \\ |
| 662 | $\id{send}(\cookie{kx-challenge}, r_A, |
| 663 | \id{cookie}(r), \id{checkval}(r))$; \-\-\\[\medskipamount] |
| 664 | Procedure $\id{msg-cookie}(\id{data})$: \+ \\ |
| 665 | \IF $\id{state} \ne \cookie{challenge}$ \THEN \RETURN; \\ |
| 666 | $(r, c_A) \gets \id{data}$; \\ |
| 667 | \IF $c_A \ne \id{cookie}(r_A)$ \THEN \RETURN; \\ |
| 668 | $\id{new-chal}(r)$; \\ |
| 669 | $\id{send}(\cookie{kx-challenge}, r_A, |
| 670 | \id{cookie}(r), \id{checkval}(r))$; \- \\[\medskipamount] |
| 671 | Procedure $\id{msg-challenge}(\id{data})$: \+ \\ |
| 672 | \IF $\id{state} \ne \cookie{challenge}$ \THEN \RETURN; \\ |
| 673 | $(r, c_A, v) \gets \id{data}$; \\ |
| 674 | \IF $c_A \ne \id{cookie}(r_A)$ \THEN \RETURN; \\ |
| 675 | $i \gets \id{check-reply}(\bot, r, v)$; \\ |
| 676 | \IF $i = \bot$ \THEN \RETURN; \\ |
| 677 | $k \gets \id{chal-tab}[i].k$; \\ |
| 678 | $y \gets \id{encrypt}(k, \cookie{kx-reply}, r^\alpha)$; \\ |
| 679 | $\id{send}(\cookie{kx-reply}, c_A, \id{cookie}(r), |
| 680 | \id{checkval}(r), y)$ |
| 681 | \next |
| 682 | Procedure $\id{msg-reply}(\id{data})$: \+ \\ |
| 683 | $(c, c_A, v, y) \gets \id{data}$; \\ |
| 684 | \IF $c_A \ne \id{cookie}(r_A)$ \THEN \RETURN; \\ |
| 685 | $i \gets \id{find-chal}(c)$; \\ |
| 686 | \IF $i = \bot$ \THEN \RETURN; \\ |
| 687 | \IF $\id{check-reply}(i, \id{chal-tab}[i].r, v) = \bot$ \THEN \\ \ind |
| 688 | \RETURN; \- \\ |
| 689 | $k \gets \id{chal-tab}[i].k$; \\ |
| 690 | $x \gets \id{decrypt}(k, \cookie{kx-reply}, y)$; \\ |
| 691 | \IF $x = \bot$ \THEN \RETURN; \\ |
| 692 | \IF $x \ne b^{\rho_A}$ \THEN \RETURN; \\ |
| 693 | $\id{state} \gets \cookie{commit}$; \\ |
| 694 | $\id{chal-commit} \gets \id{chal-tab}[i]$; \\ |
| 695 | $w \gets H(\cookie{switch-request}, c_A, c)$; \\ |
| 696 | $x \gets \id{chal-tab}[i].r^\alpha$; \\ |
| 697 | $y \gets \id{encrypt}(k, (x, \cookie{kx-switch}, w))$; \\ |
| 698 | $\id{send}(\cookie{kx-switch}, c_A, c, y)$; \-\\[\medskipamount] |
| 699 | Procedure $\id{msg-switch}(\id{data})$: \+ \\ |
| 700 | $(c, c_A, y) \gets \id{data}$; \\ |
| 701 | \IF $c_A \ne \cookie(r_A)$ \THEN \RETURN; \\ |
| 702 | $i \gets \id{find-chal}(c)$; \\ |
| 703 | \IF $i = \bot$ \THEN \RETURN; \\ |
| 704 | $k \gets \id{chal-tab}[i].k$; \\ |
| 705 | $x \gets \id{decrypt}(k, \cookie{kx-switch}, y)$; \\ |
| 706 | \IF $x = \bot$ \THEN \RETURN; \\ |
| 707 | $(x, w) \gets x$; \\ |
| 708 | \IF $\id{state} = \cookie{challenge}$ \THEN \\ \ind |
| 709 | \IF $x \ne b^{\rho_A}$ \THEN \RETURN; \\ |
| 710 | $\id{chal-commit} \gets \id{chal-tab}[i]$; \- \\ |
| 711 | \ELSE \IF $c \ne \id{chal-commit}.c$ \THEN \RETURN; \\ |
| 712 | \IF $w \ne H(\cookie{switch-request}, c, c_A)$ \THEN \RETURN; \\ |
| 713 | $w \gets H(\cookie{switch-confirm}, c_A, c)$; \\ |
| 714 | $y \gets \id{encrypt}(y, \cookie{kx-switch-ok}, w)$; \\ |
| 715 | $\id{send}(\cookie{switch-ok}, y)$; \\ |
