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74eb47db | 1 | %%% -*-latex-*- |
2 | %%% | |
74eb47db | 3 | %%% Description of the Wrestlers Protocol |
4 | %%% | |
5 | %%% (c) 2001 Mark Wooding | |
6 | %%% | |
7 | ||
c128b544 | 8 | \newif\iffancystyle\fancystyletrue |
d7d62ac0 | 9 | |
10 | \iffancystyle | |
11 | \documentclass | |
12 | [a4paper, article, 10pt, numbering, noherefloats, notitlepage] | |
13 | {strayman} | |
14 | \usepackage[palatino, helvetica, courier, maths=cmr]{mdwfonts} | |
c128b544 | 15 | \usepackage[mdwmargin]{mdwthm} |
16 | \PassOptionsToPackage{dvips}{xy} | |
d7d62ac0 | 17 | \else |
c128b544 | 18 | \documentclass{llncs} |
d7d62ac0 | 19 | \fi |
20 | ||
c128b544 | 21 | \usepackage{mdwtab, mathenv, mdwlist, mdwmath, crypto} |
d7d62ac0 | 22 | \usepackage{amssymb, amstext} |
c128b544 | 23 | \usepackage{tabularx} |
24 | \usepackage{url} | |
25 | \usepackage[all]{xy} | |
74eb47db | 26 | |
90d03a85 | 27 | \errorcontextlines=999 |
d7d62ac0 | 28 | \showboxbreadth=999 |
29 | \showboxdepth=999 | |
90d03a85 | 30 | \makeatletter |
74eb47db | 31 | |
90d03a85 | 32 | \title{The Wrestlers Protocol: proof-of-receipt and secure key exchange} |
33 | \author{Mark Wooding \and Clive Jones} | |
74eb47db | 34 | |
c128b544 | 35 | \bibliographystyle{mdwalpha} |
1a981bdb | 36 | |
37 | \newcolumntype{G}{p{0pt}} | |
c128b544 | 38 | \def\Nupto#1{\N_{<{#1}}} |
39 | \let\Bin\Sigma | |
40 | \let\epsilon\varepsilon | |
41 | \let\emptystring\lambda | |
42 | \def\bitsto{\mathbin{..}} | |
43 | \turnradius{4pt} | |
44 | \def\fixme{\marginpar{FIXME}} | |
45 | \def\messages{% | |
46 | \basedescript{% | |
47 | \desclabelwidth{2.5cm}% | |
48 | \desclabelstyle\pushlabel% | |
49 | \let\makelabel\cookie% | |
50 | }% | |
51 | } | |
52 | \let\endmessages\endbasedescript | |
1a981bdb | 53 | |
90d03a85 | 54 | \begin{document} |
74eb47db | 55 | |
90d03a85 | 56 | \maketitle |
57 | \begin{abstract} | |
c128b544 | 58 | The Wrestlers Protocol\footnote{% |
59 | `The Wrestlers' is a pub in Cambridge which serves good beer and | |
60 | excellent Thai food. It's where the authors made their first attempts at | |
61 | a secure key-exchange protocol which doesn't use signatures.} % | |
62 | is a key-exchange protocol with the interesting property that it leaves no | |
63 | evidence which could be used to convince a third party that any of the | |
64 | participants are involved. We describe the protocol and prove its security | |
65 | in the random oracle model. | |
66 | ||
67 | Almost incidentally, we provide a new security proof for the CBC encryption | |
68 | mode. Our proof is much simpler than that of \cite{Bellare:2000:CST}, and | |
69 | gives a slightly better security bound. | |
70 | ||
71 | % I've not yet decided whose key-exchange model to use, but this ought to | |
72 | % be mentioned. | |
90d03a85 | 73 | \end{abstract} |
74 | \tableofcontents | |
75 | \newpage | |
74eb47db | 76 | |
90d03a85 | 77 | %%%-------------------------------------------------------------------------- |
74eb47db | 78 | |
90d03a85 | 79 | \section{Introduction} |
c128b544 | 80 | \label{sec:intro} |
90d03a85 | 81 | % Some waffle here about the desirability of a key-exchange protocol that |
82 | % doesn't leave signatures lying around, followed by an extended report of | |
83 | % the various results. | |
74eb47db | 84 | |
90d03a85 | 85 | %%%-------------------------------------------------------------------------- |
74eb47db | 86 | |
c128b544 | 87 | \section{Preliminaries} |
88 | \label{sec:prelim} | |
89 | % Here we provide definitions of the various kinds of things we use and make, | |
90 | % and describe some of the notation we use. | |
1a981bdb | 91 | |
c128b544 | 92 | \subsection{Bit strings} |
93 | ||
94 | Most of our notation for bit strings is standard. The main thing to note is | |
95 | that everything is zero-indexed. | |
96 | ||
97 | \begin{itemize} | |
98 | \item We write $\Bin = \{0, 1\}$ for the set of binary digits. Then $\Bin^n$ | |
99 | is the set of $n$-bit strings, and $\Bin^*$ is the set of all bit strings. | |
100 | \item If $x$ is a bit string then $|x|$ is the length of $x$. If $x \in | |
101 | \Bin^n$ then $|x| = n$. | |
102 | \item If $x, y \in \Bin^n$ are strings of bits of the same length then $x | |
103 | \xor y \in \Bin^n$ is their bitwise XOR. | |
104 | \item If $x$ and $y$ are bit strings then $x \cat y$ is the result of | |
105 | concatenating $y$ to $x$. If $z = x \cat y$ then we have $|z| = |x| + | |
106 | |y|$. | |
107 | \item The empty string is denoted $\emptystring$. We have $|\emptystring| = | |
108 | 0$, and $x = x \cat \emptystring = \emptystring \cat x$ for all strings $x | |
109 | \in \Bin^*$. | |
110 | \item If $x$ is a bit string and $i$ is an integer satisfying $0 \le i < |x|$ | |
111 | then $x[i]$ is the $i$th bit of $x$. If $a$ and $b$ are integers | |
112 | satisfying $0 \le a \le b \le |x|$ then $x[a \bitsto b]$ is the substring | |
113 | of $x$ beginning with bit $a$ and ending just \emph{before} bit $b$. We | |
114 | have $|x[i]| = 1$ and $|x[a \bitsto b]| = b - a$; if $y = x[a \bitsto b]$ | |
115 | then $y[i] = x[a + i]$. | |
116 | \item If $x$ is a bit string and $n$ is a natural number then $x^n$ is the | |
117 | result of concatenating $x$ to itself $n$ times. We have $x^0 = | |
118 | \emptystring$ and if $n > 0$ then $x^n = x^{n-1} \cat x = x \cat x^{n-1}$. | |
119 | \end{itemize} | |
120 | ||
121 | \subsection{Other notation} | |
122 | ||
123 | \begin{itemize} | |
124 | \item If $n$ is any natural number, then $\Nupto{n}$ is the set $\{\, i \in | |
125 | \Z \mid 0 \le i < n \,\} = \{ 0, 1, \ldots, n \}$. | |
126 | \item The symbol $\bot$ (`bottom') is different from every bit string and | |
127 | group element. | |
128 | \item We write $\Func{l}{L}$ as the set of all functions from $\Bin^l$ to | |
129 | $\Bin^L$, and $\Perm{l}$ as the set of all permutations on $\Bin^l$. | |
130 | \end{itemize} | |
131 | ||
132 | \subsection{Algorithm descriptions} | |
133 | ||
134 | Most of the notation used in the algorithm descriptions should be obvious. | |
135 | We briefly note a few features which may be unfamiliar. | |
136 | \begin{itemize} | |
137 | \item The notation $a \gets x$ denotes the action of assigning the value $x$ | |
138 | to the variable $a$. | |
139 | \item The notation $a \getsr X$, where $X$ is a finite set, denotes the | |
140 | action of assigning to $a$ a random value $x \in X$ according to the | |
141 | uniform probability distribution on $X$; i.e., following $a \getsr X$, | |
142 | $\Pr[a = x] = 1/|X|$ for any $x \in X$. | |
143 | \end{itemize} | |
144 | The notation is generally quite sloppy about types and scopes. In | |
145 | particular, there are implicit coercions between bit strings, integers and | |
146 | group elements. Any simple injective mapping will do for handling the | |
147 | conversions. We don't think these informalities cause much confusion, and | |
148 | they greatly simplify the presentation of the algorithms. | |
149 | ||
150 | \subsection{Random oracles} | |
151 | ||
152 | We shall analyse the Wrestlers Protocol in the random oracle model | |
153 | \cite{Bellare:1993:ROP}. That is, each participant including the adversary | |
