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1 | %%% -*-latex-*- |
2 | %%% | |
3 | %%% $Id$ | |
4 | %%% | |
5 | %%% Standard block cipher modes of operation | |
6 | %%% | |
7 | %%% (c) 2003 Mark Wooding | |
8 | %%% | |
9 | ||
10 | \newif\iffancystyle\fancystylefalse | |
11 | \fancystyletrue | |
12 | \errorcontextlines=\maxdimen | |
13 | \showboxdepth=\maxdimen | |
14 | \showboxbreadth=\maxdimen | |
15 | ||
16 | \iffancystyle | |
17 | \documentclass | |
18 | [a4paper, article, 10pt, numbering, noherefloats, notitlepage] | |
19 | {strayman} | |
20 | \usepackage[T1]{fontenc} | |
21 | \usepackage[palatino, helvetica, courier, maths=cmr]{mdwfonts} | |
22 | \usepackage[within = subsection, mdwmargin]{mdwthm} | |
23 | \usepackage{mdwlist} | |
24 | \usepackage{sverb} | |
25 | \PassOptionsToPackage{dvips}{xy} | |
26 | \else | |
27 | \documentclass[a4paper]{llncs} | |
28 | \usepackage{a4wide} | |
29 | \fi | |
30 | ||
31 | \PassOptionsToPackage{show}{slowbox} | |
32 | %\PassOptionsToPackage{hide}{slowbox} | |
33 | \usepackage{mdwtab, mathenv, mdwmath, crypto} | |
34 | \usepackage{slowbox} | |
35 | \usepackage{amssymb, amstext} | |
36 | \usepackage{url, multicol} | |
37 | \DeclareUrlCommand\email{\urlstyle{tt}} | |
38 | \ifslowboxshow | |
39 | \usepackage[all]{xy} | |
40 | \turnradius{4pt} | |
41 | \fi | |
42 | ||
43 | \title{New proofs for old modes} | |
44 | \iffancystyle | |
45 | \author{Mark Wooding \\ \email{mdw@distorted.org.uk}} | |
46 | \else | |
47 | \author{Mark Wooding} | |
48 | \institute{\email{mdw@distorted.org.uk}} | |
49 | \fi | |
50 | ||
51 | \iffancystyle | |
52 | \bibliographystyle{mdwalpha} | |
53 | \let\epsilon\varepsilon | |
54 | \let\emptyset\varnothing | |
55 | \let\le\leqslant\let\leq\le | |
56 | \let\ge\geqslant\let\geq\ge | |
57 | \numberwithin{table}{section} | |
58 | \numberwithin{figure}{section} | |
59 | \else | |
60 | \bibliographystyle{plain} | |
61 | \expandafter\let\csname claim*\endcsname\claim | |
62 | \expandafter\let\csname endclaim*\endcsname\endclaim | |
63 | \fi | |
64 | ||
65 | %%\newcommand{\Nupto}[1]{\N_{<{#1}}} | |
66 | \newcommand{\Nupto}[1]{\{0, 1, \ldots, #1 - 1\}} | |
67 | \let\Bin\Sigma | |
68 | \let\emptystring\lambda | |
69 | \edef\Pr{\expandafter\noexpand\Pr\nolimits} | |
70 | \newcommand{\bitsto}{\mathbin{..}} | |
71 | \newcommand{\shift}[1]{\lsl_{#1}} | |
72 | \newcommand{\E}{{\mathcal{E}}} | |
73 | \newcommand{\M}{{\mathcal{M}}} | |
74 | \iffancystyle | |
75 | \def\description{% | |
76 | \basedescript{% | |
77 | \let\makelabel\textit% | |
78 | \desclabelstyle\multilinelabel% | |
79 | \desclabelwidth{1in}% | |
80 | }% | |
81 | } | |
82 | \fi | |
83 | \def\fixme{\marginpar{FIXME}} | |
84 | ||
85 | \newslowboxenv{cgraph}{\par$$}{\begin{graph}}{\end{graph}}{$$\par} | |
86 | \newslowboxenv{vgraph} | |
87 | {\hfil$\vcenter\bgroup\hbox\bgroup} | |
88 | {\begin{graph}} | |
89 | {\end{graph}} | |
90 | {\egroup\egroup$} | |
91 | \newenvironment{vgraphs}{\hbox to\hsize\bgroup}{\hfil\egroup} | |
92 | ||
93 | \begin{document} | |
94 | ||
95 | %%%-------------------------------------------------------------------------- | |
96 | ||
97 | \maketitle | |
98 | ||
99 | \begin{abstract} | |
100 | We study the standard block cipher modes of operation: CBC, CFB, OFB, and | |
101 | CBCMAC and analyse their security. We don't look at ECB other than briefly | |
102 | to note its insecurity, and we have no new results on counter mode. Our | |
103 | results improve over those previously published in that (a) our bounds are | |
104 | better, (b) our proofs are shorter and easier, (c) the proofs correct | |
105 | errors we discovered in previous work, or some combination of these. We | |
106 | provide a new security notion for symmetric encryption which turns out to | |
107 | be rather useful when analysing block cipher modes. Finally, we define a | |
108 | new, condition for initialization vectors, introducing the concept of a | |
109 | `generalized counter', and proving that generalized suffice for security in | |
110 | (full-width) CFB and OFB modes and that generalized counters encrypted | |
111 | using the block cipher (with the same key) suffice for all the encryption | |
112 | modes we study. | |
113 | \end{abstract} | |
114 | ||
115 | \iffancystyle | |
116 | \newpage | |
117 | \columnsep=2em \columnseprule=0pt | |
118 | \tableofcontents[\begin{multicols}{2}\raggedright][\end{multicols}] | |
119 | \listoffigures[\begin{multicols}{2}\raggedright][\end{multicols}] | |
120 | \listoftables[\begin{multicols}{2}\raggedright][\end{multicols}] | |
121 | \newpage | |
122 | \fi | |
123 | ||
124 | %%%-------------------------------------------------------------------------- | |
125 | ||
126 | \section{Introduction} | |
127 | \label{sec:intro} | |
128 | ||
129 | \subsection{Block cipher modes} | |
130 | ||
131 | Block ciphers -- keyed pseudorandom permutations -- are essential | |
132 | cryptographic tools, widely used for bulk data encryption and to an | |
133 | increasing extent for message authentication. Because the efficient block | |
134 | ciphers we have operate on fixed and relatively small strings of bits -- 64 | |
135 | or 128 bits at a time, one needs a `mode of operation' to explain how to | |
136 | process longer messages. | |
137 | ||
138 | A collection of encryption modes, named ECB, CBC, CFB and OFB, were defined | |
139 | in \cite{FIPS81}. Of these, ECB -- simply divide the message into blocks and | |
140 | process them independently with the block cipher -- is just insecure and not | |
141 | to be recommended for anything much. We describe the other three, and | |
142 | analyse their security using the standard quantitative provable-security | |
143 | approach. All three require an `initialization vector' or `IV' which | |
144 | diversifies the output making it hard to correlate ciphertexts with | |
145 | plaintexts. We investigate which conditions on these IVs suffice for secure | |
146 | encryption. | |
147 | ||
148 | We also examine the CBC-MAC message-authentication scheme, because it's | |
149 | intimately related to the CBC encryption scheme and the same techniques we | |
150 | used in the analysis of the latter apply to the former. | |
151 | ||
152 | \subsection{Previous work} | |