| 716 | $\id{done}(k)$; \- \\[\medskipamount] |
| 717 | Procedure $\id{msg-switch-ok}(\id{data})$ \+ \\ |
| 718 | \IF $\id{state} \ne \cookie{commit}$ \THEN \RETURN; \\ |
| 719 | $y \gets \id{data}$; \\ |
| 720 | $k \gets \id{chal-commit}.k$; \\ |
| 721 | $w \gets \id{decrypt}(k, \cookie{kx-switch-ok}, y)$; \\ |
| 722 | \IF $w = \bot$ \THEN \RETURN; \\ |
| 723 | $c \gets \id{chal-commit}.c$; \\ |
| 724 | $c_A \gets \id{cookie}(r_A)$; \\ |
| 725 | \IF $w \ne H(\cookie{switch-confirm}, c, c_A)$ \THEN \RETURN; \\ |
| 726 | $\id{done}(k)$; |
| 727 | \end{program} |
| 728 | |
| 729 | \caption{The key-exchange protocol: message handling} |
| 730 | \label{fig:kx-messages} |
| 731 | \end{figure} |
| 732 | |
| 733 | \begin{figure} |
| 734 | \begin{program} |
| 735 | Structure $\id{chal-slot}$: \+ \\ |
| 736 | $r$; $c$; $\id{replied}$; $k$; \- \\[\medskipamount] |
| 737 | Function $\id{find-chal}(c)$: \+ \\ |
| 738 | \FOR $i = 0$ \TO $\Xid{n}{chal}$ \DO \\ \ind |
| 739 | \IF $\id{chal-tab}[i].c = c$ \THEN \RETURN $i$; \- \\ |
| 740 | \RETURN $\bot$; \- \\[\medskipamount] |
| 741 | Function $\id{cookie}(r)$: \+ \\ |
| 742 | \RETURN $H(\cookie{cookie}, r)$; \- \\[\medskipamount] |
| 743 | Function $\id{check-reply}(i, r, v)$: \+ \\ |
| 744 | \IF $i \ne \bot \land \id{chal-tab}[i].\id{replied} = 1$ \THEN \\ \ind |
| 745 | \RETURN $i$; \- \\ |
| 746 | $\rho \gets v \xor H(\cookie{expected-reply}, r, r_A, r^\alpha)$; \\ |
| 747 | \IF $g^\rho \ne r$ \THEN \RETURN $\bot$; \\ |
| 748 | \IF $i = \bot$ \THEN $i \gets \id{new-chal}(r)$; \\ |
| 749 | $\id{chal-tab}[i].k \gets \id{gen-keys}(r_A, r, r^{\rho_A})$; \\ |
| 750 | $\id{chal-tab}[i].\id{replied} \gets 1$; \\ |
| 751 | \RETURN $i$; |
| 752 | \next |
| 753 | Function $\id{checkval}(r)$: \\ \ind |
| 754 | \RETURN $\rho_A \xor H(\cookie{expected-reply}, |
| 755 | r_A,r, b^{\rho_A})$; \- \\[\medskipamount] |
| 756 | Function $\id{new-chal}(r)$: \+ \\ |
| 757 | $c \gets \id{cookie}(r)$; \\ |
| 758 | $i \gets \id{find-chal}(c)$; \\ |
| 759 | \IF $i \ne \bot$ \THEN \RETURN $i$; \\ |
| 760 | \IF $\Xid{n}{chal} < \Xid{n}{chal-max}$ \THEN \\ \ind |
| 761 | $i \gets \Xid{n}{chal}$; \\ |
| 762 | $\id{chal-tab}[i] \gets \NEW \id{chal-slot}$; \\ |
| 763 | $\Xid{n}{chal} \gets \Xid{n}{chal} + 1$; \- \\ |
| 764 | \ELSE \\ \ind |
| 765 | $i \getsr [\Xid{n}{chal-max}]$; \- \\ |
| 766 | $\id{chal-tab}[i].r \gets r$; \\ |
| 767 | $\id{chal-tab}[i].c \gets c$; \\ |
| 768 | $\id{chal-tab}[i].\id{replied} \gets 0$; \\ |
| 769 | $\id{chal-tab}[i].k \gets \bot$; \\ |
| 770 | \RETURN $i$; |
| 771 | \end{program} |
| 772 | |
| 773 | \caption{The key-exchange protocol: support functions} |
| 774 | \label{fig:kx-support} |
| 775 | \end{figure} |
| 776 | |
| 777 | %%%-------------------------------------------------------------------------- |
| 778 | |