154 | is given oracle access (only) to a uniformly-distributed random function | |
155 | $H\colon \Bin^* \to \Bin^\infty$ chosen at the beginning of the game: for any | |
156 | input string $x$, the oracle can produce, on demand, any prefix of an | |
157 | infinitely long random answer $y = H(x)$. Repeating a query yields a prefix | |
158 | of the same random result string; asking a new query yields a prefix of a new | |
159 | randomly-chosen string. | |
160 | ||
161 | We shan't need either to query the oracle on very long input strings nor | |
162 | shall we need outputs much longer than a representation of a group index. | |
163 | Indeed, since all the programs we shall be dealing with run in finite time, | |
164 | and can therefore make only a finite number of oracle queries, each with a | |
165 | finitely long result, we can safely think about the random oracle as a finite | |
166 | object. | |
167 | ||
168 | Finally, we shall treat the oracle as a function of multiple inputs and | |
169 | expect it to operate on some unambiguous encoding of all of the arguments in | |
170 | order. | |
171 | ||
172 | \subsection{Symmetric encryption} | |
173 | ||
174 | \begin{definition}[Symmetric encryption] | |
175 | \label{def:sym-enc} | |
176 | A \emph{symmetric encryption scheme} $\mathcal{E} = (E, D)$ is a pair of | |
177 | algorithms: | |
178 | \begin{itemize} | |
179 | \item a randomized \emph{encryption algorithm} $E\colon \keys \mathcal{E} | |
180 | \times \Bin^* \to \Bin^*$; and | |
181 | \item a deterministic \emph{decryption algorithm} $E\colon \keys | |
182 | \mathcal{E} \times \Bin^* \to \Bin^* \cup \{ \bot \}$ | |
183 | \end{itemize} | |
184 | with the property that, for any $K \in \keys \mathcal{E}$, any plaintext | |
185 | message $x$, and any ciphertext $y$ returned as a result of $E_K(x)$, we | |
186 | have $D_K(y) = x$. | |
e04c2d50 | 187 | \end{definition} |
c128b544 | 188 | |
189 | \begin{definition}[Chosen plaintext security for symmetric encryption] | |
190 | \label{def:sym-cpa} | |
191 | Let $\mathcal{E} = (E, D)$ be a symmetric encryption scheme. Let $A$ be | |
192 | any algorithm. Define | |
193 | \begin{program} | |
194 | Experiment $\Expt{lor-cpa-$b$}{\mathcal{E}}(A)$: \+ \\ | |
195 | $K \getsr \keys \mathcal{E}$; \\ | |
196 | $b' \getsr A^{E_K(\id{lr}(b, \cdot, \cdot))}$; \\ | |
197 | \RETURN $b'$; | |
198 | \next | |
199 | Function $\id{lr}(b, x_0, x_1)$: \+ \\ | |
200 | \RETURN $x_b$; | |
201 | \end{program} | |
202 | An adversary $A$ is forbidden from querying its encryption oracle | |
203 | $E_K(\id{lr}(b, \cdot, \cdot))$ on a pair of strings with differing | |
204 | lengths. We define the adversary's \emph{advantage} in this game by | |
205 | \begin{equation} | |
206 | \Adv{lor-cpa}{\mathcal{E}}(A) = | |
207 | \Pr[\Expt{lor-cpa-$1$}{\mathcal{E}}(A) = 1] - | |
208 | \Pr[\Expt{lor-cpa-$0$}{\mathcal{E}}(A) = 1] | |
209 | \end{equation} | |
210 | and the \emph{left-or-right insecurity of $\mathcal{E}$ under | |
211 | chosen-plaintext attack} is given by | |
212 | \begin{equation} | |
213 | \InSec{lor-cpa}(\mathcal{E}; t, q_E, \mu_E) = | |
214 | \max_A \Adv{lor-cpa}{\mathcal{E}}(A) | |
215 | \end{equation} | |
216 | where the maximum is taken over all adversaries $A$ running in time $t$ and | |
217 | making at most $q_E$ encryption queries, totalling most $\mu_E$ bits of | |
218 | plaintext. | |
219 | \end{definition} | |
220 | ||
221 | \subsection{The decision Diffie-Hellman problem} | |
222 | ||
223 | Let $G$ be some cyclic group. The standard \emph{Diffie-Hellman problem} | |
224 | \cite{Diffie:1976:NDC} is to compute $g^{\alpha\beta}$ given $g^\alpha$ and | |
225 | $g^\beta$. We need a slightly stronger assumption: that, given $g^\alpha$ | |
226 | and $g^\beta$, it's hard to tell the difference between the correct | |
227 | Diffie-Hellman value $g^{\alpha\beta}$ and a randomly-chosen group element | |
228 | $g^\gamma$. This is the \emph{decision Diffie-Hellman problem} | |
229 | \cite{Boneh:1998:DDP}. | |
230 | ||
231 | \begin{definition} | |
232 | \label{def:ddh} | |
233 | Let $G$ be a cyclic group of order $q$, and let $g$ be a generator of $G$. | |
234 | Let $A$ be any algorithm. Then $A$'s \emph{advantage in solving the | |
235 | decision Diffie-Hellman problem in $G$} is | |
236 | \begin{equation} | |
237 | \begin{eqnalign}[rl] | |
238 | \Adv{ddh}{G}(A) = | |
239 | & \Pr[\alpha \getsr \Nupto{q}; \beta \getsr \Nupto{q} : | |
e04c2d50 | 240 | A(g^\alpha, g^\beta, g^{\alpha\beta}) = 1] - {} \\ |
c128b544 | 241 | & \Pr[\alpha \getsr \Nupto{q}; \beta \getsr \Nupto{q}; |
e04c2d50 MW |
242 | \gamma \getsr \Nupto{q} : |
243 | A(g^\alpha, g^\beta, g^\gamma) = 1]. | |
c128b544 | 244 | \end{eqnalign} |
245 | \end{equation} | |
246 | The \emph{insecurity function of the decision Diffie-Hellman problem in | |
247 | $G$} is | |
248 | \begin{equation} | |
249 | \InSec{ddh}(G; t) = \max_A \Adv{ddh}{G}(A) | |
250 | \end{equation} | |
251 | where the maximum is taken over all algorithms $A$ which run in time $t$. | |
252 | \end{definition} | |
253 | ||
254 | %%%-------------------------------------------------------------------------- | |
255 | ||
256 | \section{The protocol} | |
257 | \label{sec:protocol} | |
258 | ||
259 | The Wrestlers Protocol is parameterized. We need the following things: | |
260 | \begin{itemize} | |
261 | \item A cyclic group $G$ whose order~$q$ is prime. Let $g$ be a generator | |
262 | of~$G$. We require that the (decision?\fixme) Diffie-Hellman problem be | |
263 | hard in~$G$. The group operation is written multiplicatively. | |
264 | \item A symmetric encryption scheme $\mathcal{E} = (E, D)$. We require that | |
265 | $\mathcal{E}$ be secure against adaptive chosen-plaintext attacks. Our | |
266 | implementation uses Blowfish \cite{Schneier:1994:BEA} in CBC mode with | |
267 | ciphertext stealing. See section~\ref{sec:cbc} for a description of | |
268 | ciphertext stealing and an analysis of its security. | |
269 | \item A message authentication scheme $\mathcal{M} = (T, V)$. We require | |
270 | that $\mathcal{M}$ be (strongly) existentially unforgeable under | |
271 | chosen-message attacks. Our implementation uses RIPEMD-160 | |
272 | \cite{Dobbertin:1996:RSV} in the HMAC \cite{Bellare:1996:HC} construction. | |
273 | \item An instantiation for the random oracle. We use RIPEMD-160 again, | |
274 | either on its own, if the output is long enough, or in the MGF-1 | |
275 | \cite{RFC2437} construction, if we need a larger output.\footnote{% | |
276 | The use of the same hash function in the MAC as for instantiating the | |
277 | random oracle is deliberate, with the aim of reducing the number of | |
278 | primitives whose security we must assume. In an application of HMAC, the | |
279 | message to be hashed is prefixed by a secret key padded out to the hash | |
280 | function's block size. In a `random oracle' query, the message is | |
281 | prefixed by a fixed identification string and not padded. Interference | |
282 | between the two is then limited to the case where one of the HMAC keys | |
283 | matches a random oracle prefix, which happens only with very tiny | |