153 | ||
154 | The first quantitative security proof for a block cipher mode is the analysis | |
155 | of CBCMAC of \cite{Bellare:1994:SCB}. Security proofs for the encryption | |
156 | modes CBC and CTR appeared in \cite{Bellare:2000:CST}, which also defines and | |
157 | relates the standard security notions of symmetric encryption. The authors | |
158 | of \cite{Alkassar:2001:OSS} offer a proof of CFB mode, though we believe it | |
159 | to be flawed in a number of respects. | |
160 | ||
161 | \subsection{Our contribution} | |
162 | ||
163 | We introduce a new security notion for symmetric encryption, named | |
164 | `result-or-garbage', or `ROG-CPA', which generalizes the `real-or-random' | |
165 | notion of \cite{Bellare:2000:CST} and the `random-string' notion of | |
166 | \cite{Rogaway:2001:OCB}. Put simply, it states that an encryption scheme is | |
167 | secure if an adversary has difficulty distinguishing true ciphertexts from | |
168 | strings chosen by an algorithm which is given only the \emph{length} of the | |
169 | adversary's plaintext. This turns out to be just the right tool for | |
170 | analysing our encryption modes. We relate this notion to the standard | |
171 | `left-or-right' notion and, thereby, all the others. | |
172 | ||
173 | Our bound for CBC mode improves over the `tight' bound proven in | |
174 | \cite{Bellare:2000:CST} by almost a factor of two. The difference comes | |
175 | because they analyse the construction as if it were built from a PRF and add | |
176 | in a `PRP-used-as-a-PRF' correction term: our analysis considers the effect | |
177 | of a permutation directly. We prove that CBC mode is still secure if an | |
178 | encrypted counter is used in place of a random string as the IV for each | |
179 | message. Finally, we show that the `ciphertext stealing' technique is | |
180 | secure. | |
181 | ||
182 | For CFB, we first discuss the work of \cite{Alkassar:2001:OSS}, who offer a | |
183 | proof for both CFB mode and an optimized variant which enhances the | |
184 | error-recovery properties of standard CFB. We believe that their proof is | |
185 | defective in a number of ways. We then offer our own proof. Our bound is a | |
186 | factor of two worse than theirs; however, we believe that fixing their proof | |
187 | introduces this missing factor of two: that is, that our `poorer' bound | |
188 | reflects the true security of CFB mode more accurately. We show that | |
189 | full-width CFB is secure if the IV is any `generalized counter', and that | |
190 | both full-width and truncated $t$-bit CFB are secure if the IV is an | |
191 | encrypted counter. We also show that, unlike CBC mode, it is safe to `carry | |
192 | over' the final shift-register value from the previous message as the IV for | |
193 | the next message. | |
194 | ||
195 | OFB mode is in fact a simple modification to CFB mode, and we prove the | |
196 | security of OFB by relating it to CFB. | |
197 | ||
198 | Finally, for CBCMAC, we analyse it using \emph{both} pseudorandom functions | |
199 | \emph{and} pseudorandom permutations, showing that, in fact, using a block | |
200 | cipher rather than a PRF reduces the security hardly at all. Also, we | |
201 | improve on the (groundbreaking) work of \cite{Bellare:1994:SCB} firstly by | |
202 | improving the security bound by a factor of almost four, and secondly by | |
203 | extending the message space from a space of fixed-length messages to | |
204 | \emph{any} prefix-free set of strings. | |
205 | ||
206 | As a convenient guide, our security bounds are summarized in | |
207 | table~\ref{tab:summary}. | |
208 | ||
209 | \begin{table} | |
210 | \def\lower#1{% | |
211 | \vbox to\baselineskip{\vskip\baselineskip\vskip2pt\hbox{#1}\vss}} | |
212 | \def\none{\multicolumn{1}{c|}{---}} | |
213 | \let\hack=\relax | |
214 | \begin{tabular}[C] | |
215 | {| c | ?>{\hack}c | c | >{\displaystyle} Mc | >{\displaystyle} Mc |} | |
216 | \hlx{hv[4]} | |
217 | \multicolumn{1}{|c|}{\lower{\bfseries Mode}} & | |
218 | \multicolumn{1}{c|}{\lower{\bfseries Section}} & | |
219 | \multicolumn{1}{c|}{\lower{\bfseries Notion}} & | |
220 | \multicolumn{2}{c|}{\bfseries Security with} \\ \hlx{v[4]zc{4-5}v} | |
221 | & & & | |
222 | \multicolumn{1}{c|}{\bfseries $(t, q, \epsilon)$-PRF} & | |
223 | \multicolumn{1}{c|}{\bfseries $(t, q, \epsilon)$-PRP} | |
224 | \\ \hlx{vhvv} | |
225 | CBC & \ref{sec:cbc} & LOR-CPA & | |
226 | \none & | |
227 | 2\epsilon + \frac{q (q - 1)}{2^\ell - q} \\ \hlx{vvhvv} | |
228 | CFB & \ref{sec:cfb} & LOR-CPA & | |
229 | 2 \epsilon + \frac{q (q - 1)}{2^\ell} & | |
230 | 2 \epsilon + \frac{q (q - 1)}{2^{\ell-1}} \\ \hlx{vvhvv} | |
231 | OFB & \ref{sec:ofb} & LOR-CPA & | |
232 | 2 \epsilon + \frac{q (q - 1)}{2^\ell} & | |
233 | 2 \epsilon + \frac{q (q - 1)}{2^{\ell-1}} \\ \hlx{vvhvv} | |
234 | CBCMAC & \ref{sec:cbcmac} & SUF-CMA & | |
235 | \epsilon + \frac{q (q - 1) + 2 q_V}{2^{\ell+1}} & | |
236 | \epsilon + | |
237 | \frac{q (q - 1)}{2 \cdot (2^\ell - q)} + | |
238 | \frac{q_V}{2^\ell - q_T} \\ \hlx{vvh} | |
239 | \end{tabular} | |
240 | ||
241 | \caption[Summary of our results] | |
242 | {Summary of our results. In all cases, $q$ is the number of block | |
243 | cipher applications used in the game.} | |
244 | \label{tab:summary} | |
245 | \end{table} | |
246 | ||
247 | \subsection{The rest of the paper} | |
248 | ||
249 | In section~\ref{sec:prelim} we define the various bits of notation and | |
250 | terminology we'll need in the rest of the paper. The formal definitions are | |
251 | given for our new `result-or-garbage' security notion, and for our | |
252 | generalized counters. In section~\ref{sec:cbc} we study CBC mode, and | |
253 | ciphertext stealing. In section~\ref{sec:cfb} we study CFB mode. In | |
254 | section~\ref{sec:ofb} we study OFB mode. In section~\ref{sec:cbcmac} we | |
255 | study the CBCMAC message authentication scheme. | |
256 | ||
257 | %%%-------------------------------------------------------------------------- | |
258 | ||
259 | \section{Notation and definitions} | |
260 | \label{sec:prelim} | |
261 | ||
262 | \subsection{Bit strings} | |
263 | \label{sec:bitstrings} | |
264 | ||
265 | Most of our notation for bit strings is standard. The main thing to note is | |