| 779 | \section{CBC mode encryption} |
| 780 | \label{sec:cbc} |
| 781 | |
| 782 | Our implementation of the Wrestlers Protocol uses Blowfish |
| 783 | \cite{Schneier:1994:BEA} in CBC mode. However, rather than pad plaintext |
| 784 | messages to a block boundary, with the ciphertext expansion that entails, we |
| 785 | use a technique called \emph{ciphertext stealing} |
| 786 | \cite[section 9.3]{Schneier:1996:ACP}. |
| 787 | |
| 788 | \subsection{Standard CBC mode} |
| 789 | |
| 790 | Suppose $E$ is an $\ell$-bit pseudorandom permutation. Normal CBC mode works |
| 791 | as follows. Given a message $X$, we divide it into blocks $x_0, x_1, \ldots, |
| 792 | x_{n-1}$. Choose a random \emph{initialization vector} $I \inr \Bin^\ell$. |
| 793 | Before passing each $x_i$ through $E$, we XOR it with the previous |
| 794 | ciphertext, with $I$ standing in for the first block: |
| 795 | \begin{equation} |
| 796 | y_0 = E_K(x_0 \xor I) \qquad |
| 797 | y_i = E_K(x_i \xor y_{i-1} \ \text{(for $1 \le i < n$)}. |
| 798 | \end{equation} |
| 799 | The ciphertext is then the concatenation of $I$ and the $y_i$. Decryption is |
| 800 | simple: |
| 801 | \begin{equation} |
| 802 | x_0 = E^{-1}_K(y_0) \xor I \qquad |
| 803 | x_i = E^{-1}_K(y_i) \xor y_{i-1} \ \text{(for $1 \le i < n$)} |
| 804 | \end{equation} |
| 805 | See figure~\ref{fig:cbc} for a diagram of CBC encryption. |
| 806 | |
| 807 | \begin{figure} |
| 808 | \[ \begin{graph} |
| 809 | []!{0; <0.85cm, 0cm>: <0cm, 0.5cm>::} |
| 810 | *+=(1, 0)+[F]{\mathstrut x_0}="x" |
| 811 | :[dd] *{\xor}="xor" |
| 812 | [ll] *+=(1, 0)+[F]{I} :"xor" |
| 813 | :[dd] *+[F]{E}="e" :[ddd] *+=(1, 0)+[F]{\mathstrut y_0}="i" |
| 814 | "e" [l] {K} :"e" |
| 815 | [rrruuuu] *+=(1, 0)+[F]{\mathstrut x_1}="x" |
| 816 | :[dd] *{\xor}="xor" |
| 817 | "e" [d] :`r [ru] `u "xor" "xor" |
| 818 | :[dd] *+[F]{E}="e" :[ddd] |
| 819 | *+=(1, 0)+[F]{\mathstrut y_1}="i" |
| 820 | "e" [l] {K} :"e" |
| 821 | [rrruuuu] *+=(1, 0)+[F--]{\mathstrut x_i}="x" |
| 822 | :@{-->}[dd] *{\xor}="xor" |
| 823 | "e" [d] :@{-->}`r [ru] `u "xor" "xor" |
| 824 | :@{-->}[dd] *+[F]{E}="e" :@{-->}[ddd] |
| 825 | *+=(1, 0)+[F--]{\mathstrut y_i}="i" |
| 826 | "e" [l] {K} :@{-->}"e" |
| 827 | [rrruuuu] *+=(1, 0)+[F]{\mathstrut x_{n-1}}="x" |
| 828 | :[dd] *{\xor}="xor" |
| 829 | "e" [d] :@{-->}`r [ru] `u "xor" "xor" |
| 830 | :[dd] *+[F]{E}="e" :[ddd] |
| 831 | *+=(1, 0)+[F]{\mathstrut y_{n-1}}="i" |
| 832 | "e" [l] {K} :"e" |
| 833 | \end{graph} \] |
| 834 | |
| 835 | \caption{Encryption using CBC mode} |
| 836 | \label{fig:cbc} |
| 837 | \end{figure} |
| 838 | |
| 839 | \begin{definition}[CBC mode] |
| 840 | \label{def:cbc} |
| 841 | Let $P\colon \keys P \times \Bin^\ell to \Bin^\ell$ be a pseudorandom |
| 842 | permutation. We define the symmetric encryption scheme |