284 | probability.}% | |
285 | \end{itemize} | |
286 | ||
287 | An authenticated encryption scheme with associated data (AEAD) | |
288 | \cite{Rogaway:2002:AEAD, Rogaway:2001:OCB, Kohno:2003:CWC} could be used | |
289 | instead of a separate symmetric encryption scheme and MAC. | |
290 | ||
291 | \subsection{Symmetric encryption} | |
292 | ||
293 | The same symmetric encryption subprotocol is used both within the key | |
294 | exchange, to ensure secrecy and binding, and afterwards for message | |
295 | transfer. It provides a secure channel between two players, assuming that | |
296 | the key was chosen properly. | |
297 | ||
298 | A \id{keyset} contains the state required for communication between the two | |
299 | players. In particular it maintains: | |
300 | \begin{itemize} | |
301 | \item separate encryption and MAC keys in each direction (four keys in | |
302 | total), chosen using the random oracle based on an input key assumed to be | |
303 | unpredictable by the adversary and a pair of nonces chosen by the two | |
304 | players; and | |
305 | \item incoming and outgoing sequence numbers, to detect and prevent replay | |
306 | attacks. | |
307 | \end{itemize} | |
308 | ||
309 | The operations involved in the symmetric encryption protocol are shown in | |
310 | figure~\ref{fig:keyset}. | |
311 | ||
312 | The \id{keygen} procedure initializes a \id{keyset}, resetting the sequence | |
313 | numbers, and selecting keys for the encryption scheme and MAC using the | |
314 | random oracle. It uses the nonces $r_A$ and $r_B$ to ensure that with high | |
315 | probability the keys are different for the two directions: assuming that | |
316 | Alice chose her nonce $r_A$ at random, and that the keys and nonce are | |
317 | $\kappa$~bits long, the probability that the keys in the two directions are | |
318 | the same is at most $2^{\kappa - 2}$. | |
319 | ||
320 | The \id{encrypt} procedure constructs a ciphertext from a message $m$ and a | |
321 | \emph{message type} $\id{ty}$. It encrypts the message giving a ciphertext | |
322 | $y$, and computes a MAC tag $\tau$ for the triple $(\id{ty}, i, y)$, where | |
323 | $i$ is the next available outgoing sequence number. The ciphertext message | |
324 | to send is then $(i, y, \tau)$. The message type codes are used to | |
325 | separate ciphertexts used by the key-exchange protocol itself from those sent | |
326 | by the players later. | |
327 | ||
328 | The \id{decrypt} procedure recovers the plaintext from a ciphertext triple | |
329 | $(i, y, \tau)$, given its expected type code $\id{ty}$. It verifies that the | |
330 | tag $\tau$ is valid for the message $(\id{ty}, i, y)$, checks that the | |
331 | sequence number $i$ hasn't been seen before,\footnote{% | |
332 | The sequence number checking shown in the figure is simple but obviously | |
333 | secure. The actual implementation maintains a window of 32 previous | |
334 | sequence numbers, to allow out-of-order reception of messages while still | |
335 | preventing replay attacks. This doesn't affect our analysis.}% | |
336 | and then decrypts the ciphertext $y$. | |
337 | ||
338 | \begin{figure} | |
339 | \begin{program} | |
340 | Structure $\id{keyset}$: \+ \\ | |
341 | $\Xid{K}{enc-in}$; $\Xid{K}{enc-out}$; \\ | |
342 | $\Xid{K}{mac-in}$; $\Xid{K}{mac-out}$; \\ | |
343 | $\id{seq-in}$; $\id{seq-out}$; \- \\[\medskipamount] | |
344 | Function $\id{gen-keys}(r_A, r_B, K)$: \+ \\ | |
345 | $k \gets \NEW \id{keyset}$; \\ | |
346 | $k.\Xid{K}{enc-in} \gets H(\cookie{encryption}, r_A, r_B, K)$; \\ | |
347 | $k.\Xid{K}{enc-out} \gets H(\cookie{encryption}, r_B, r_A, K)$; \\ | |
348 | $k.\Xid{K}{mac-in} \gets H(\cookie{integrity}, r_A, r_B, K)$; \\ | |
349 | $k.\Xid{K}{mac-out} \gets H(\cookie{integrity}, r_B, r_A, K)$; \\ | |
350 | $k.\id{seq-in} \gets 0$; \\ | |
351 | $k.\id{seq-out} \gets 0$; \\ | |
352 | \RETURN $k$; | |
353 | \next | |
354 | Function $\id{encrypt}(k, \id{ty}, m)$: \+ \\ | |
355 | $y \gets (E_{k.\Xid{K}{enc-out}}(m))$; \\ | |
356 | $i \gets k.\id{seq-out}$; \\ | |
357 | $\tau \gets T_{k.\Xid{K}{mac-out}}(\id{ty}, i, y)$; \\ | |
358 | $k.\id{seq-out} \gets i + 1$; \\ | |
359 | \RETURN $(i, y, \tau)$; \- \\[\medskipamount] | |
360 | Function $\id{decrypt}(k, \id{ty}, c)$: \+ \\ | |
361 | $(i, y, \tau) \gets c$; \\ | |
362 | \IF $V_{k.\Xid{K}{mac-in}}((\id{ty}, i, y), \tau) = 0$ \THEN \\ \ind | |
e04c2d50 | 363 | \RETURN $\bot$; \- \\ |
c128b544 | 364 | \IF $i < k.\id{seq-in}$ \THEN \RETURN $\bot$; \\ |
365 | $m \gets D_{k.\Xid{K}{enc-in}}(y)$; \\ | |
366 | $k.\id{seq-in} \gets i + 1$; \\ | |
367 | \RETURN $m$; | |
368 | \end{program} | |
369 | ||
370 | \caption{Symmetric-key encryption functions} | |
371 | \label{fig:keyset} | |
372 | \end{figure} | |
373 | ||
374 | \subsection{The key-exchange} | |
375 | ||
376 | The key-exchange protocol is completely symmetrical. Either party may | |
377 | initiate, or both may attempt to converse at the same time. We shall | |
378 | describe the protocol from the point of view of Alice attempting to exchange | |
379 | a key with Bob. | |
380 | ||
381 | Alice's private key is a random index $\alpha \inr \Nupto{q}$. Her public | |
382 | key is $a = g^\alpha$. Bob's public key is $b \in G$. We'll subscript the | |
383 | variables Alice computes with an~$A$, and the values Bob has sent with a~$B$. | |
384 | Of course, if Bob is following the protocol correctly, he will have computed | |
385 | his $B$ values in a completely symmetrical way. | |
386 | ||
387 | There are six messages in the protocol, and we shall briefly discuss the | |
388 | purpose of each before embarking on the detailed descriptions. At the | |
389 | beginning of the protocol, Alice chooses a new random index $\rho_A$ and | |
390 | computes her \emph{challenge} $r_A = g^{\rho_A}$. Eventually, the shared | |
391 | secret key will be computed as $K = r_B^{\rho_A} = r_A^{\rho_B} = | |
392 | g^{\rho_A\rho_B}$, as for standard Diffie-Hellman key agreement. | |
393 | ||
394 | Throughout, we shall assume that messages are implicitly labelled with the | |
395 | sender's identity. If Alice is actually trying to talk to several other | |
396 | people she'll need to run multiple instances of the protocol, each with its | |
397 | own state, and she can use the sender label to decide which instance a | |
398 | message should be processed by. There's no need for the implicit labels to | |
399 | be attached securely. | |
400 | ||
401 | We'll summarize the messages and their part in the scheme of things before we | |
402 | start on the serious detail. For a summary of the names and symbols used in | |
403 | these descriptions, see table~\ref{tab:kx-names}. The actual message | |
404 | contents are summarized in table~\ref{tab:kx-messages}. A state-transition | |
405 | diagram of the protocol is shown in figure~\ref{fig:kx-states}. If reading | |
406 | pesudocode algorithms is your thing then you'll find message-processing | |
407 | procedures in figure~\ref{fig:kx-messages} with the necessary support procedures | |
408 | in figure~\ref{fig:kx-support}. | |
409 | ||
410 | \begin{table} | |
411 | \begin{tabularx}{\textwidth}{Mr X} | |
e04c2d50 MW |