266 | that everything is zero-indexed. | |
267 | ||
268 | \begin{itemize} | |
269 | \item We write $\Bin = \{0, 1\}$ for the set of binary digits. Then $\Bin^n$ | |
270 | is the set of $n$-bit strings, and $\Bin^*$ is the set of all (finite) bit | |
271 | strings. | |
272 | \item If $x$ is a bit string then $|x|$ is the length of $x$. If $x \in | |
273 | \Bin^n$ then $|x| = n$. | |
274 | \item If $x, y \in \Bin^n$ are strings of bits of the same length then $x | |
275 | \xor y \in \Bin^n$ is their bitwise XOR. | |
276 | \item If $x$ is a bit string and $i$ is an integer satisfying $0 \le i < |x|$ | |
277 | then $x[i]$ is the $i$th bit of $x$. If $a$ and $b$ are integers | |
278 | satisfying $0 \le a \le b \le |x|$ then $x[a \bitsto b]$ is the substring | |
279 | of $x$ beginning with bit $a$ and ending just \emph{before} bit $b$. We | |
280 | have $|x[i]| = 1$ and $|x[a \bitsto b]| = b - a$; if $y = x[a \bitsto b]$ | |
281 | then $y[i] = x[a + i]$. | |
282 | \item If $x$ and $y$ are bit strings then $x y$ is the result of | |
283 | concatenating $y$ to $x$. If $z = x y$ then $|z| = |x| + |y|$; $z[i] = | |
284 | x[i]$ if $0 \le i < |x|$ and $z[i] = y[i - |x|]$ if $|x| \le i < |x| + | |
285 | |y|$. Sometimes, for clarity (e.g., to distinguish from integer | |
286 | multiplication) we write $x \cat y$ instead of $x y$. | |
287 | \item The empty string is denoted $\emptystring$. We have $|\emptystring| = | |
288 | 0$, and $x = x \emptystring = \emptystring x$ for all strings $x | |
289 | \in \Bin^*$. | |
290 | \item If $x$ is a bit string and $n$ is a natural number then $x^n$ is the | |
291 | result of concatenating $x$ to itself $n$ times. We have $x^0 = | |
292 | \emptystring$ and if $n > 0$ then $x^n = x^{n-1} \cat x = x \cat x^{n-1}$. | |
293 | \item If $x$ and $y$ are bit strings, $|x| = \ell$, and $|y| = t$, then we | |
294 | define $x \shift{t} y$ as: | |
295 | \[ x \shift{t} y = (x y)[t \bitsto t + \ell] = \begin{cases} | |
296 | x[t \bitsto \ell] \cat y & if $t < \ell$, or \\ | |
297 | y[t - \ell \bitsto t] & if $t \ge \ell$. | |
298 | \end{cases} \] | |
299 | Observe that, if $z = x \shift{t} y$ then $|z| = |x| = \ell$ and | |
300 | \[ z[i] = (x y)[i + t] = \begin{cases} | |
301 | x[i + t] & if $0 \le i < \ell - t$, or \\ | |
302 | y[i + t - \ell] & if $\min(0, \ell - t) \le i < \ell$. | |
303 | \end{cases} \] | |
304 | Obviously $x \shift{0} \emptystring = x$, and if $|x| = |y| = t$ then $x | |
305 | \shift{t} y = y$. Finally, if $|y| = t$ and $|z| = t'$ then $(x \shift{t} | |
306 | y) \shift{t'} z = x \shift{t + t'} (y z)$. | |
307 | \end{itemize} | |
308 | ||
309 | \subsection{Other notation} | |
310 | \label{sec:miscnotation} | |
311 | ||
312 | \begin{itemize} | |
313 | \iffalse | |
314 | \item If $n$ is any natural number, then $\Nupto{n}$ is the set $\{\, i \in | |
315 | \Z \mid 0 \le i < n \,\} = \{ 0, 1, \ldots, n \}$. | |
316 | \fi | |
317 | \item The symbol $\bot$ (`bottom') is a value different from every bit | |
318 | string. | |
319 | \item We write $\Func{l}{L}$ as the set of all functions from $\Bin^l$ to | |
320 | $\Bin^L$, and $\Perm{l}$ as the set of all permutations on $\Bin^l$. | |
321 | \end{itemize} | |
322 | ||
323 | \subsection{Algorithm descriptions} | |
324 | \label{sec:algorithms} | |
325 | ||
326 | An \emph{adversary} is a probabilistic algorithm which attempts (possibly) to | |
327 | `break' a cryptographic scheme. We will often provide adversaries with | |
328 | oracles which compute values with secret data. The \emph{running time} of an | |
329 | adversary conventionally includes the size of the adversary's description: | |
330 | this is an attempt to `charge' the adversary for having large precomputed | |
331 | tables. | |
332 | ||
333 | Most of the notation used in the algorithm descriptions should be obvious. | |
334 | We briefly note a few features which may be unfamiliar. | |
335 | \begin{itemize} | |
336 | \item The notation $a \gets x$ denotes the action of assigning the value $x$ | |
337 | to the variable $a$. | |
338 | \item We write oracles as superscripts, with dots marking where inputs to | |
339 | the oracle go, e.g., $A^{O(\cdot)}$. | |
340 | \item The notation $a \getsr X$, where $X$ is a finite set, denotes the | |
341 | action of assigning to $a$ a random value $x \in X$ according to the | |
342 | uniform probability distribution on $X$; i.e., following $a \getsr X$, we | |
343 | have $\Pr[a = x] = 1/|X|$ for any $x \in X$. | |
344 | \end{itemize} | |
345 | The notation is generally quite sloppy about types and scopes. We don't | |
346 | think these informalities cause much confusion, and they greatly simplify the | |
347 | presentation of the algorithms. | |
348 | ||
349 | \subsection{Pseudorandom functions and permutations} | |
350 | \label{sec:prfs-and-prps} | |
351 | ||
352 | Our definitions of pseudorandom functions and permutations are standard. We | |
353 | provide them for the sake of completeness. | |
354 | ||
355 | \begin{definition}[Pseudorandom function family] | |
356 | \label{def:prf} | |
357 | A \emph{pseudorandom function family (PRF)} $F = \{F_K\}_K$ is a collection | |
358 | of functions $F_K\colon \Bin^\ell \to \Bin^L$ indexed by a \emph{key} $K | |
359 | \in \keys F$. If $A$ is any adversary, we define $A$'s \emph{advantage in | |
360 | distinguishing $F$ from a random function} to be | |
361 | \[ \Adv{prf}{F}(A) = | |
362 | \Pr[K \getsr \keys F: A^{F_K(\cdot)} = 1] - | |
363 | \Pr[R \getsr \Func{\ell}{L}: A^{R(\cdot)} = 1] | |
364 | \] | |
365 | where the probability is taken over all choices of keys, random functions, | |
366 | and the internal coin-tosses of $A$. The \emph{insecurity of $F$} is given | |
367 | by | |
368 | \[ \InSec{prf}(F; t, q) = \max_A \Adv{prf}{F}(A) \] | |
369 | where the maximum is taken over all adversaries which run in time~$t$ and | |
370 | issue at most $q$ oracle queries. If $\InSec{prf}(F; t, q) \le \epsilon$ | |
371 | then we say that $F$ is a $(t, q, \epsilon)$-PRF. | |
372 | \end{definition} | |
373 | ||
374 | \begin{definition}[Pseudorandom permutation family] | |
375 | \label{def:prp} | |
376 | A \emph{pseudorandom permutation family (PRP)} $E = \{E_K\}_K$ is a | |