| 843 | $\Xid{\mathcal{E}}{CBC}^P = (\Xid{E}{CBC}^P, \Xid{D}{CBC}^P)$ for messages |
| 844 | in $\Bin^{\ell\Z}$ by setting $\keys \Xid{\mathcal{E}}{CBC} = \keys P$ and |
| 845 | defining the encryption and decryption algorithms as follows: |
| 846 | \begin{program} |
| 847 | Algorithm $\Xid{E}{CBC}^P_K(x)$: \+ \\ |
| 848 | $I \getsr \Bin^\ell$; \\ |
| 849 | $y \gets I$; \\ |
| 850 | \FOR $i = 0$ \TO $|x|/\ell$ \DO \\ \ind |
| 851 | $x_i \gets x[\ell i \bitsto \ell (i + 1)]$; \\ |
| 852 | $y_i \gets P_K(x_i \xor I)$; \\ |
| 853 | $I \gets y_i$; \\ |
| 854 | $y \gets y \cat y_i$; \- \\ |
| 855 | \RETURN $y$; |
| 856 | \next |
| 857 | Algorithm $\Xid{D}{CBC}^P_K(y)$: \+ \\ |
| 858 | $I \gets y[0 \bitsto \ell]$; \\ |
| 859 | $x \gets \emptystring$; \\ |
| 860 | \FOR $1 = 0$ \TO $|y|/\ell$ \DO \\ \ind |
| 861 | $y_i \gets y[\ell i \bitsto \ell (i + 1)]$; \\ |
| 862 | $x_i \gets P^{-1}_K(y_i) \xor I$; \\ |
| 863 | $I \gets y_i$; \\ |
| 864 | $x \gets x \cat x_i$; \- \\ |
| 865 | \RETURN $x$; |
| 866 | \end{program} |
| 867 | \end{definition} |
| 868 | |
| 869 | \begin{theorem}[Security of standard CBC mode] |
| 870 | \label{thm:cbc} |
| 871 | Let $P\colon \keys P \times \Bin^\ell \to \Bin^\ell$ be a pseudorandom |
| 872 | permutation. Then, |
| 873 | \begin{equation} |
| 874 | \InSec{lor-cpa}(\Xid{\mathcal{E}}{CBC}; t, q_E + \mu_E) \le |
| 875 | 2 \cdot \InSec{prp}(P; t + q t_P, q) + |
| 876 | \frac{q (q - 1)}{2^\ell - 2^{\ell/2}} |
| 877 | \end{equation} |
| 878 | where $q = \mu_E/\ell$ and $t_P$ is some small constant. |
| 879 | \end{theorem} |
| 880 | |
| 881 | \begin{note} |
| 882 | Our security bound is slightly better than that of \cite[theorem |
| 883 | 17]{Bellare:2000:CST}. Their theorem statement contains a term $3 \cdot q |
| 884 | (q - 1) 2^{-\ell-1}$. Our result lowers the factor from 3 to just over 2. |
| 885 | Our proof is also much shorter and considerably more comprehensible. |
| 886 | \end{note} |
| 887 | |
| 888 | The proof of this theorem is given in section~\ref{sec:cbc-proof} |
| 889 | |
| 890 | \subsection{Ciphertext stealing} |
| 891 | |
| 892 | Ciphertext stealing allows us to encrypt any message in $\Bin^*$ and make the |
| 893 | ciphertext exactly $\ell$ bits longer than the plaintext. See |
| 894 | figure~\ref{fig:cbc-steal} for a diagram. |
| 895 | |
| 896 | \begin{figure} |
| 897 | \[ \begin{graph} |
| 898 | []!{0; <0.85cm, 0cm>: <0cm, 0.5cm>::} |
| 899 | *+=(1, 0)+[F]{\mathstrut x_0}="x" |
| 900 | :[dd] *{\xor}="xor" |
| 901 | [ll] *+=(1, 0)+[F]{I} :"xor" |
| 902 | :[dd] *+[F]{E}="e" :[ddddd] *+=(1, 0)+[F]{\mathstrut y_0}="i" |
| 903 | "e" [l] {K} :"e" |
| 904 | [rrruuuu] *+=(1, 0)+[F]{\mathstrut x_1}="x" |
| 905 | :[dd] *{\xor}="xor" |
| 906 | "e" [d] :`r [ru] `u "xor" "xor" |
| 907 | :[dd] *+[F]{E}="e" :[ddddd] |
| 908 | *+=(1, 0)+[F]{\mathstrut y_1}="i" |
| 909 | "e" [l] {K} :"e" |