412 | G & A cyclic group known by all participants \\ |
413 | q = |G| & The prime order of $G$ \\ | |
414 | g & A generator of $G$ \\ | |
415 | E_K(\cdot) & Encryption under key $K$, here used to denote | |
416 | application of the $\id{encrypt}(K, \cdot)$ | |
417 | operation \\ | |
c128b544 | 418 | \alpha \inr \Nupto{q} & Alice's private key \\ |
e04c2d50 | 419 | a = g^{\alpha} & Alice's public key \\ |
c128b544 | 420 | \rho_A \inr \Nupto{q} & Alice's secret Diffie-Hellman value \\ |
e04c2d50 | 421 | r_A = g^{\rho_A} & Alice's public \emph{challenge} \\ |
c128b544 | 422 | c_A = H(\cookie{cookie}, r_A) |
e04c2d50 | 423 | & Alice's \emph{cookie} \\ |
c128b544 | 424 | v_A = \rho_A \xor H(\cookie{expected-reply}, r_A, r_B, b^{\rho_A}) |
e04c2d50 | 425 | & Alice's challenge \emph{check value} \\ |
c128b544 | 426 | r_B^\alpha = a^{\rho_B} |
e04c2d50 | 427 | & Alice's reply \\ |
c128b544 | 428 | K = r_B^{\rho_A} = r_B^{\rho_A} = g^{\rho_A\rho_B} |
e04c2d50 | 429 | & Alice and Bob's shared secret key \\ |
c128b544 | 430 | w_A = H(\cookie{switch-request}, c_A, c_B) |
e04c2d50 | 431 | & Alice's \emph{switch request} value \\ |
c128b544 | 432 | u_A = H(\cookie{switch-confirm}, c_A, c_B) |
e04c2d50 | 433 | & Alice's \emph{switch confirm} value \\ |
c128b544 | 434 | \end{tabularx} |
435 | ||
436 | \caption{Names used during key-exchange} | |
437 | \label{tab:kx-names} | |
438 | \end{table} | |
439 | ||
440 | \begin{table} | |
441 | \begin{tabular}[C]{Ml} | |
442 | \cookie{kx-pre-challenge}, r_A \\ | |
443 | \cookie{kx-cookie}, r_A, c_B \\ | |
444 | \cookie{kx-challenge}, r_A, c_B, v_A \\ | |
445 | \cookie{kx-reply}, c_A, c_B, v_A, E_K(r_B^\alpha)) \\ | |
446 | \cookie{kx-switch}, c_A, c_B, E_K(r_B^\alpha, w_A)) \\ | |
447 | \cookie{kx-switch-ok}, E_K(u_A)) | |
448 | \end{tabular} | |
449 | ||
450 | \caption{Message contents, as sent by Alice} | |
451 | \label{tab:kx-messages} | |
452 | \end{table} | |
e04c2d50 | 453 | |
c128b544 | 454 | \begin{messages} |
455 | \item[kx-pre-challenge] Contains a plain statement of Alice's challenge. | |
456 | This is Alice's first message of a session. | |
457 | \item[kx-cookie] A bare acknowledgement of a received challenge: it restates | |
458 | Alice's challenge, and contains a hash of Bob's challenge. This is an | |
459 | engineering measure (rather than a cryptographic one) which prevents | |
460 | trivial denial-of-service attacks from working. | |
461 | \item[kx-challenge] A full challenge, with a `check value' which proves the | |
462 | challenge's honesty. Bob's correct reply to this challenge informs Alice | |
463 | that she's received his challenge correctly. | |
464 | \item[kx-reply] A reply. This contains a `check value', like the | |
465 | \cookie{kx-challenge} message above, and an encrypted reply which confirms | |
466 | to Bob Alice's successful receipt of his challenge and lets Bob know he | |
467 | received Alice's challenge correctly. | |
468 | \item[kx-switch] Acknowledges Alice's receipt of Bob's \cookie{kx-reply} | |
469 | message, including Alice's own reply to Bob's challenge. Tells Bob that | |
470 | she can start using the key they've agreed. | |
471 | \item[kx-switch-ok] Acknowlegement to Bob's \cookie{kx-switch} message. | |
472 | \end{messages} | |
473 | ||
474 | \begin{figure} | |
475 | \small | |
476 | \let\ns\normalsize | |
477 | \let\c\cookie | |
478 | \[ \begin{graph} | |
479 | []!{0; <4.5cm, 0cm>: <0cm, 1.5cm>::} | |
480 | *++[F:<4pt>]\txt{\ns Start \\ Choose $\rho_A$} ="start" | |
481 | :[dd] | |
482 | *++[F:<4pt>]\txt{ | |
483 | \ns State \c{challenge} \\ | |
484 | Send $(\c{pre-challenge}, r_A)$} | |
485 | ="chal" | |
486 | [] "chal" !{!L(0.5)} ="chal-cookie" | |
487 | :@(d, d)[l] | |
488 | *+\txt{Send $(\c{cookie}, r_A, c_B)$} | |
489 | ="cookie" | |
490 | |*+\txt{Receive \\ $(\c{pre-challenge}, r_B)$ \\ (no spare slot)} | |
491 | :@(u, u)"chal-cookie" | |
492 | "chal" :@/_0.8cm/ [ddddl] | |
493 | *+\txt{Send \\ $(\c{challenge}, $\\$ r_A, c_B, v_A)$} | |
494 | ="send-chal" | |
495 | |<>(0.67) *+\txt\small{ | |
496 | Receive \\ $(\c{pre-challenge}, r_B)$ \\ (spare slot)} | |
497 | "chal" :@/^0.8cm/ "send-chal" |<>(0.33) | |
498 | *+\txt{Receive \\ $(\c{cookie}, r_B, c_A)$} | |
499 | :[rr] | |
500 | *+\txt{Send \\ $(\c{reply}, c_A, c_B, $\\$ v_A, E_K(r_B^\alpha))$} | |
501 | ="send-reply" | |
502 | |*+\txt{Receive \\ $(\c{challenge}, $\\$ r_B, c_A, v_B)$} | |
503 | "chal" :"send-reply" | |
504 | |*+\txt{Receive \\ $(\c{challenge}, $\\$ r_B, c_A, v_B)$} | |
505 | "send-chal" :[ddd] | |
506 | *++[F:<4pt>]\txt{ | |
507 | \ns State \c{commit} \\ | |
508 | Send \\ $(\c{switch}, c_A, c_B, $\\$ E_K(r_B^\alpha, w_A))$} | |
509 | ="commit" | |
510 | |*+\txt{Receive \\ $(\c{reply}, c_B, c_A, $\\$ v_B, E_K(b^{\rho_A}))$} | |
511 | :[rr] | |
512 | *+\txt{Send \\ $(\c{switch-ok}, E_K(u_A))$} | |
513 | ="send-switch-ok" | |
514 | |*+\txt{Receive \\ $(\c{switch}, c_B, c_A, $\\$ E_K(b^{\rho_A}, w_B))$} | |
515 | "send-reply" :"commit" | |
516 | |*+\txt{Receive \\ $(\c{reply}, c_B, c_A, $\\$ v_B, E_K(b^{\rho_A}))$} | |
517 | "send-reply" :"send-switch-ok" | |
518 | |*+\txt{Receive \\ $(\c{switch}, c_B, c_A, $\\$ E_K(b^{\rho_A}, w_B))$} | |
519 | :[dddl] | |
520 | *++[F:<4pt>]\txt{\ns Done} | |
521 | ="done" | |
522 | "commit" :"done" | |
523 | |*+\txt{Receive \\ $(\c{switch-ok}, E_K(u_B))$} | |
524 | "send-chal" [r] !{+<0cm, 0.75cm>} | |
525 | *\txt\itshape{For each outstanding challenge} | |
526 | ="for-each" | |
527 | !{"send-chal"+DL-<8pt, 8pt> ="p0", | |
528 | "for-each"+U+<8pt> ="p1", | |
529 | "send-reply"+UR+<8pt, 8pt> ="p2", | |
530 | "send-reply"+DR+<8pt, 8pt> ="p3", | |
531 | "p0" !{"p1"-"p0"} !{"p2"-"p1"} !{"p3"-"p2"} | |
532 | *\frm<8pt>{--}} | |
533 | \end{graph} \] | |
534 | ||
535 | \caption{State-transition diagram for key-exchange protocol} | |
536 | \label{fig:kx-states} | |
537 | \end{figure} | |
538 | ||
539 | We now describe the protocol message by message, and Alice's actions when she | |
540 | receives each. Since the protocol is completely symmetrical, Bob should do | |
541 | the same, only swapping round $A$ and $B$ subscripts, the public keys $a$ and | |
542 | $b$, and using his private key $\beta$ instead of $\alpha$. | |
543 | ||
544 | \subsubsection{Starting the protocol} | |
545 | ||
546 | As described above, at the beginning of the protocol Alice chooses a random | |
547 | $\rho_A \inr \Nupto q$, and computes her \emph{challenge} $r_A = g^{\rho_A}$ | |
548 | and her \emph{cookie} $c_A = H(\cookie{cookie}, r_A)$. She sends her | |
549 | announcement of her challenge as | |
550 | \begin{equation} | |
551 | \label{eq:kx-pre-challenge} | |
552 | \cookie{kx-pre-challenge}, r_A | |
553 | \end{equation} | |
554 | and enters the \cookie{challenge} state. | |
555 | ||
556 | \subsubsection{The \cookie{kx-pre-challenge} message} | |
557 | ||
558 | If Alice receieves a \cookie{kx-pre-challenge}, she ensures that she's in the | |
559 | \cookie{challenge} state: if not, she rejects the message. | |
560 | ||
561 | She must first calculate Bob's cookie $c_B = H(\cookie{cookie}, r_B)$. Then | |
562 | she has a choice: either she can send a full challenge, or she can send the | |