377 | collection of permutations $E_K\colon \Bin^\ell \to \Bin^\ell$ indexed by a | |
378 | \emph{key} $K \in \keys E$. If $A$ is any adversary, we define $A$'s | |
379 | \emph{advantage in distinguishing $E$ from a random permutation} to be | |
380 | \[ \Adv{prp}{F}(A) = | |
381 | \Pr[K \getsr \keys E: A^{E_K(\cdot)} = 1] - | |
382 | \Pr[P \getsr \Perm{\ell}: A^{P(\cdot)} = 1] | |
383 | \] | |
384 | where the probability is taken over all choices of keys, random | |
385 | permutations, and the internal coin-tosses of $A$. Note that the adversary | |
386 | is not allowed to query the inverse permutation $E^{-1}_K(\cdot)$ or | |
387 | $P^{-1}(\cdot)$. The \emph{insecurity of $E$} is given by | |
388 | \[ \InSec{prp}(E; t, q) = \max_A \Adv{prf}{E}(A) \] | |
389 | where the maximum is taken over all adversaries which run in time~$t$ and | |
390 | issue at most $q$ oracle queries. If $\InSec{prp}(E; t, q) \le \epsilon$ | |
391 | then we say that $E$ is a $(t, q, \epsilon)$-PRP. | |
392 | \end{definition} | |
393 | ||
394 | The following result is standard; we shall require it for the security proofs | |
395 | of CFB and OFB modes. The proof is given as an introduction to our general | |
396 | approach. | |
397 | ||
398 | \begin{proposition} | |
399 | \label{prop:prps-are-prfs} | |
400 | Suppose $E$ is a PRP over $\Bin^\ell$. Then | |
401 | \[ \InSec{prf}(E; t, q) | |
402 | \le \InSec{prp}(E; t, q) + \frac{q (q - 1)}{2^{\ell+1}}. | |
403 | \] | |
404 | \end{proposition} | |
405 | \begin{proof} | |
406 | We claim | |
407 | \[ \InSec{prf}(\Perm{\ell}; t, q) \le \frac{q (q - 1)}{2^{\ell+1}}, \] | |
408 | i.e., that a \emph{perfect} $\ell$-bit random permutation is a PRF with the | |
409 | stated bounds. The proposition follows immediately from this claim and the | |
410 | definition of a PRP. | |
411 | ||
412 | We now prove the claim. Consider any adversary~$A$. Let $x_i$ be $A$'s | |
413 | queries, and let $y_i$ be the responses, for $0 \le i < q$. Assume, | |
414 | without loss of generality, that the $x_i$ are distinct. Let $C_n$ be the | |
415 | event in the random-function game $\Expt{prf-$0$}{\Perm{\ell}}(A)$ that | |
416 | $y_i = y_j$ for some $i$ and $j$ where $0 \le i < j < n$. Then | |
417 | \begin{equation} | |
418 | \Pr[C_n] \le \sum_{0\le i<n} \frac{i}{2^\ell} | |
419 | = \frac{n (n - 1)}{2^{\ell+1}}. | |
420 | \end{equation} | |
421 | It's clear that the two games proceed identically if $C_q$ doesn't occur in | |
422 | the random-function game. The claim follows. | |
423 | \end{proof} | |
424 | ||
425 | \subsection{Symmetric encryption} | |
426 | \label{sec:sym-enc} | |
427 | ||
428 | We begin with a purely syntactic description of a symmetric encryption | |
429 | scheme, and then define our two notions of security. | |
430 | ||
431 | \begin{definition}[Symmetric encryption] | |
432 | \label{def:symm-enc} | |
433 | A \emph{symmetric encryption scheme} is a triple of algorithms $\E = (G, E, | |
434 | D)$, with three (implicitly) associated sets: a keyspace, a plaintext | |
435 | space, and a ciphertext space. | |
436 | \begin{itemize} | |
437 | \item $G$ is a probabilistic \emph{key-generation algorithm}. It is | |
438 | invoked with no arguments, and returns a key $K$ which can be used with | |
439 | the other two algorithms. We write $K \gets G()$. | |
440 | \item $E$ is a probabilistic \emph{encryption algorithm}. It is invoked | |
441 | with a key $K$ and a \emph{plaintext} $x$ in the plaintext space, and it | |
442 | returns a \emph{ciphertext} $y$ in the ciphertext space. We write $y | |
443 | \gets E_K(x)$. | |
444 | \item $D$ is a deterministic \emph{decryption algorithm}. It is invoked | |
445 | with a key $K$ and a ciphertext $y$, and it returns either a plaintext | |
446 | $x$ or the distinguished symbol $\bot$. We write $x \gets D_K(y)$. | |
447 | \end{itemize} | |
448 | For correctness, we require that whenever $y$ is a possible result of | |
449 | computing $E_K(x)$, then $x = D_K(y)$. | |
450 | \end{definition} | |
451 | ||
452 | Our primary notion of security is \emph{left-or-right indistinguishability | |
453 | under chosen-plaintext attack} (LOR-CPA), since it offers the best reductions | |
454 | to the other common notions. (We can't acheive security against chosen | |
455 | ciphertext attack using any of our modes, so we don't even try.) See | |
456 | \cite{Bellare:2000:CST} for a complete discussion of LOR-CPA, and how it | |
457 | relates to other security notions for symmetric encryption. | |
458 | ||
459 | \begin{definition}[Left-or-right indistinguishability] | |
460 | \label{def:lor-cpa} | |
461 | Let $\E = (G, E, D)$ be a symmetric encryption scheme. Define the function | |
462 | $\id{lr}(b, x_0, x_1) = x_b$. Then for any adversary $A$, we define $A$'s | |
463 | \emph{advantage against the LOR-CPA security of $\E$} as | |
464 | \[ \Adv{lor-cpa}{\E}(A) = | |
465 | \Pr[K \gets G(): A^{E_K(\id{lr}(1, \cdot, \cdot))} = 1] - | |
466 | \Pr[K \gets G(): A^{E_K(\id{lr}(0, \cdot, \cdot))} = 1]. | |
467 | \] | |
468 | We define the \emph{LOR-CPA insecurity of $\E$} to be | |
469 | \[ \InSec{lor-cpa}(\E; t, q_E, \mu_E) = | |
470 | \max_A \Adv{lor-cpa}{\E}(A) | |
471 | \] | |
472 | where the maximum is taken over all adversaries which run in time~$t$ and | |
473 | issue at most $q_E$ encryption queries totalling at most $\mu_E$ bits. If | |
474 | $\InSec{lor-cpa}(\E; t, q_E, \mu_E) \le \epsilon$ then we say that $\E$ is | |
475 | $(t, q_E, \mu_E, \epsilon)$-LOR-CPA. | |
476 | \end{definition} | |
477 | ||
478 | Our second notion is named \emph{result-or-garbage} and abbreviated ROG-CPA. | |
479 | It is related to the notion used by \cite{Rogaway:2001:OCB}, though different | |
480 | in important ways: for example, there are reductions both ways between | |
481 | ROG-CPA and LOR-CPA (and hence the other standard notions of security for | |
482 | symmetric encryption), whereas the notion of \cite{Rogaway:2001:OCB} is | |
483 | strictly stronger than LOR-CPA. Our idea is that an encryption scheme is | |
484 | secure if ciphertexts of given plaintexts -- \emph{results} -- hard to | |
485 | distinguish from strings constructed independently of any plaintexts -- | |
486 | \emph{garbage}. We formalize this notion in terms of a | |