| 910 | [rrruuuu] *+=(1, 0)+[F--]{\mathstrut x_i}="x" |
| 911 | :@{-->}[dd] *{\xor}="xor" |
| 912 | "e" [d] :@{-->}`r [ru] `u "xor" "xor" |
| 913 | :@{-->}[dd] *+[F]{E}="e" :@{-->}[ddddd] |
| 914 | *+=(1, 0)+[F--]{\mathstrut y_i}="i" |
| 915 | "e" [l] {K} :@{-->}"e" |
| 916 | [rrruuuu] *+=(1, 0)+[F]{\mathstrut x_{n-2}}="x" |
| 917 | :[dd] *{\xor}="xor" |
| 918 | "e" [d] :@{-->}`r [ru] `u "xor" "xor" |
| 919 | :[dd] *+[F]{E}="e" |
| 920 | "e" [l] {K} :"e" |
| 921 | [rrruuuu] *+=(1, 0)+[F]{\mathstrut x_{n-1} \cat 0^{\ell-t}}="x" |
| 922 | :[dd] *{\xor}="xor" |
| 923 | "e" [d] :`r [ru] `u "xor" "xor" |
| 924 | "e" [dddddrrr] *+=(1, 0)+[F]{\mathstrut y_{n-1}[0 \bitsto t]}="i" |
| 925 | "e" [dd] ="x" |
| 926 | "i" [uu] ="y" |
| 927 | []!{"x"; "e" **{}, "x"+/4pt/ ="p", |
| 928 | "x"; "y" **{}, "x"+/4pt/ ="q", |
| 929 | "y"; "x" **{}, "y"+/4pt/ ="r", |
| 930 | "y"; "i" **{}, "y"+/4pt/ ="s", |
| 931 | "e"; |
| 932 | "p" **\dir{-}; |
| 933 | "q" **\crv{"x"}; |
| 934 | "r" **\dir{-}; |
| 935 | "s" **\crv{"y"}; |
| 936 | "i" **\dir{-}?>*\dir{>}} |
| 937 | "xor" :[dd] *+[F]{E}="e" |
| 938 | "e" [l] {K} :"e" |
| 939 | "e" [dddddlll] *+=(1, 0)+[F]{\mathstrut y_{n-2}}="i" |
| 940 | "e" [dd] ="x" |
| 941 | "i" [uu] ="y" |
| 942 | []!{"x"; "e" **{}, "x"+/4pt/ ="p", |
| 943 | "x"; "y" **{}, "x"+/4pt/ ="q", |
| 944 | "y"; "x" **{}, "y"+/4pt/ ="r", |
| 945 | "y"; "i" **{}, "y"+/4pt/ ="s", |
| 946 | "x"; "y" **{} ?="c" ?(0.5)/-4pt/ ="cx" ?(0.5)/4pt/ ="cy", |
| 947 | "e"; |
| 948 | "p" **\dir{-}; |
| 949 | "q" **\crv{"x"}; |
| 950 | "cx" **\dir{-}; |
| 951 | "c" *[@]\cir<4pt>{d^u}; |
| 952 | "cy"; |
| 953 | "r" **\dir{-}; |
| 954 | "s" **\crv{"y"}; |
| 955 | "i" **\dir{-}?>*\dir{>}} |
| 956 | \end{graph} \] |
| 957 | |
| 958 | \caption{Encryption using CBC mode with ciphertext stealing} |
| 959 | \label{fig:cbc-steal} |
| 960 | \end{figure} |
| 961 | |
| 962 | \begin{definition}[CBC stealing] |
| 963 | \label{def:cbc-steal} |
| 964 | Let $P\colon \keys P \times \Bin^\ell \to \Bin^\ell$ be a pseudorandom |
| 965 | permutation. We define the symmetric encryption scheme |
| 966 | $\Xid{\mathcal{E}}{CBC-steal}^P = (\Xid{G}{CBC}^P, \Xid{E}{CBC-steal}^P, |
| 967 | \Xid{D}{CBC-steal}^P)$ for messages in $\Bin^{\ell\Z}$ by setting $\keys |
| 968 | \Xid{\mathcal{E}}{CBC-steal} = \keys P$ and defining the encryption and |
| 969 | decryption algorithms as follows: |
| 970 | \begin{program} |
| 971 | Algorithm $\Xid{E}{CBC-steal}^P_K(x)$: \+ \\ |
| 972 | $I \getsr \Bin^\ell$; \\ |
| 973 | $y \gets I$; \\ |
| 974 | $t = |x| \bmod \ell$; \\ |
| 975 | \IF $t \ne 0$ \THEN $x \gets x \cat 0^{\ell-t}$; \\ |
| 976 | \FOR $i = 0$ \TO $|x|/\ell$ \DO \\ \ind |
| 977 | $x_i \gets x[\ell i \bitsto \ell (i + 1)]$; \\ |
| 978 | $y_i \gets P_K(x_i \xor I)$; \\ |
| 979 | $I \gets y_i$; \\ |
| 980 | $y \gets y \cat y_i$; \- \\ |