563 | cookie back. | |
564 | ||
565 | Suppose she decides to send a full challenge. She must compute a \emph{check | |
566 | value} | |
567 | \begin{equation} | |
568 | \label{eq:v_A} | |
569 | v_A = \rho_A \xor H(\cookie{expected-reply}, r_A, r_B, b^{\rho_A}) | |
570 | \end{equation} | |
571 | and sends | |
572 | \begin{equation} | |
573 | \label{eq:kx-challenge} | |
574 | \cookie{kx-challenge}, r_A, c_B, v_A | |
575 | \end{equation} | |
576 | to Bob. Then she remembers Bob's challenge for later use, and awaits his | |
577 | reply. | |
578 | ||
579 | If she decides to send only a cookie, she just transmits | |
580 | \begin{equation} | |
581 | \label{eq:kx-cookie} | |
582 | \cookie{kx-cookie}, r_A, c_B | |
583 | \end{equation} | |
584 | to Bob and forgets all about it. | |
585 | ||
586 | Why's this useful? Well, if Alice sends off a full \cookie{kx-challenge} | |
587 | message, she must remember Bob's $r_B$ so she can check his reply and that | |
588 | involves using up a table slot. That means that someone can send Alice | |
589 | messages purporting to come from Bob which will chew up Alice's memory, and | |
590 | they don't even need to be able to read Alice's messages to Bob to do that. | |
591 | If this protocol were used over the open Internet, script kiddies from all | |
592 | over the world might be flooding Alice with bogus \cookie{kx-pre-challenge} | |
593 | messages and she'd never get around to talking to Bob. | |
594 | ||
595 | By sending a cookie intead, she avoids committing a table slot until Bob (or | |
596 | someone) sends either a cookie or a full challenge, thus proving, at least, | |
597 | that he can read her messages. This is the best we can do at this stage in | |
598 | the protocol. Against an adversary as powerful as the one we present in | |
599 | section~\fixme\ref{sec:formal} this measure provides no benefit (but we have | |
600 | to analyse it anyway); but it raises the bar too sufficiently high to | |
601 | eliminate a large class of `nuisance' attacks in the real world. | |
602 | ||
603 | Our definition of the Wrestlers Protocol doesn't stipulate when Alice should | |
604 | send a full challenge or just a cookie: we leave this up to individual | |
605 | implementations, because it makes no difference to the security of the | |
606 | protocol against powerful adversaries. But we recommend that Alice proceed | |
607 | `optimistically' at first, sending full challenges until her challenge table | |
608 | looks like it's running out, and then generating cookies only if it actually | |
609 | looks like she's under attack. This is what our pseudocode in | |
610 | figure~\ref{fig:kx-messages} does. | |
1a981bdb | 611 | |
c128b544 | 612 | \subsubsection{The \cookie{kx-cookie} message} |
1a981bdb | 613 | |
c128b544 | 614 | When Alice receives a \cookie{kx-cookie} message, she must ensure that she's |
615 | in the \cookie{challenge} state: if not, she rejects the message. She checks | |
616 | the cookie in the message against the value of $c_A$ she computed earlier. | |
617 | If all is well, Alice sends a \cookie{kx-challenge} message, as in | |
618 | equation~\ref{eq:kx-challenge} above. | |
1a981bdb | 619 | |
c128b544 | 620 | This time, she doesn't have a choice about using up a table slot to remember |
621 | Bob's $r_B$. If her table size is fixed, she must choose a slot to recycle. | |
622 | We suggest simply recycling slots at random: this means there's no clever | |
623 | pattern of \cookie{kx-cookie} messages an attacker might be able to send to | |
624 | clog up all of Alice's slots. | |
1a981bdb | 625 | |
c128b544 | 626 | \subsubsection{The \cookie{kx-challenge} message} |
1a981bdb | 627 | |
c128b544 | 628 | |
629 | ||
630 | \begin{figure} | |
631 | \begin{program} | |
632 | Procedure $\id{kx-initialize}$: \+ \\ | |
633 | $\rho_A \getsr [q]$; \\ | |
634 | $r_a \gets g^{\rho_A}$; \\ | |
635 | $\id{state} \gets \cookie{challenge}$; \\ | |
636 | $\Xid{n}{chal} \gets 0$; \\ | |
637 | $k \gets \bot$; \\ | |
638 | $\id{chal-commit} \gets \bot$; \\ | |
639 | $\id{send}(\cookie{kx-pre-challenge}, r_A)$; \- \\[\medskipamount] | |
640 | Procedure $\id{kx-receive}(\id{type}, \id{data})$: \\ \ind | |
641 | \IF $\id{type} = \cookie{kx-pre-challenge}$ \THEN \\ \ind | |
e04c2d50 | 642 | \id{msg-pre-challenge}(\id{data}); \- \\ |
c128b544 | 643 | \ELSE \IF $\id{type} = \cookie{kx-cookie}$ \THEN \\ \ind |
e04c2d50 | 644 | \id{msg-cookie}(\id{data}); \- \\ |
c128b544 | 645 | \ELSE \IF $\id{type} = \cookie{kx-challenge}$ \THEN \\ \ind |
e04c2d50 | 646 | \id{msg-challenge}(\id{data}); \- \\ |
c128b544 | 647 | \ELSE \IF $\id{type} = \cookie{kx-reply}$ \THEN \\ \ind |
e04c2d50 | 648 | \id{msg-reply}(\id{data}); \- \\ |
c128b544 | 649 | \ELSE \IF $\id{type} = \cookie{kx-switch}$ \THEN \\ \ind |
e04c2d50 | 650 | \id{msg-switch}(\id{data}); \- \\ |
c128b544 | 651 | \ELSE \IF $\id{type} = \cookie{kx-switch-ok}$ \THEN \\ \ind |
e04c2d50 | 652 | \id{msg-switch-ok}(\id{data}); \-\- \\[\medskipamount] |
c128b544 | 653 | Procedure $\id{msg-pre-challenge}(\id{data})$: \+ \\ |
654 | \IF $\id{state} \ne \cookie{challenge}$ \THEN \RETURN; \\ | |
655 | $r \gets \id{data}$; \\ | |
656 | \IF $\Xid{n}{chal} \ge \Xid{n}{chal-thresh}$ \THEN \\ \ind | |
e04c2d50 | 657 | $\id{send}(\cookie{kx-cookie}, r_A, \id{cookie}(r_A)))$; \- \\ |
c128b544 | 658 | \ELSE \+ \\ |
e04c2d50 MW |
659 | $\id{new-chal}(r)$; \\ |
660 | $\id{send}(\cookie{kx-challenge}, r_A, | |
661 | \id{cookie}(r), \id{checkval}(r))$; \-\-\\[\medskipamount] | |
c128b544 | 662 | Procedure $\id{msg-cookie}(\id{data})$: \+ \\ |
663 | \IF $\id{state} \ne \cookie{challenge}$ \THEN \RETURN; \\ | |
664 | $(r, c_A) \gets \id{data}$; \\ | |
665 | \IF $c_A \ne \id{cookie}(r_A)$ \THEN \RETURN; \\ | |
666 | $\id{new-chal}(r)$; \\ | |
667 | $\id{send}(\cookie{kx-challenge}, r_A, | |
e04c2d50 | 668 | \id{cookie}(r), \id{checkval}(r))$; \- \\[\medskipamount] |
c128b544 | 669 | Procedure $\id{msg-challenge}(\id{data})$: \+ \\ |
670 | \IF $\id{state} \ne \cookie{challenge}$ \THEN \RETURN; \\ | |
671 | $(r, c_A, v) \gets \id{data}$; \\ | |
672 | \IF $c_A \ne \id{cookie}(r_A)$ \THEN \RETURN; \\ | |
673 | $i \gets \id{check-reply}(\bot, r, v)$; \\ | |
674 | \IF $i = \bot$ \THEN \RETURN; \\ | |
675 | $k \gets \id{chal-tab}[i].k$; \\ | |
676 | $y \gets \id{encrypt}(k, \cookie{kx-reply}, r^\alpha)$; \\ | |
677 | $\id{send}(\cookie{kx-reply}, c_A, \id{cookie}(r), | |
e04c2d50 | 678 | \id{checkval}(r), y)$ |
c128b544 | 679 | \next |
680 | Procedure $\id{msg-reply}(\id{data})$: \+ \\ | |
681 | $(c, c_A, v, y) \gets \id{data}$; \\ | |
682 | \IF $c_A \ne \id{cookie}(r_A)$ \THEN \RETURN; \\ | |
683 | $i \gets \id{find-chal}(c)$; \\ | |
684 | \IF $i = \bot$ \THEN \RETURN; \\ | |
685 | \IF $\id{check-reply}(i, \id{chal-tab}[i].r, v) = \bot$ \THEN \\ \ind | |
e04c2d50 | 686 | \RETURN; \- \\ |
c128b544 | 687 | $k \gets \id{chal-tab}[i].k$; \\ |
688 | $x \gets \id{decrypt}(k, \cookie{kx-reply}, y)$; \\ | |
689 | \IF $x = \bot$ \THEN \RETURN; \\ | |
690 | \IF $x \ne b^{\rho_A}$ \THEN \RETURN; \\ | |
691 | $\id{state} \gets \cookie{commit}$; \\ | |
692 | $\id{chal-commit} \gets \id{chal-tab}[i]$; \\ | |