487 | \emph{garbage-emission algorithm} $W$ which is given only the length of the | |
488 | plaintext. The algorithm $W$ will usually be probabilistic, and may maintain | |
489 | state. Unlike \cite{Rogaway:2001:OCB}, we don't require that $W$'s output | |
490 | `look random' in any way, just that it be chosen independently of the | |
491 | adversary's plaintext selection. | |
492 | ||
493 | \begin{definition}[Result-or-garbage indistinguishability] | |
494 | \label{def:rog-cpa} | |
495 | Let $\E = (G, E, D)$ be a symmetric encryption scheme, and let $W$ be a | |
496 | possibly-stateful, possibly-probabilistic \emph{garbage-emission} | |
497 | algorithm. Then for any adversary $A$, we define $A$'s \emph{advantage | |
498 | against the ROG-CPA-$W$ security of $\E$} as | |
499 | \[ \Adv{rog-cpa-$W$}{\E}(A) = | |
500 | \Pr[K \gets G(): A^{E_K(\cdot)} = 1] - \Pr[A^{W(|\cdot|)} = 1]. \] | |
501 | We define the \emph{ROG-CPA insecurity of $\E$} to be | |
502 | \[ \InSec{lor-cpa}(\E; t, q_E, \mu_E) = | |
503 | \max_A \Adv{lor-cpa}{\E}(A) \] | |
504 | where the maximum is taken over all adversaries which run in time~$t$ and | |
505 | issue at most $q_E$ encryption queries totalling at most $\mu_E$ bits. If | |
506 | $\InSec{rog-cpa-$W$}(\E; t, q_E, \mu_E) \le \epsilon$ for some $W$ then we | |
507 | say that $\E$ is $(t, q_E, \mu_E, \epsilon)$-ROG-CPA. | |
508 | \end{definition} | |
509 | ||
510 | The following proposition relates our new notion to the existing known | |
511 | notions of security. | |
512 | ||
513 | \begin{proposition}[ROG $\Leftrightarrow$ LOR] | |
514 | \label{prop:rog-and-lor} | |
515 | Let $\E$ be a symmetric encryption scheme. Then, | |
516 | \begin{enumerate} | |
517 | \item for all garbage-emission algorithms $W$, | |
518 | \[ \InSec{lor-cpa}(\E; t, q_E, \mu_E) | |
519 | \le 2 \cdot | |
520 | \InSec{rog-cpa-$W$}(\E; t + t_E \mu_E, q_E, \mu_E) | |
521 | \] | |
522 | and | |
523 | \item there exists a garbage-emission algorithm $W$ for which | |
524 | \[ \InSec{rog-cpa-$W$}(\E; t, q_E, \mu_E) | |
525 | \le \InSec{lor-cpa}(\E; t + t_E \mu_E, q_E, \mu_E) | |
526 | \] | |
527 | \end{enumerate} | |
528 | for some fairly small constant $t_E$. | |
529 | \end{proposition} | |
530 | \begin{proof} | |
531 | \begin{enumerate} | |
532 | \item Let $W$ and $\E$ be given, and let $A$ be an adversary attacking the | |
533 | LOR-CPA security of $\E$. Consider adversary $B$ attacking $\E$'s | |
534 | ROG-CPA-$W$ security. | |
535 | \begin{program} | |
536 | Adversary $B^E(\cdot)$: \+ \\ | |
537 | $b^* \getsr \Bin$; \\ | |
538 | $b \gets A^{E(\id{lr}(b^*, \cdot, \cdot))}$; \\ | |
539 | \IF $b = b^*$ \THEN \RETURN $1$ \ELSE \RETURN $0$; | |
540 | \next | |
541 | Function $\id{lr}(b, x_0, x_1)$: \+ \\ | |
542 | \IF $b = 0$ \THEN \RETURN $x_0$ \ELSE \RETURN $x_1$; | |
543 | \end{program} | |
544 | If $E(\cdot)$ is the `result' encryption oracle, then $B$ simulates the | |
545 | left-or-right game for the benefit of $A$, and therefore returns $1$ with | |
546 | probability $(\Adv{lor-cpa}{\E}(A) + 1)/2$. On the other hand, if | |
547 | $E(\cdot)$ returns `garbage' then the oracle responses are entirely | |
548 | independent of $b^*$. This follows because $A$ is constrained to query | |
549 | only on pairs of plaintexts with equal lengths, and the responses are | |
550 | dependent only on these (common) lengths and any internal state and coin | |
551 | tosses of $W$. So $b$ is independent of $b^*$ and $\Pr[b = b^*] = | |
552 | \frac{1}{2}$. The result follows. | |
553 | \item Let $\E = (G, E, D)$ be given. Our garbage emitter simulates the | |
554 | real-or-random game of \cite{Bellare:2000:CST}. Let $K_W = \bot$ | |
555 | initially: we define our emitter $W$ thus: | |
556 | \begin{program} | |
557 | Garbage emitter $W(n)$: \+ \\ | |
558 | \IF $K_W = \bot$ \THEN $K_W \gets G()$; \\ | |
559 | $x \getsr \Bin^n$; \\ | |
560 | \RETURN $E_K(x)$; | |
561 | \end{program} | |
562 | We now show that $\InSec{rog-cpa-$W$}(\E; t, q_E, \mu_E) \le | |
563 | \InSec{lor-cpa}(\E; t + t_E \mu_E, q_E, \mu_E)$ for our $W$. Let $A$ be | |
564 | an adversary attacking the ROG-CPA-$W$ security of $\E$. Consider | |
565 | adversary $B$ attacking $\E$'s LOR-CPA security: | |
566 | \begin{program} | |
567 | Adversary $B^{E(\cdot, \cdot)}$: \+ \\ | |
568 | $b \gets A^{\id{lorify}(\cdot)}$; \\ | |
569 | \RETURN $b$; | |
570 | \next | |
571 | Function $\id{lorify}(x)$: \+ \\ | |
572 | $x' \getsr \Bin^{|x|}$; \\ | |
573 | \RETURN $E(x', x)$; | |
574 | \end{program} | |
575 | The adversary simulates the ROG-CPA-$W$ games perfectly for our chosen | |
576 | $W$, since the game has chosen the random $K_W$ for us already: the | |
577 | `left' game returns only the results of encrypting random `garbage' | |
578 | plaintexts $x'$, while the right game returns correct results of | |
579 | encrypting the given plaintexts $x$. The result follows. | |
580 | \qed | |
581 | \end{enumerate} | |
582 | \end{proof} | |
583 | ||
584 | ||
585 | ||
586 | \subsection{Message authentication} | |
587 | \label{sec:mac} | |
588 | ||
589 | Our definitions for message authentication are standard; little needs to be | |
590 | said of them. As with symmetric encryption, we begin with a syntactic | |
591 | definition, and then describe our notion of security. | |
592 | ||
593 | \begin{definition}[Message authentication code] | |
594 | \label{def:mac} | |
595 | A \emph{message authentication code (MAC)} is a triple of algorithms $\M = | |
596 | (G, T, V)$ with three (implicitly) associated sets: a keyspace, a message | |
597 | space, and a tag space. | |
598 | \begin{itemize} | |
599 | \item $G$ is a probabilistic \emph{key-generation algorithm}. It is | |
600 | invoked with no arguments, and returns a key $K$ which can be used with | |
601 | the other two algorithms. We write $K \gets G()$. | |
602 | \item $T$ is a probabilistic \emph{tagging algorithm}. It is invoked with | |
603 | a key $K$ and a \emph{message} $x$ in the message space, and it returns a | |
604 | \emph{tag} $\tau$ in the tag space. We write $\tau \gets T_K(x)$. | |
605 | \item $V$ is a deterministic \emph{verification algorithm}. It is invoked | |
606 | with a key $K$, a message $x$ and a tag $\tau$, and returns a bit $b \in | |