| 981 | \IF $t \ne 0$ \THEN \\ \ind |
| 982 | $b \gets |y| - 2\ell$; \\ |
| 983 | $y \gets $\=$y[0 \bitsto b] \cat |
| 984 | y[b + \ell \bitsto |y|] \cat {}$ \\ |
| 985 | \>$y[b \bitsto b + t]$; \- \\ |
| 986 | \RETURN $y$; |
| 987 | \next |
| 988 | Algorithm $\Xid{D}{CBC-steal}^P_K(y)$: \+ \\ |
| 989 | $I \gets y[0 \bitsto \ell]$; \\ |
| 990 | $t = |y| \bmod \ell$; \\ |
| 991 | \IF $t \ne 0$ \THEN \\ \ind |
| 992 | $b \gets |y| - t - \ell$; \\ |
| 993 | $z \gets P^{-1}_K(y[b \bitsto b + \ell])$; \\ |
| 994 | $y \gets $\=$y[0 \bitsto b] \cat |
| 995 | y[b + \ell \bitsto |y|] \cat {}$ \\ |
| 996 | \>$z[t \bitsto \ell]$; \- \\ |
| 997 | $x \gets \emptystring$; \\ |
| 998 | \FOR $1 = 0$ \TO $|y|/\ell$ \DO \\ \ind |
| 999 | $y_i \gets y[\ell i \bitsto \ell (i + 1)]$; \\ |
| 1000 | $x_i \gets P^{-1}_K(y_i) \xor I$; \\ |
| 1001 | $I \gets y_i$; \\ |
| 1002 | $x \gets x \cat x_i$; \- \\ |
| 1003 | \IF $t \ne 0$ \THEN \\ \ind |
| 1004 | $x \gets x \cat z[0 \bitsto t] \xor y[b \bitsto b + t]$; \- \\ |
| 1005 | \RETURN $x$; |
| 1006 | \end{program} |
| 1007 | \end{definition} |
| 1008 | |
| 1009 | \begin{theorem}[Security of CBC with ciphertext stealing] |
| 1010 | \label{thm:cbc-steal} |
| 1011 | Let $P\colon \keys P \times \Bin^\ell \to \Bin^\ell$ be a pseudorandom |
| 1012 | permutation. Then |
| 1013 | \begin{equation} |
| 1014 | \InSec{lor-cpa}(\Xid{\mathcal{E}}{CBC-steal}; t, q_E, \mu_E) \le |
| 1015 | 2 \cdot \InSec{prp}(P; t + q t_P, q) + |
| 1016 | \frac{q (q - 1)}{2^\ell - 2^{\ell/2}} |
| 1017 | \end{equation} |
| 1018 | where $q = \mu_E/\ell$ and $t_P$ is some small constant. |
| 1019 | \end{theorem} |
| 1020 | |
| 1021 | \begin{proof} |
| 1022 | This is an easy reducibility argument. Let $A$ be an adversary attacking |
| 1023 | $\Xid{\mathcal{E}}{CBC-steal}^P$. We construct an adversary which attacks |
| 1024 | $\Xid{\mathcal{E}}{CBC}^P$: |
| 1025 | \begin{program} |
| 1026 | Adversary $A'^{E(\cdot)}$: \+ \\ |
| 1027 | $b \gets A^{\Xid{E}{steal}(\cdot)}$; \\ |
| 1028 | \RETURN $b$; |
| 1029 | \- \\[\medskipamount] |
| 1030 | Oracle $\Xid{E}{steal}(x_0, x_1)$: \+ \\ |
| 1031 | \IF $|x_0| \ne |x_1|$ \THEN \ABORT; \\ |
| 1032 | \RETURN $\id{steal}(|x_0|, E(\id{pad}(x_0), \id{pad}(x_1)))$; |
| 1033 | \next |
| 1034 | Function $\id{pad}(x)$: \+ \\ |
| 1035 | $t \gets |x| \bmod \ell$; \\ |
| 1036 | \RETURN $x \cat 0^{\ell-t}$; |
| 1037 | \- \\[\medskipamount] |
| 1038 | Function $\id{steal}(l, y)$: \+ \\ |
| 1039 | $t \gets l \bmod \ell$; \\ |
| 1040 | \IF $t \ne 0$ \THEN \\ \ind |
| 1041 | $b \gets |y| - 2\ell$; \\ |
| 1042 | $y \gets $\=$y[0 \bitsto b] \cat |
| 1043 | y[b + \ell \bitsto |y|] \cat y[b \bitsto b + t]$; \- \\ |
| 1044 | \RETURN $y$; |
| 1045 | \end{program} |
| 1046 | Comparing this to definition~\ref{def:cbc-steal} shows that $A'$ simlates |
