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1 | %%% -*-latex-*- |
2 | %%% |
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3 | %%% $Id: wrestlers.tex,v 1.7 2004/04/08 01:36:17 mdw Exp $ |
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4 | %%% |
5 | %%% Description of the Wrestlers Protocol |
6 | %%% |
7 | %%% (c) 2001 Mark Wooding |
8 | %%% |
9 | |
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10 | \newif\iffancystyle\fancystyletrue |
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11 | |
12 | \iffancystyle |
13 | \documentclass |
14 | [a4paper, article, 10pt, numbering, noherefloats, notitlepage] |
15 | {strayman} |
16 | \usepackage[palatino, helvetica, courier, maths=cmr]{mdwfonts} |
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17 | \usepackage[mdwmargin]{mdwthm} |
18 | \PassOptionsToPackage{dvips}{xy} |
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19 | \else |
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20 | \documentclass{llncs} |
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21 | \fi |
22 | |
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23 | \usepackage{mdwtab, mathenv, mdwlist, mdwmath, crypto} |
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24 | \usepackage{amssymb, amstext} |
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25 | \usepackage{tabularx} |
26 | \usepackage{url} |
27 | \usepackage[all]{xy} |
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28 | |
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29 | \errorcontextlines=999 |
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30 | \showboxbreadth=999 |
31 | \showboxdepth=999 |
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32 | \makeatletter |
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33 | |
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34 | \title{The Wrestlers Protocol: proof-of-receipt and secure key exchange} |
35 | \author{Mark Wooding \and Clive Jones} |
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36 | |
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37 | \bibliographystyle{mdwalpha} |
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38 | |
39 | \newcolumntype{G}{p{0pt}} |
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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 |
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55 | |
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56 | \begin{document} |
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57 | |
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58 | \maketitle |
59 | \begin{abstract} |
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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. |
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75 | \end{abstract} |
76 | \tableofcontents |
77 | \newpage |
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78 | |
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79 | %%%-------------------------------------------------------------------------- |
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80 | |
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81 | \section{Introduction} |
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82 | \label{sec:intro} |
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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. |
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86 | |
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87 | %%%-------------------------------------------------------------------------- |
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88 | |
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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. |
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93 | |
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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} |
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455 | |
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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. |
1a981bdb |
613 | |
c128b544 |
614 | \subsubsection{The \cookie{kx-cookie} message} |
1a981bdb |
615 | |
c128b544 |
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. |
1a981bdb |
621 | |
c128b544 |
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. |
1a981bdb |
627 | |
c128b544 |
628 | \subsubsection{The \cookie{kx-challenge} message} |
1a981bdb |
629 | |
c128b544 |
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} |
1a981bdb |
776 | |
d7d62ac0 |
777 | %%%-------------------------------------------------------------------------- |
1a981bdb |
778 | |
c128b544 |
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} |
1a981bdb |
1050 | |
c128b544 |
1051 | \subsection{Proof of theorem~\ref{thm:cbc}} |
1052 | \label{sec:cbc-proof} |
1a981bdb |
1053 | |
c128b544 |
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. |
1a981bdb |
1061 | |
c128b544 |
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. |
874aed51 |
1074 | |
c128b544 |
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} |
874aed51 |
1101 | |
c128b544 |
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. |
74eb47db |
1107 | |
c128b544 |
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 $2 |
1119 | W_q$. |
74eb47db |
1120 | |
c128b544 |
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 |
74eb47db |
1138 | |
1139 | %%%----- That's all, folks -------------------------------------------------- |
1140 | |
c128b544 |
1141 | \bibliography{mdw-crypto,cryptography,cryptography2000,rfc} |
74eb47db |
1142 | \end{document} |
1143 | |
1144 | %%% Local Variables: |
1145 | %%% mode: latex |
1146 | %%% TeX-master: "wrestlers" |
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1147 | %%% End: |