607 | \Bin$. We write $b \gets V_K(x, \tau)$. | |
608 | \end{itemize} | |
609 | For correctness, we require that whenever $\tau$ is a possible result of | |
610 | computing $T_K(x)$, then $V_K(x, \tau) = 1$. | |
611 | \end{definition} | |
612 | ||
613 | Our notion of security is the strong unforgeability of | |
614 | \cite{Abdalla:2001:DHIES,Bellare:2000:AER}. | |
615 | ||
616 | \begin{definition}[Strong unforgeability] | |
617 | Let $\M = (G, T, V)$ be a message authentication code, and let $A$ | |
618 | be an adversary. We perform the following experiment. | |
619 | \begin{program} | |
620 | Experiment $\Expt{suf-cma}{\M}(A)$: \+ \\ | |
621 | $K \gets G()$; \\ | |
622 | $\Xid{T}{list} \gets \emptyset$; \\ | |
623 | $\id{good} \gets 0$; \\ | |
624 | $A^{\id{tag}(\cdot), \id{verify}(\cdot, \cdot)}$; \\ | |
625 | \RETURN $\id{good}$; | |
626 | \newline | |
627 | Oracle $\id{tag}(x)$: \+ \\ | |
628 | $\tau \gets T_K(x)$; \\ | |
629 | $\Xid{T}{list} \gets \Xid{T}{list} \cup \{(x, \tau)\}$; \\ | |
630 | \RETURN $\tau$; | |
631 | \next | |
632 | Oracle $\id{verify}(x, \tau)$: \+ \\ | |
633 | $b \gets V_K(x, \tau)$; \\ | |
634 | \IF $b = 1 \land (x, \tau) \notin \Xid{T}{list}$ \THEN | |
635 | $\id{good} \gets 1$; \\ | |
636 | \RETURN $b$; | |
637 | \end{program} | |
638 | That is, the adversary `wins' if it submits a query to its verification | |
639 | oracle which is \emph{new} -- doesn't match any message/tag pair from the | |
640 | tagging oracle -- and \emph{valid} -- the verification oracle returned | |
641 | success. We define the adversary's \emph{success probability} as | |
642 | \[ \Succ{suf-cma}{\M}(A) = | |
643 | \Pr[\Expt{suf-cma}{\M}(A) = 1]. \] | |
644 | We define the \emph{SUF-CMA insecurity of $\M$} to be | |
645 | \[ \InSec{suf-cma}(\M; t, q_T, \mu_T, q_V, \mu_V) = | |
646 | \max_A \Adv{suf-cma}{\M}(A) \] | |
647 | where the maximum is taken over all adversaries which run in time~$t$, | |
648 | issue at most $q_T$ tagging queries totalling at most $\mu_T$ bits, and | |
649 | issue at most $q_V$ verification queries totalling at most $\mu_V$ bits. | |
650 | If $\InSec{suf-cma}(\M; t, q_T, \mu_T, q_V, \mu_V) \le \epsilon$ | |
651 | then we say that $\E$ is $(t, q_T, \mu_T, q_V, \mu_V)$-SUF-CMA. | |
652 | \end{definition} | |
653 | ||
654 | \subsection{Initialization vectors and encryption modes} | |
655 | \label{sec:iv} | |
656 | ||
657 | In order to reduce the number of definitions in this paper to a tractable | |
658 | level, we will describe the basic modes independently of how initialization | |
659 | vectors (IVs) are chosen, and then construct the actual encryption schemes by | |
660 | applying various IV selection methods from the modes. | |
661 | ||
662 | We consider the following IV selection methods. | |
663 | \begin{description} | |
664 | \item[Random selection] An IV is chosen uniformly at random just before | |
665 | encrypting each message. | |
666 | \item[Counter] The IV for each message is a \emph{generalized counter} (see | |
667 | discussion below, and definition~\ref{def:genctr}). | |
668 | \item[Encrypted counter] The IV for a message is the result of applying the | |
669 | block cipher to a generalized counter, using the same key as for message | |
670 | encryption. | |
671 | \item[Carry-over] The IV for the first message is fixed; the IV for | |
672 | subsequent messages is some function of the previous plaintexts or | |
673 | ciphertexts (e.g., the last ciphertext block of the previous message). | |
674 | \end{description} | |
675 | Not all of these methods are secure for all of the modes we consider. | |
676 | ||
677 | \begin{definition}[Generalized counters] | |
678 | \label{def:genctr} | |
679 | If $S$ is a finite set, then a \emph{generalized counter in $S$} is an | |
680 | bijection $c\colon \Nupto{|S|} \leftrightarrow S$. For brevity, we shall | |
681 | refer simply to `counters', leaving implicit the generalization. | |
682 | \end{definition} | |
683 | ||
684 | \begin{remark}[Examples of generalized counters] \leavevmode | |
685 | \begin{itemize} | |
686 | \item There is a `natural' binary representation of the natural numbers | |
687 | $\Nupto{2^\ell}$ as $\ell$-bit strings: for any $n \in \Nupto{2^\ell}$, | |
688 | let $R(n)$ be the unique $r \in \Bin^\ell$ such that $\smash{n = | |
689 | \sum_{0\le i<\ell} 2^i r[i]}$. Then $R(\cdot)$ is a generalized counter | |
690 | in $\Bin^\ell$. | |
691 | \item We can represent elements of the finite field $\gf{2^\ell}$ as | |
692 | $\ell$-bit strings. Let $p(x) \in \gf{2}[x]$ be a primitive polynomial | |
693 | of degree $\ell$; then represent $\gf{2^\ell}$ by $\gf{2}[x]/(p(x))$. | |
694 | Now for any $a \in \gf{2^\ell}$, let $R(a)$ be the unique $r \in | |
695 | \Bin^\ell$ such that $\smash{a = \sum_{0\le i<\ell} x^i r[i]}$. Because | |
696 | $p(x)$ is primitive, $x$ generates the multiplicative group | |
697 | $\gf{2^\ell}^{\,*}$, so define $c(n) = R(x^n)$ for $0 \le n < 2^\ell - 1$ | |
698 | and $c(2^{\ell - 1}) = 0^\ell$; then $c(\cdot)$ is a generalized counter | |
699 | in $\Bin^\ell$. This counter can be implemented efficiently in hardware | |
700 | using a linear feedback shift register. | |
701 | \end{itemize} | |
702 | \end{remark} | |
703 | ||
704 | \begin{definition}[Encryption modes] | |
705 | \label{def:enc-modes} | |
706 | ||
707 | A \emph{block cipher encryption mode} $m_P = (\id{encrypt}, \id{decrypt})$ | |
708 | is a pair of deterministic oracle algorithms (and implicitly-defined | |
709 | plaintext and ciphertext spaces) which satisfy the following conditions: | |
710 | \begin{enumerate} | |
711 | \item The algorithm $\id{encrypt}$ runs with oracle access to a permutation | |
712 | $P\colon \Bin^\ell \leftrightarrow \Bin^\ell$; on input a plaintext $x$ | |
713 | and an initialization vector $v \in \Bin^\ell$, it returns a ciphertext | |
714 | $y$ and a \emph{chaining value} $v' \in \Bin^\ell$. We write $(v', y) = | |
715 | \id{encrypt}^{P(\cdot)}(v, x)$. | |
716 | \item The algorithm $\id{decrypt}$ runs with oracle access to a permutation | |
717 | $P\colon \Bin^\ell \leftrightarrow \Bin^\ell$ and its inverse | |
718 | $P^{-1}(\cdot)$; on input a ciphertext $y$ and an initialization vector | |