| 1047 | the LOR-CPA game for $\Xid{\mathcal{E}}{CBC-steal}$ perfectly. The theorem |
| 1048 | follows. |
| 1049 | \end{proof} |
| 1050 | |
| 1051 | \subsection{Proof of theorem~\ref{thm:cbc}} |
| 1052 | \label{sec:cbc-proof} |
| 1053 | |
| 1054 | Consider an adversary $A$ attacking CBC encryption using an ideal random |
| 1055 | permutation $P(\cdot)$. Pick some point in the attack game when we're just |
| 1056 | about to encrypt the $n$th plaintext block. For each $i \in \Nupto{n}$, |
| 1057 | let $x_i$ be the $i$th block of plaintext we've processed; let $y_i$ be the |
| 1058 | corresponding ciphertext; and let $z_i = P^{-1}(y_i)$, i.e., $z_i = x_i \xor |
| 1059 | I$ for the first block of a message, and $z_i = x_i \xor y_{i-1}$ for the |
| 1060 | subsequent blocks. |
| 1061 | |
| 1062 | Say that `something went wrong' if any $z_i = z_j$ for $i \ne j$. This is |
| 1063 | indeed a disaster, because it means that $y_i = y_j$ , so he can detect it, |
| 1064 | and $x_i \xor y_{i-1} = x_j \xor y_{j-1}$, so he can compute an XOR |
| 1065 | difference between two plaintext blocks from the ciphertext and thus |
| 1066 | (possibly) reveal whether he's getting his left or right plaintexts |
| 1067 | encrypted. The alternative, `everything is fine', is much better. If all |
| 1068 | the $z_i$ are distinct, then because $y_i = P(z_i)$, the $y_i$ are all |
| 1069 | generated by $P(\cdot)$ on inputs it's never seen before, so they're all |
| 1070 | random subject to the requirement that they be distinct. If everything is |
| 1071 | fine, then, the adversary has no better way of deciding whether he has a left |
| 1072 | oracle or a right oracle than tossing a coin, and his advantage is therefore |
| 1073 | zero. Thus, we must bound the probability that something went wrong. |
| 1074 | |
| 1075 | Assume that, at our point in the game so far, everything is fine. But we're |
| 1076 | just about to encrypt $x^* = x_n$. There are two cases: |
| 1077 | \begin{itemize} |
| 1078 | \item If $x_n$ is the first block in a new message, we've just invented a new |
| 1079 | random IV $I \in \Bin^\ell$ which is unknown to $A$, and $z_n = x_n \xor |
| 1080 | I$. Let $y^* = I$. |
| 1081 | \item If $x_n$ is \emph{not} the first block, then $z_n = x_n \xor y_{n-1}$, |
| 1082 | but the adversary doesn't yet know $y_{n-1}$, except that because $P$ is a |
| 1083 | permutation and all the $z_i$ are distinct, $y_{n-1} \ne y_i$ for any $0 |
| 1084 | \le i < n - 1$. Let $y^* = y_{n-1}$. |
| 1085 | \end{itemize} |
| 1086 | Either way, the adversary's choice of $x^*$ is independent of $y^*$. Let |
| 1087 | $z^* = x^* \xor y^*$. We want to know the probability that something goes |
| 1088 | wrong at this point, i.e., that $z^* = z_i$ for some $0 \le i < n$. Let's |
| 1089 | call this event $C_n$. Note first that, in the first case, there are |