693 | $w \gets H(\cookie{switch-request}, c_A, c)$; \\ | |
694 | $x \gets \id{chal-tab}[i].r^\alpha$; \\ | |
695 | $y \gets \id{encrypt}(k, (x, \cookie{kx-switch}, w))$; \\ | |
696 | $\id{send}(\cookie{kx-switch}, c_A, c, y)$; \-\\[\medskipamount] | |
697 | Procedure $\id{msg-switch}(\id{data})$: \+ \\ | |
698 | $(c, c_A, y) \gets \id{data}$; \\ | |
699 | \IF $c_A \ne \cookie(r_A)$ \THEN \RETURN; \\ | |
700 | $i \gets \id{find-chal}(c)$; \\ | |
701 | \IF $i = \bot$ \THEN \RETURN; \\ | |
702 | $k \gets \id{chal-tab}[i].k$; \\ | |
703 | $x \gets \id{decrypt}(k, \cookie{kx-switch}, y)$; \\ | |
704 | \IF $x = \bot$ \THEN \RETURN; \\ | |
705 | $(x, w) \gets x$; \\ | |
706 | \IF $\id{state} = \cookie{challenge}$ \THEN \\ \ind | |
e04c2d50 MW |
707 | \IF $x \ne b^{\rho_A}$ \THEN \RETURN; \\ |
708 | $\id{chal-commit} \gets \id{chal-tab}[i]$; \- \\ | |
c128b544 | 709 | \ELSE \IF $c \ne \id{chal-commit}.c$ \THEN \RETURN; \\ |
710 | \IF $w \ne H(\cookie{switch-request}, c, c_A)$ \THEN \RETURN; \\ | |
711 | $w \gets H(\cookie{switch-confirm}, c_A, c)$; \\ | |
712 | $y \gets \id{encrypt}(y, \cookie{kx-switch-ok}, w)$; \\ | |
713 | $\id{send}(\cookie{switch-ok}, y)$; \\ | |
714 | $\id{done}(k)$; \- \\[\medskipamount] | |
715 | Procedure $\id{msg-switch-ok}(\id{data})$ \+ \\ | |
716 | \IF $\id{state} \ne \cookie{commit}$ \THEN \RETURN; \\ | |
717 | $y \gets \id{data}$; \\ | |
718 | $k \gets \id{chal-commit}.k$; \\ | |
719 | $w \gets \id{decrypt}(k, \cookie{kx-switch-ok}, y)$; \\ | |
720 | \IF $w = \bot$ \THEN \RETURN; \\ | |
721 | $c \gets \id{chal-commit}.c$; \\ | |
722 | $c_A \gets \id{cookie}(r_A)$; \\ | |
723 | \IF $w \ne H(\cookie{switch-confirm}, c, c_A)$ \THEN \RETURN; \\ | |
724 | $\id{done}(k)$; | |
725 | \end{program} | |
726 | ||
727 | \caption{The key-exchange protocol: message handling} | |
728 | \label{fig:kx-messages} | |
729 | \end{figure} | |
730 | ||
731 | \begin{figure} | |
732 | \begin{program} | |
733 | Structure $\id{chal-slot}$: \+ \\ | |
734 | $r$; $c$; $\id{replied}$; $k$; \- \\[\medskipamount] | |
735 | Function $\id{find-chal}(c)$: \+ \\ | |
736 | \FOR $i = 0$ \TO $\Xid{n}{chal}$ \DO \\ \ind | |
e04c2d50 | 737 | \IF $\id{chal-tab}[i].c = c$ \THEN \RETURN $i$; \- \\ |
c128b544 | 738 | \RETURN $\bot$; \- \\[\medskipamount] |
739 | Function $\id{cookie}(r)$: \+ \\ | |
740 | \RETURN $H(\cookie{cookie}, r)$; \- \\[\medskipamount] | |
741 | Function $\id{check-reply}(i, r, v)$: \+ \\ | |
742 | \IF $i \ne \bot \land \id{chal-tab}[i].\id{replied} = 1$ \THEN \\ \ind | |
e04c2d50 | 743 | \RETURN $i$; \- \\ |
c128b544 | 744 | $\rho \gets v \xor H(\cookie{expected-reply}, r, r_A, r^\alpha)$; \\ |
745 | \IF $g^\rho \ne r$ \THEN \RETURN $\bot$; \\ | |
746 | \IF $i = \bot$ \THEN $i \gets \id{new-chal}(r)$; \\ | |
747 | $\id{chal-tab}[i].k \gets \id{gen-keys}(r_A, r, r^{\rho_A})$; \\ | |
748 | $\id{chal-tab}[i].\id{replied} \gets 1$; \\ | |
749 | \RETURN $i$; | |
750 | \next | |
751 | Function $\id{checkval}(r)$: \\ \ind | |
752 | \RETURN $\rho_A \xor H(\cookie{expected-reply}, | |
e04c2d50 | 753 | r_A,r, b^{\rho_A})$; \- \\[\medskipamount] |
c128b544 | 754 | Function $\id{new-chal}(r)$: \+ \\ |
755 | $c \gets \id{cookie}(r)$; \\ | |
756 | $i \gets \id{find-chal}(c)$; \\ | |
757 | \IF $i \ne \bot$ \THEN \RETURN $i$; \\ | |
758 | \IF $\Xid{n}{chal} < \Xid{n}{chal-max}$ \THEN \\ \ind | |
e04c2d50 MW |
759 | $i \gets \Xid{n}{chal}$; \\ |
760 | $\id{chal-tab}[i] \gets \NEW \id{chal-slot}$; \\ | |
761 | $\Xid{n}{chal} \gets \Xid{n}{chal} + 1$; \- \\ | |
c128b544 | 762 | \ELSE \\ \ind |
e04c2d50 | 763 | $i \getsr [\Xid{n}{chal-max}]$; \- \\ |
c128b544 | 764 | $\id{chal-tab}[i].r \gets r$; \\ |
765 | $\id{chal-tab}[i].c \gets c$; \\ | |
766 | $\id{chal-tab}[i].\id{replied} \gets 0$; \\ | |
767 | $\id{chal-tab}[i].k \gets \bot$; \\ | |
768 | \RETURN $i$; | |
769 | \end{program} | |
770 | ||
771 | \caption{The key-exchange protocol: support functions} | |
772 | \label{fig:kx-support} | |
773 | \end{figure} | |
1a981bdb | 774 | |
d7d62ac0 | 775 | %%%-------------------------------------------------------------------------- |
1a981bdb | 776 | |
c128b544 | 777 | \section{CBC mode encryption} |
778 | \label{sec:cbc} | |
779 | ||
780 | Our implementation of the Wrestlers Protocol uses Blowfish | |
781 | \cite{Schneier:1994:BEA} in CBC mode. However, rather than pad plaintext | |
782 | messages to a block boundary, with the ciphertext expansion that entails, we | |
783 | use a technique called \emph{ciphertext stealing} | |
784 | \cite[section 9.3]{Schneier:1996:ACP}. | |
785 | ||
786 | \subsection{Standard CBC mode} | |
787 | ||
788 | Suppose $E$ is an $\ell$-bit pseudorandom permutation. Normal CBC mode works | |
789 | as follows. Given a message $X$, we divide it into blocks $x_0, x_1, \ldots, | |
790 | x_{n-1}$. Choose a random \emph{initialization vector} $I \inr \Bin^\ell$. | |
791 | Before passing each $x_i$ through $E$, we XOR it with the previous | |
792 | ciphertext, with $I$ standing in for the first block: | |
793 | \begin{equation} | |
794 | y_0 = E_K(x_0 \xor I) \qquad | |
795 | y_i = E_K(x_i \xor y_{i-1} \ \text{(for $1 \le i < n$)}. | |
796 | \end{equation} | |
797 | The ciphertext is then the concatenation of $I$ and the $y_i$. Decryption is | |
798 | simple: | |
799 | \begin{equation} | |
800 | x_0 = E^{-1}_K(y_0) \xor I \qquad | |
801 | x_i = E^{-1}_K(y_i) \xor y_{i-1} \ \text{(for $1 \le i < n$)} | |
802 | \end{equation} | |
803 | See figure~\ref{fig:cbc} for a diagram of CBC encryption. | |
804 | ||
805 | \begin{figure} | |
806 | \[ \begin{graph} | |
807 | []!{0; <0.85cm, 0cm>: <0cm, 0.5cm>::} | |
808 | *+=(1, 0)+[F]{\mathstrut x_0}="x" | |
809 | :[dd] *{\xor}="xor" | |
810 | [ll] *+=(1, 0)+[F]{I} :"xor" | |
811 | :[dd] *+[F]{E}="e" :[ddd] *+=(1, 0)+[F]{\mathstrut y_0}="i" | |
812 | "e" [l] {K} :"e" | |
813 | [rrruuuu] *+=(1, 0)+[F]{\mathstrut x_1}="x" | |
814 | :[dd] *{\xor}="xor" | |
815 | "e" [d] :`r [ru] `u "xor" "xor" | |
816 | :[dd] *+[F]{E}="e" :[ddd] | |
817 | *+=(1, 0)+[F]{\mathstrut y_1}="i" | |
818 | "e" [l] {K} :"e" | |
819 | [rrruuuu] *+=(1, 0)+[F--]{\mathstrut x_i}="x" | |
820 | :@{-->}[dd] *{\xor}="xor" | |
821 | "e" [d] :@{-->}`r [ru] `u "xor" "xor" | |
822 | :@{-->}[dd] *+[F]{E}="e" :@{-->}[ddd] | |
823 | *+=(1, 0)+[F--]{\mathstrut y_i}="i" | |
824 | "e" [l] {K} :@{-->}"e" | |
825 | [rrruuuu] *+=(1, 0)+[F]{\mathstrut x_{n-1}}="x" | |
826 | :[dd] *{\xor}="xor" | |
827 | "e" [d] :@{-->}`r [ru] `u "xor" "xor" | |
828 | :[dd] *+[F]{E}="e" :[ddd] | |
829 | *+=(1, 0)+[F]{\mathstrut y_{n-1}}="i" | |
830 | "e" [l] {K} :"e" | |
831 | \end{graph} \] | |
832 | ||
833 | \caption{Encryption using CBC mode} | |
834 | \label{fig:cbc} | |
835 | \end{figure} | |
e04c2d50 | 836 | |
c128b544 | 837 | \begin{definition}[CBC mode] |
838 | \label{def:cbc} | |
839 | Let $P\colon \keys P \times \Bin^\ell to \Bin^\ell$ be a pseudorandom | |
840 | permutation. We define the symmetric encryption scheme | |
841 | $\Xid{\mathcal{E}}{CBC}^P = (\Xid{E}{CBC}^P, \Xid{D}{CBC}^P)$ for messages | |
842 | in $\Bin^{\ell\Z}$ by setting $\keys \Xid{\mathcal{E}}{CBC} = \keys P$ and | |
843 | defining the encryption and decryption algorithms as follows: | |
844 | \begin{program} | |