719 | $v \in \Bin^\ell$, it returns a plaintext $x$. We write that $x = | |
720 | \id{decrypt}^{P(\cdot), P^{-1}(\cdot)}(v, y)$. | |
721 | \item For all permutations $P\colon \Bin^\ell \leftrightarrow \Bin^\ell$, | |
722 | all plaintexts $x$ and all initialization vectors $v$, if $(v', y) = | |
723 | \id{encrypt}^{P(\cdot)}(v, x)$ then $x = \id{decrypt}^{P(\cdot), | |
724 | P^{-1}(\cdot)}(v, y)$. | |
725 | \item There exists an efficient algorithm which, given a ciphertext $y$ and | |
726 | the initialization vector but \emph{not} access to $P$, computes the | |
727 | chaining value $v'$ such that $(v', y) = \id{encrypt}^P(v, x)$. | |
728 | \end{enumerate} | |
729 | Similarly, a \emph{PRF encryption mode} $m_F = (\id{encrypt}, | |
730 | \id{decrypt})$ is a pair of deterministic oracle algorithms (and | |
731 | implicitly-defined plaintext and ciphertext spaces) which satisfy the | |
732 | following conditions: | |
733 | \begin{enumerate} | |
734 | \item The algorithm $\id{encrypt}$ runs with oracle access to a function | |
735 | $F\colon \Bin^\ell \to \Bin^L$; on input a plaintext $x$ and an | |
736 | initialization vector $v \in \Bin^\ell$, it returns a ciphertext $y$ and | |
737 | a \emph{chaining value} $v' \in \Bin^\ell$. We write $(v', y) = | |
738 | \id{encrypt}^{F(\cdot)}(v, x)$. | |
739 | \item The algorithm $\id{decrypt}$ runs with oracle access to a function | |
740 | $F\colon \Bin^\ell \to \Bin^L$; on input a ciphertext $y$ and an | |
741 | initialization vector $v \in \Bin^\ell$, it returns a plaintext $x$. We | |
742 | write that $(v', x) = \id{decrypt}^{F(\cdot)}(v, y)$. | |
743 | \item For all functions $F\colon \Bin^\ell \to \Bin^L$, all plaintexts $x$ | |
744 | and all initialization vectors $v$, if $(v', y) = | |
745 | \id{encrypt}^{F(\cdot)}(v, x)$ then $x = \id{decrypt}^{F(\cdot)}(v, y)$. | |
746 | \item There exists an efficient algorithm which, given a ciphertext $y$ and | |
747 | the initialization vector but \emph{not} access to $F$, computes the | |
748 | chaining value $v'$ such that $(v', y) = \id{encrypt}^F(v, x)$. | |
749 | \qed | |
750 | \end{enumerate} | |
751 | \end{definition} | |
752 | ||
753 | \begin{definition}[Symmetric encryption schemes from modes] | |
754 | \label{def:enc-scheme} | |
755 | Let $F$ be a pseudorandom permutation on $\Bin^\ell$ (resp.\ a | |
756 | pseudorandom function from $\Bin^\ell$ to $\Bin^L$); let $m = | |
757 | (\id{encrypt}, \id{decrypt})$ be a block cipher (resp.\ PRF) encryption | |
758 | mode. (To save on repetition, if $F$ is a PRF then define $F_K^{-1}(x) = | |
759 | \bot$ for all keys $K$ and inputs $x$.) We define the following symmetric | |
760 | encryption schemes according to how IVs are selected. | |
761 | ||
762 | \begin{itemize} | |
763 | \def\Enc{\Xid{\E}{$m$\what}^{\super}} | |
764 | \def\GG{\Xid{G}{$m$\what}^{\super}} | |
765 | \def\EE{\Xid{E}{$m$\what}^{\super}} | |
766 | \def\DD{\Xid{D}{$m$\what}^{\super}} | |
767 | ||
768 | \def\what{$\$$} | |
769 | \def\super{F} | |
770 | \item Randomized selection: define $\Enc = (\GG, \EE, \DD)$, where | |
771 | \begin{program} | |
772 | Algorithm $\GG()$: \+ \\ | |
773 | $K \getsr \keys F$; \\ | |
774 | \RETURN $K$; | |
775 | \next | |
776 | Algorithm $\EE(K, x)$: \+ \\ | |
777 | $v \getsr \Bin^\ell$; \\ | |
778 | $(v', x) \gets \id{encrypt}^{F_K(\cdot)}(v, x)$; \\ | |
779 | \RETURN $(v, y)$; | |
780 | \next | |
781 | Algorithm $\DD(K, y')$; \+ \\ | |
782 | $(v, y) \gets y'$; \\ | |
783 | $(v', x) \gets {}$ \\ | |
784 | \qquad $\id{decrypt}^{F_K(\cdot), F^{-1}_K(\cdot)}(v, y)$; \\ | |
785 | \RETURN $x$; | |
786 | \end{program} | |
787 | ||
788 | \def\what{C} | |
789 | \def\super{F, c} | |
790 | \def\imsg{\Xid{i}{msg}} | |
791 | \item Generalized counters: define $\Enc = (\GG, \EE, \DD)$, where $c$ is a | |
792 | generalized counter in $\Bin^\ell$, and | |
793 | \begin{program} | |
794 | Algorithm $\GG()$: \+ \\ | |
795 | $K \getsr \keys F$; \\ | |
796 | $\imsg \gets 0$; \\ | |
797 | \RETURN $K$; | |
798 | \next | |
799 | Algorithm $\EE(K, x)$: \+ \\ | |
800 | $i \gets c(\imsg)$; \\ | |
801 | $(v', x) \gets \id{encrypt}^{F_K(\cdot)}(i, x)$; \\ | |
802 | $\imsg \gets \imsg + 1$; \\ | |
803 | \RETURN $(i, y)$; | |
804 | \next | |
805 | Algorithm $\DD(K, y')$; \+ \\ | |
806 | $(i, y) \gets y'$; \\ | |
807 | $(v', x) \gets {}$ \\ | |
808 | \qquad $\id{decrypt}^{F_K(\cdot), F^{-1}_K(\cdot)}(i, y)$; \\ | |
809 | \RETURN $x$; | |
810 | \end{program} | |
811 | ||
812 | \def\what{E} | |
813 | \def\super{F, c} | |
814 | \def\imsg{\Xid{i}{msg}} | |
815 | \item Encrypted counters: if $L \ge \ell$, then define $\Enc = (\GG, \EE, | |
816 | \DD)$, where $c$ is a generalized counter in $\Bin^\ell$, and | |
817 | \begin{program} | |
818 | Algorithm $\GG()$: \+ \\ | |
819 | $K \getsr \keys F$; \\ | |
820 | $\imsg \gets 0$; \\ | |
821 | \RETURN $K$; | |
822 | \next | |
823 | Algorithm $\EE(K, x)$: \+ \\ | |
824 | $i \gets c(\imsg)$; \\ | |
825 | $v \gets F_K(i)[0 \bitsto \ell]$; \\ | |
826 | $(v', x) \gets \id{encrypt}^{F_K(\cdot)}(v, x)$; \\ | |
827 | $\imsg \gets \imsg + 1$; \\ | |
828 | \RETURN $(i, y)$; | |
829 | \next | |
830 | Algorithm $\DD(K, y')$; \+ \\ | |
831 | $(i, y) \gets y'$; \\ | |
832 | $v \gets F_K(i)[0 \bitsto \ell]$; \\ | |
833 | $(v', x) \gets {}$ \\ | |
834 | \qquad $\id{decrypt}^{F_K(\cdot), F^{-1}_K(\cdot)}(v, y)$; \\ | |
835 | \RETURN $x$; | |
836 | \end{program} | |
837 | (We require $L \ge \ell$ for this to be well-defined; otherwise the | |
838 | encrypted counter value is too short.) | |
839 | ||
840 | \def\what{L} | |
841 | \def\super{F, V_0} | |
842 | \def\vnext{\Xid{v}{next}} | |
843 | \item Carry-over: define $\Enc = (\GG, \EE, \DD)$ where $V_0 \in \Bin^\ell$ | |
844 | is the initialization vector for the first message, and | |
845 | \begin{program} | |
846 | Algorithm $\GG()$: \+ \\ | |
847 | $K \getsr \keys F$; \\ | |
848 | $\vnext \gets V_0$; \\ | |
849 | \RETURN $K$; | |
850 | \next | |
851 | Algorithm $\EE(K, x)$: \+ \\ | |
852 | $v \gets \vnext$; \\ | |
853 | $(v', x) \gets \id{encrypt}^{F_K(\cdot)}(v, x)$; \\ | |
854 | $\vnext \gets v'$; \\ | |
855 | \RETURN $(v, y)$; | |
856 | \next | |
857 | Algorithm $\DD(K, y')$; \+ \\ | |
858 | $(v, y) \gets y'$; \\ | |
859 | $(v', x) \gets {}$ \\ | |