| 1090 | $2^\ell$ possible values for $y^*$ and in the second there are $2^\ell - n + |
| 1091 | 1$ possibilities for $y^*$. Then |
| 1092 | \begin{eqnarray}[rl] |
| 1093 | \Pr[C_n] |
| 1094 | & = \sum_{x \in \Bin^\ell} \Pr[C_n \mid x^* = x] \Pr[x^* = x] \\ |
| 1095 | & = \sum_{x \in \Bin^\ell} |
| 1096 | \Pr[x^* = x] \sum_{0\le i<n} \Pr[y^* = z_i \xor x] \\ |
| 1097 | & \le \sum_{0\le i<n} \frac{1}{2^\ell - n} |
| 1098 | \sum_{x \in \Bin^\ell} \Pr[x^* = x] \\ |
| 1099 | & = \frac{n}{2^\ell - n} |
| 1100 | \end{eqnarray} |
| 1101 | |
| 1102 | Having bounded the probability that something went wrong for any particular |
| 1103 | block, we can proceed to bound the probability of something going wrong in |
| 1104 | the course of the entire game. Let's suppose that $q = \mu_E/\ell \le |
| 1105 | 2^{\ell/2}$; for if not, $q (q - 1) > 2^\ell$ and the theorem is trivially |
| 1106 | true, since no adversary can achieve advantage greater than one. |
| 1107 | |
| 1108 | Let's give the name $W_i$ to the probability that something went wrong after |
| 1109 | encrypting $i$ blocks. We therefore want to bound $W_q$ from above. |
| 1110 | Armed with the knowledge that $q \le 2^{\ell/2}$, we have |
| 1111 | \begin{eqnarray}[rl] |
| 1112 | W_q &\le \sum_{0\le i<q} \Pr[C_i] |
| 1113 | \le \sum_{0\le i<q} \frac{i}{2^\ell - i} \\ |
| 1114 | &\le \frac{1}{2^\ell - 2^{\ell/2}} \sum_{0\le i<q} i \\ |
| 1115 | &= \frac{q (q - 1)}{2 \cdot (2^\ell - 2^{\ell/2})} |
| 1116 | \end{eqnarray} |
| 1117 | Working through the definition of LOR-CPA security, we can see that $A$'s |
| 1118 | (and hence any adversary's) advantage against the ideal system is at most $2 |
| 1119 | W_q$. |
| 1120 | |
| 1121 | By using an adversary attacking CBC encryption as a statistical test in an |
| 1122 | attempt to distinguish $P_K(\cdot)$ from a pseudorandom permutation, we see |
| 1123 | that |
| 1124 | \begin{equation} |
| 1125 | \InSec{prp}(P; t + q t_P, q) \ge |
| 1126 | \frac{1}{2} \cdot |
| 1127 | \InSec{lor-cpa}(\Xid{\mathcal{E}}{CBC}; t, q_E, \mu_E) - |
| 1128 | \frac{q (q - 1)}{2 \cdot (2^\ell - 2^{\ell/2})} |
| 1129 | \end{equation} |
| 1130 | where $t_P$ expresses the overhead of doing the XORs and other care and |
| 1131 | feeding of the CBC adversary; whence |
| 1132 | \begin{equation} |
| 1133 | \InSec{lor-cpa}(\Xid{\mathcal{E}}{CBC}; t, q_E, \mu_E) \le |
| 1134 | 2 \cdot \InSec{prp}(P; t, q) + \frac{q (q - 1)}{2^\ell - 2^{\ell/2}} |
| 1135 | \end{equation} |
| 1136 | as required. |
| 1137 | \qed |
| 1138 | |
| 1139 | %%%----- That's all, folks -------------------------------------------------- |
| 1140 | |
| 1141 | \bibliography{mdw-crypto,cryptography,cryptography2000,rfc} |
| 1142 | \end{document} |
| 1143 | |
| 1144 | %%% Local Variables: |
| 1145 | %%% mode: latex |
| 1146 | %%% TeX-master: "wrestlers" |
| 1147 | %%% End: |