845 | Algorithm $\Xid{E}{CBC}^P_K(x)$: \+ \\ | |
846 | $I \getsr \Bin^\ell$; \\ | |
847 | $y \gets I$; \\ | |
848 | \FOR $i = 0$ \TO $|x|/\ell$ \DO \\ \ind | |
e04c2d50 MW |
849 | $x_i \gets x[\ell i \bitsto \ell (i + 1)]$; \\ |
850 | $y_i \gets P_K(x_i \xor I)$; \\ | |
851 | $I \gets y_i$; \\ | |
852 | $y \gets y \cat y_i$; \- \\ | |
c128b544 | 853 | \RETURN $y$; |
854 | \next | |
855 | Algorithm $\Xid{D}{CBC}^P_K(y)$: \+ \\ | |
856 | $I \gets y[0 \bitsto \ell]$; \\ | |
857 | $x \gets \emptystring$; \\ | |
858 | \FOR $1 = 0$ \TO $|y|/\ell$ \DO \\ \ind | |
e04c2d50 MW |
859 | $y_i \gets y[\ell i \bitsto \ell (i + 1)]$; \\ |
860 | $x_i \gets P^{-1}_K(y_i) \xor I$; \\ | |
861 | $I \gets y_i$; \\ | |
862 | $x \gets x \cat x_i$; \- \\ | |
c128b544 | 863 | \RETURN $x$; |
864 | \end{program} | |
865 | \end{definition} | |
866 | ||
867 | \begin{theorem}[Security of standard CBC mode] | |
868 | \label{thm:cbc} | |
869 | Let $P\colon \keys P \times \Bin^\ell \to \Bin^\ell$ be a pseudorandom | |
870 | permutation. Then, | |
871 | \begin{equation} | |
872 | \InSec{lor-cpa}(\Xid{\mathcal{E}}{CBC}; t, q_E + \mu_E) \le | |
873 | 2 \cdot \InSec{prp}(P; t + q t_P, q) + | |
874 | \frac{q (q - 1)}{2^\ell - 2^{\ell/2}} | |
875 | \end{equation} | |
876 | where $q = \mu_E/\ell$ and $t_P$ is some small constant. | |
877 | \end{theorem} | |
878 | ||
879 | \begin{note} | |
880 | Our security bound is slightly better than that of \cite[theorem | |
881 | 17]{Bellare:2000:CST}. Their theorem statement contains a term $3 \cdot q | |
882 | (q - 1) 2^{-\ell-1}$. Our result lowers the factor from 3 to just over 2. | |
883 | Our proof is also much shorter and considerably more comprehensible. | |
884 | \end{note} | |
885 | ||
886 | The proof of this theorem is given in section~\ref{sec:cbc-proof} | |
887 | ||
888 | \subsection{Ciphertext stealing} | |
889 | ||
890 | Ciphertext stealing allows us to encrypt any message in $\Bin^*$ and make the | |
891 | ciphertext exactly $\ell$ bits longer than the plaintext. See | |
892 | figure~\ref{fig:cbc-steal} for a diagram. | |
893 | ||
894 | \begin{figure} | |
895 | \[ \begin{graph} | |
896 | []!{0; <0.85cm, 0cm>: <0cm, 0.5cm>::} | |
897 | *+=(1, 0)+[F]{\mathstrut x_0}="x" | |
898 | :[dd] *{\xor}="xor" | |
899 | [ll] *+=(1, 0)+[F]{I} :"xor" | |
900 | :[dd] *+[F]{E}="e" :[ddddd] *+=(1, 0)+[F]{\mathstrut y_0}="i" | |
901 | "e" [l] {K} :"e" | |
902 | [rrruuuu] *+=(1, 0)+[F]{\mathstrut x_1}="x" | |
903 | :[dd] *{\xor}="xor" | |
904 | "e" [d] :`r [ru] `u "xor" "xor" | |
905 | :[dd] *+[F]{E}="e" :[ddddd] | |
906 | *+=(1, 0)+[F]{\mathstrut y_1}="i" | |
907 | "e" [l] {K} :"e" | |
908 | [rrruuuu] *+=(1, 0)+[F--]{\mathstrut x_i}="x" | |
909 | :@{-->}[dd] *{\xor}="xor" | |
910 | "e" [d] :@{-->}`r [ru] `u "xor" "xor" | |
911 | :@{-->}[dd] *+[F]{E}="e" :@{-->}[ddddd] | |
912 | *+=(1, 0)+[F--]{\mathstrut y_i}="i" | |
913 | "e" [l] {K} :@{-->}"e" | |
914 | [rrruuuu] *+=(1, 0)+[F]{\mathstrut x_{n-2}}="x" | |
915 | :[dd] *{\xor}="xor" | |
916 | "e" [d] :@{-->}`r [ru] `u "xor" "xor" | |
917 | :[dd] *+[F]{E}="e" | |
918 | "e" [l] {K} :"e" | |
919 | [rrruuuu] *+=(1, 0)+[F]{\mathstrut x_{n-1} \cat 0^{\ell-t}}="x" | |
920 | :[dd] *{\xor}="xor" | |
921 | "e" [d] :`r [ru] `u "xor" "xor" | |
922 | "e" [dddddrrr] *+=(1, 0)+[F]{\mathstrut y_{n-1}[0 \bitsto t]}="i" | |
923 | "e" [dd] ="x" | |
924 | "i" [uu] ="y" | |
925 | []!{"x"; "e" **{}, "x"+/4pt/ ="p", | |
e04c2d50 MW |
926 | "x"; "y" **{}, "x"+/4pt/ ="q", |
927 | "y"; "x" **{}, "y"+/4pt/ ="r", | |
928 | "y"; "i" **{}, "y"+/4pt/ ="s", | |
929 | "e"; | |
930 | "p" **\dir{-}; | |
931 | "q" **\crv{"x"}; | |
932 | "r" **\dir{-}; | |
933 | "s" **\crv{"y"}; | |
934 | "i" **\dir{-}?>*\dir{>}} | |
c128b544 | 935 | "xor" :[dd] *+[F]{E}="e" |
936 | "e" [l] {K} :"e" | |
937 | "e" [dddddlll] *+=(1, 0)+[F]{\mathstrut y_{n-2}}="i" | |
938 | "e" [dd] ="x" | |
939 | "i" [uu] ="y" | |
940 | []!{"x"; "e" **{}, "x"+/4pt/ ="p", | |
e04c2d50 MW |
941 | "x"; "y" **{}, "x"+/4pt/ ="q", |
942 | "y"; "x" **{}, "y"+/4pt/ ="r", | |
943 | "y"; "i" **{}, "y"+/4pt/ ="s", | |
944 | "x"; "y" **{} ?="c" ?(0.5)/-4pt/ ="cx" ?(0.5)/4pt/ ="cy", | |
945 | "e"; | |
946 | "p" **\dir{-}; | |
947 | "q" **\crv{"x"}; | |
948 | "cx" **\dir{-}; | |
949 | "c" *[@]\cir<4pt>{d^u}; | |
950 | "cy"; | |
951 | "r" **\dir{-}; | |
952 | "s" **\crv{"y"}; | |
953 | "i" **\dir{-}?>*\dir{>}} | |
c128b544 | 954 | \end{graph} \] |
955 | ||
956 | \caption{Encryption using CBC mode with ciphertext stealing} | |
957 | \label{fig:cbc-steal} | |
e04c2d50 | 958 | \end{figure} |
c128b544 | 959 | |
960 | \begin{definition}[CBC stealing] | |
961 | \label{def:cbc-steal} | |
962 | Let $P\colon \keys P \times \Bin^\ell \to \Bin^\ell$ be a pseudorandom | |
963 | permutation. We define the symmetric encryption scheme | |
964 | $\Xid{\mathcal{E}}{CBC-steal}^P = (\Xid{G}{CBC}^P, \Xid{E}{CBC-steal}^P, | |
965 | \Xid{D}{CBC-steal}^P)$ for messages in $\Bin^{\ell\Z}$ by setting $\keys | |
966 | \Xid{\mathcal{E}}{CBC-steal} = \keys P$ and defining the encryption and | |
967 | decryption algorithms as follows: | |
968 | \begin{program} | |
969 | Algorithm $\Xid{E}{CBC-steal}^P_K(x)$: \+ \\ | |
970 | $I \getsr \Bin^\ell$; \\ | |
971 | $y \gets I$; \\ | |
972 | $t = |x| \bmod \ell$; \\ | |
973 | \IF $t \ne 0$ \THEN $x \gets x \cat 0^{\ell-t}$; \\ | |
974 | \FOR $i = 0$ \TO $|x|/\ell$ \DO \\ \ind | |
e04c2d50 MW |
975 | $x_i \gets x[\ell i \bitsto \ell (i + 1)]$; \\ |
976 | $y_i \gets P_K(x_i \xor I)$; \\ | |
977 | $I \gets y_i$; \\ | |
978 | $y \gets y \cat y_i$; \- \\ | |
c128b544 | 979 | \IF $t \ne 0$ \THEN \\ \ind |
e04c2d50 MW |
980 | $b \gets |y| - 2\ell$; \\ |
981 | $y \gets $\=$y[0 \bitsto b] \cat | |
982 | y[b + \ell \bitsto |y|] \cat {}$ \\ | |
983 | \>$y[b \bitsto b + t]$; \- \\ | |
c128b544 | 984 | \RETURN $y$; |
985 | \next | |
986 | Algorithm $\Xid{D}{CBC-steal}^P_K(y)$: \+ \\ | |
987 | $I \gets y[0 \bitsto \ell]$; \\ | |
988 | $t = |y| \bmod \ell$; \\ | |
989 | \IF $t \ne 0$ \THEN \\ \ind | |
e04c2d50 MW |
990 | $b \gets |y| - t - \ell$; \\ |
991 | $z \gets P^{-1}_K(y[b \bitsto b + \ell])$; \\ | |
992 | $y \gets $\=$y[0 \bitsto b] \cat | |
993 | y[b + \ell \bitsto |y|] \cat {}$ \\ | |
994 | \>$z[t \bitsto \ell]$; \- \\ | |
c128b544 | 995 | $x \gets \emptystring$; \\ |
996 | \FOR $1 = 0$ \TO $|y|/\ell$ \DO \\ \ind | |
e04c2d50 MW |
997 | $y_i \gets y[\ell i \bitsto \ell (i + 1)]$; \\ |
998 | $x_i \gets P^{-1}_K(y_i) \xor I$; \\ | |
999 | $I \gets y_i$; \\ | |
1000 | $x \gets x \cat x_i$; \- \\ | |
c128b544 | 1001 | \IF $t \ne 0$ \THEN \\ \ind |
e04c2d50 | 1002 | $x \gets x \cat z[0 \bitsto t] \xor y[b \bitsto b + t]$; \- \\ |
c128b544 | 1003 | \RETURN $x$; |
1004 | \end{program} | |
1005 | \end{definition} | |
1006 | ||
1007 | \begin{theorem}[Security of CBC with ciphertext stealing] | |
1008 | \label{thm:cbc-steal} | |
1009 | Let $P\colon \keys P \times \Bin^\ell \to \Bin^\ell$ be a pseudorandom | |
1010 | permutation. Then | |
1011 | \begin{equation} | |
1012 | \InSec{lor-cpa}(\Xid{\mathcal{E}}{CBC-steal}; t, q_E, \mu_E) \le | |
1013 | 2 \cdot \InSec{prp}(P; t + q t_P, q) + | |
1014 | \frac{q (q - 1)}{2^\ell - 2^{\ell/2}} | |
1015 | \end{equation} | |
1016 | where $q = \mu_E/\ell$ and $t_P$ is some small constant. | |
1017 | \end{theorem} | |
1018 | ||
1019 | \begin{proof} | |
1020 | This is an easy reducibility argument. Let $A$ be an adversary attacking | |
1021 | $\Xid{\mathcal{E}}{CBC-steal}^P$. We construct an adversary which attacks | |
1022 | $\Xid{\mathcal{E}}{CBC}^P$: | |
1023 | \begin{program} | |
1024 | Adversary $A'^{E(\cdot)}$: \+ \\ | |
1025 | $b \gets A^{\Xid{E}{steal}(\cdot)}$; \\ | |
1026 | \RETURN $b$; | |
1027 | \- \\[\medskipamount] | |
1028 | Oracle $\Xid{E}{steal}(x_0, x_1)$: \+ \\ | |
1029 | \IF $|x_0| \ne |x_1|$ \THEN \ABORT; \\ | |
1030 | \RETURN $\id{steal}(|x_0|, E(\id{pad}(x_0), \id{pad}(x_1)))$; | |
1031 | \next | |
1032 | Function $\id{pad}(x)$: \+ \\ | |
1033 | $t \gets |x| \bmod \ell$; \\ | |
1034 | \RETURN $x \cat 0^{\ell-t}$; | |
1035 | \- \\[\medskipamount] | |
1036 | Function $\id{steal}(l, y)$: \+ \\ | |
1037 | $t \gets l \bmod \ell$; \\ | |
1038 | \IF $t \ne 0$ \THEN \\ \ind | |
e04c2d50 MW |
1039 | $b \gets |y| - 2\ell$; \\ |
1040 | $y \gets $\=$y[0 \bitsto b] \cat | |
1041 | y[b + \ell \bitsto |y|] \cat y[b \bitsto b + t]$; \- \\ | |
c128b544 | 1042 | \RETURN $y$; |
1043 | \end{program} | |
1044 | Comparing this to definition~\ref{def:cbc-steal} shows that $A'$ simlates | |
1045 | the LOR-CPA game for $\Xid{\mathcal{E}}{CBC-steal}$ perfectly. The theorem | |
1046 | follows. | |
1047 | \end{proof} | |
1a981bdb | 1048 | |
c128b544 | 1049 | \subsection{Proof of theorem~\ref{thm:cbc}} |
1050 | \label{sec:cbc-proof} | |
1a981bdb | 1051 | |
c128b544 | 1052 | Consider an adversary $A$ attacking CBC encryption using an ideal random |
1053 | permutation $P(\cdot)$. Pick some point in the attack game when we're just | |
1054 | about to encrypt the $n$th plaintext block. For each $i \in \Nupto{n}$, | |
1055 | let $x_i$ be the $i$th block of plaintext we've processed; let $y_i$ be the | |
1056 | corresponding ciphertext; and let $z_i = P^{-1}(y_i)$, i.e., $z_i = x_i \xor | |
1057 | I$ for the first block of a message, and $z_i = x_i \xor y_{i-1}$ for the | |
1058 | subsequent blocks. | |
1a981bdb | 1059 | |
c128b544 | 1060 | Say that `something went wrong' if any $z_i = z_j$ for $i \ne j$. This is |
1061 | indeed a disaster, because it means that $y_i = y_j$ , so he can detect it, | |
1062 | and $x_i \xor y_{i-1} = x_j \xor y_{j-1}$, so he can compute an XOR | |
1063 | difference between two plaintext blocks from the ciphertext and thus | |
1064 | (possibly) reveal whether he's getting his left or right plaintexts | |
1065 | encrypted. The alternative, `everything is fine', is much better. If all | |
1066 | the $z_i$ are distinct, then because $y_i = P(z_i)$, the $y_i$ are all | |
1067 | generated by $P(\cdot)$ on inputs it's never seen before, so they're all | |
1068 | random subject to the requirement that they be distinct. If everything is | |
1069 | fine, then, the adversary has no better way of deciding whether he has a left | |
1070 | oracle or a right oracle than tossing a coin, and his advantage is therefore | |
1071 | zero. Thus, we must bound the probability that something went wrong. | |
874aed51 | 1072 | |
c128b544 | 1073 | Assume that, at our point in the game so far, everything is fine. But we're |
1074 | just about to encrypt $x^* = x_n$. There are two cases: | |
1075 | \begin{itemize} | |
1076 | \item If $x_n$ is the first block in a new message, we've just invented a new | |
1077 | random IV $I \in \Bin^\ell$ which is unknown to $A$, and $z_n = x_n \xor | |
1078 | I$. Let $y^* = I$. | |
1079 | \item If $x_n$ is \emph{not} the first block, then $z_n = x_n \xor y_{n-1}$, | |
1080 | but the adversary doesn't yet know $y_{n-1}$, except that because $P$ is a | |
1081 | permutation and all the $z_i$ are distinct, $y_{n-1} \ne y_i$ for any $0 | |
1082 | \le i < n - 1$. Let $y^* = y_{n-1}$. | |
1083 | \end{itemize} | |
1084 | Either way, the adversary's choice of $x^*$ is independent of $y^*$. Let | |
1085 | $z^* = x^* \xor y^*$. We want to know the probability that something goes | |
1086 | wrong at this point, i.e., that $z^* = z_i$ for some $0 \le i < n$. Let's | |
1087 | call this event $C_n$. Note first that, in the first case, there are | |
1088 | $2^\ell$ possible values for $y^*$ and in the second there are $2^\ell - n + | |
1089 | 1$ possibilities for $y^*$. Then | |
1090 | \begin{eqnarray}[rl] | |
1091 | \Pr[C_n] | |
1092 | & = \sum_{x \in \Bin^\ell} \Pr[C_n \mid x^* = x] \Pr[x^* = x] \\ | |
1093 | & = \sum_{x \in \Bin^\ell} | |
e04c2d50 | 1094 | \Pr[x^* = x] \sum_{0\le i<n} \Pr[y^* = z_i \xor x] \\ |
c128b544 | 1095 | & \le \sum_{0\le i<n} \frac{1}{2^\ell - n} |
e04c2d50 | 1096 | \sum_{x \in \Bin^\ell} \Pr[x^* = x] \\ |
c128b544 | 1097 | & = \frac{n}{2^\ell - n} |
1098 | \end{eqnarray} | |
874aed51 | 1099 | |
c128b544 | 1100 | Having bounded the probability that something went wrong for any particular |
1101 | block, we can proceed to bound the probability of something going wrong in | |
1102 | the course of the entire game. Let's suppose that $q = \mu_E/\ell \le | |
1103 | 2^{\ell/2}$; for if not, $q (q - 1) > 2^\ell$ and the theorem is trivially | |
1104 | true, since no adversary can achieve advantage greater than one. | |
74eb47db | 1105 | |
c128b544 | 1106 | Let's give the name $W_i$ to the probability that something went wrong after |
1107 | encrypting $i$ blocks. We therefore want to bound $W_q$ from above. | |
1108 | Armed with the knowledge that $q \le 2^{\ell/2}$, we have | |
1109 | \begin{eqnarray}[rl] | |
1110 | W_q &\le \sum_{0\le i<q} \Pr[C_i] | |
1111 | \le \sum_{0\le i<q} \frac{i}{2^\ell - i} \\ | |
1112 | &\le \frac{1}{2^\ell - 2^{\ell/2}} \sum_{0\le i<q} i \\ | |
1113 | &= \frac{q (q - 1)}{2 \cdot (2^\ell - 2^{\ell/2})} | |
1114 | \end{eqnarray} | |
1115 | Working through the definition of LOR-CPA security, we can see that $A$'s | |
1116 | (and hence any adversary's) advantage against the ideal system is at most $2 | |
1117 | W_q$. | |
74eb47db | 1118 | |
c128b544 | 1119 | By using an adversary attacking CBC encryption as a statistical test in an |
1120 | attempt to distinguish $P_K(\cdot)$ from a pseudorandom permutation, we see | |
1121 | that | |
1122 | \begin{equation} | |
1123 | \InSec{prp}(P; t + q t_P, q) \ge | |
1124 | \frac{1}{2} \cdot | |
1125 | \InSec{lor-cpa}(\Xid{\mathcal{E}}{CBC}; t, q_E, \mu_E) - | |
1126 | \frac{q (q - 1)}{2 \cdot (2^\ell - 2^{\ell/2})} | |
1127 | \end{equation} | |
1128 | where $t_P$ expresses the overhead of doing the XORs and other care and | |
1129 | feeding of the CBC adversary; whence | |
1130 | \begin{equation} | |
1131 | \InSec{lor-cpa}(\Xid{\mathcal{E}}{CBC}; t, q_E, \mu_E) \le | |
1132 | 2 \cdot \InSec{prp}(P; t, q) + \frac{q (q - 1)}{2^\ell - 2^{\ell/2}} | |
1133 | \end{equation} | |
1134 | as required. | |
1135 | \qed | |
74eb47db | 1136 | |
1137 | %%%----- That's all, folks -------------------------------------------------- | |
1138 | ||
c128b544 | 1139 | \bibliography{mdw-crypto,cryptography,cryptography2000,rfc} |
74eb47db | 1140 | \end{document} |
1141 | ||
e04c2d50 | 1142 | %%% Local Variables: |
74eb47db | 1143 | %%% mode: latex |
1144 | %%% TeX-master: "wrestlers" | |
d7d62ac0 | 1145 | %%% End: |