860 | \qquad $\id{decrypt}^{F_K(\cdot), F^{-1}_K(\cdot)}(v, y)$; \\ | |
861 | \RETURN $x$; | |
862 | \end{program} | |
863 | \end{itemize} | |
864 | ||
865 | Note that, while the encryption algorithms of the above schemes are either | |
866 | randomized or stateful, the decryption algorithms are simple and | |
867 | deterministic. | |
868 | \end{definition} | |
869 | ||
870 | The following simple and standard result will be very useful in our proofs. | |
871 | ||
872 | \begin{proposition} | |
873 | \label{prop:enc-info-to-real} | |
874 | \leavevmode | |
875 | \begin{enumerate} | |
876 | \item Suppose that $\E^P = (G^P, E^P, D^P)$ is one of the symmetric | |
877 | encryption schemes of definition~\ref{def:enc-scheme}, constructed from a | |
878 | pseudorandom permutation $P\colon \Bin^\ell \leftrightarrow \Bin^\ell$. | |
879 | If $q$ is an upper bound on the number of PRP applications required for | |
880 | the encryption $q_E$ messages totalling $\mu_E$ bits, and $t'$ is some | |
881 | small constant, then | |
882 | \[ \InSec{lor-cpa}(\E^P; t, q_E, \mu_E) \le | |
883 | \InSec{lor-cpa}(\E^{\Perm{\ell}}; t, q_E, \mu_E) + | |
884 | 2 \cdot \InSec{prp}(P; t + q t', q) . | |
885 | \] | |
886 | \item Similarly, suppose that $\E^F = (G^F, E^F, D^F)$ is one of the | |
887 | symmetric encryption schemes of definition~\ref{def:enc-scheme}, | |
888 | constructed from a pseudorandom function $F\colon \Bin^\ell \to \Bin^L$. | |
889 | If $q$ is an upper bound on the number of PRP applications required for | |
890 | the encryption $q_E$ messages totalling $\mu_E$ bits, and $t'$ is some | |
891 | small constant, then | |
892 | \[ \InSec{lor-cpa}(\E^F; t, q_E, \mu_E) \le | |
893 | \InSec{lor-cpa}(\E^{\Func{\ell}{L}}; t, q_E, \mu_E) + | |
894 | 2 \cdot \InSec{prf}(F; t + q t', q) . | |
895 | \] | |
896 | \end{enumerate} | |
897 | \end{proposition} | |
898 | \begin{proof} | |
899 | \begin{enumerate} | |
900 | \item Let $A$ be an adversary attacking the LOR-CPA security of $\E^P$, | |
901 | which takes time $t$ and issues $q_E$ encryption queries totalling | |
902 | $\mu_E$ bits. We construct an adversary $B$ attacking the security of | |
903 | the PRP $P$ as follows. $B$ selects a random $b^* \inr \Bin$. It then | |
904 | runs $A$, simulating the LOR-CPA game by using $b^*$ to decide whether to | |
905 | encrypt the left or right plaintext, and using its oracle access to $P$ | |
906 | to do the encryption. Eventually, $A$ returns a bit $b$. If $b = b^*$, | |
907 | $B$ returns $1$ (indicating `pseudorandom'); otherwise it returns $0$. | |
908 | ||
909 | If $B$'s oracle is selected from the PRP $P$, then $B$ correctly | |
910 | simulates the LOR-CPA game for $\E^P$, and $B$ returns $1$ with | |
911 | probability precisely $(\Adv{lor-cpa}{\E^P}(A) + 1)/2$. Conversely, if | |
912 | $B$'s oracle is a random permutation, then $B$ correctly simulates the | |
913 | LOR-CPA game for $\E^{\Perm{\ell}}$, so $B$ returns $1$ with probability | |
914 | $(\Adv{lor-cpa}{\E^P}(A) + 1)/2$. Thus, we have | |
915 | \begin{eqnarray}[rl] | |
916 | \Adv{prp}{P}(B) | |
917 | & = (\Adv{lor-cpa}{\E^P}(A) + 1)/2 | |
918 | - (\Adv{lor-cpa}{\E^{\Perm{\ell}}}(A) + 1)/2 \\ | |
919 | & = (\Adv{lor-cpa}{\E^P}(A) | |
920 | - \Adv{lor-cpa}{\E^{\Perm{\ell}}}(A))/2 . | |
921 | \end{eqnarray} | |
922 | Note that the extra work which $B$ does over $A$ -- initialization, | |
923 | tidying up and encrypting messages -- is bounded by some small constant | |
924 | $t_P$ multiplied by the number of oracle queries~$q$ made by~$B$, and the | |
925 | required result follows by multiplying through by~$2$ and rearranging. | |
926 | \item The proof for this case is almost identical: merely substitute $F$ | |
927 | for $P$, `PRF' for `PRP' and $\Func{\ell}{L}$ for $\Perm{\ell}$ in the | |
928 | above. \qed | |
929 | \end{enumerate} | |
930 | \end{proof} | |
931 | ||
932 | Of course, proving theorems about each of the above schemes individually will | |
933 | be very tedious. We therefore define a `hybrid' scheme which switches | |
934 | between the above selection methods. This isn't a practical encryption | |
935 | scheme -- just a `trick' to reduce the number of complicated proofs we need | |
936 | to give. | |
937 | ||
938 | \begin{definition}[Hybrid encryption modes] | |
939 | \label{def:enc-hybrid} | |
940 | \def\Enc{\Xid{\E}{$m$\what}^{\super}_{\sub}} | |
941 | \def\GG{\Xid{G}{$m$\what}^{\super}_{\sub}} | |
942 | \def\EE{\Xid{E}{$m$\what}^{\super}_{\sub}} | |
943 | \def\DD{\Xid{D}{$m$\what}^{\super}_{\sub}} | |
944 | \def\what{H} | |
945 | \def\super{F, V_0, c} | |
946 | \def\sub{n_L, n_C, n_E} | |
947 | \def\imsg{\Xid{i}{msg}} | |
948 | \def\vnext{\Xid{v}{next}} | |
949 | Let $n_L$, $n_C$ and $n_E$ be nonnegative integers, with $n_L + n_C + n_E | |
950 | \le 2^{\ell}$; let $F$ be a pseudorandom permutation on $\Bin^\ell$ (resp.\ | |
951 | a pseudorandom function from $\Bin^\ell$ to $\Bin^L$); let $m = | |
952 | (\id{encrypt}, \id{decrypt})$ be a block cipher (resp.\ PRF) encryption | |
953 | mode let $V_0 \in \Bin^\ell$ be an initialization vector; and let $c\colon | |
954 | \Nupto{2^\ell} \to \Bin^\ell$ be a generalized counter. (Again, if $F$ is | |
955 | a PRF, we set $F_K(x) = \bot$ for all $K$ and $x$.) We define the scheme | |
956 | $\Enc = (\GG, \EE, \DD)$ as follows. | |
957 | \begin{program} | |
958 | Algorithm $\GG()$: \+ \\ | |
959 | $K \getsr \keys F$; \\ | |
960 | $\imsg \gets 0$; \\ | |
961 | $\vnext \gets V_0$; \\ | |
962 | \RETURN $K$; | |
963 | \next | |
964 | Algorithm $\DD(K, y')$; \+ \\ | |
965 | $(v, y) \gets y'$; \\ | |
966 | $(v', x) \gets \id{decrypt}^{F_K(\cdot), F^{-1}_K(\cdot)}(v, y)$; \\ | |
967 | \RETURN $x$; | |
968 | \newline | |
969 | Algorithm $\EE(K, x)$: \+ \\ | |
970 | \IF $\imsg < n_L$ \THEN $v \gets \vnext$; \\ | |
971 | \ELSE\IF $\imsg < n_L + n_C$ \THEN $v \gets c(\imsg)$; \\ | |
972 | \ELSE\IF $\imsg < n_L + n_C + n_E$ \THEN | |
973 | $v \gets F_K(c(\imsg)[0 \bitsto \ell])$; \\ | |
974 | \ELSE $v \getsr \Bin^\ell$; \\ | |
975 | $(v', x) \gets \id{encrypt}^{F_K(\cdot)}(v, x)$; \\ | |
976 | $\vnext \gets v'$; \\ | |
977 | $\imsg \gets \imsg + 1$; \\ | |
978 | \RETURN $(v, y)$; | |
979 | \end{program} | |
980 | For this to be well-defined, we require that $L \ge \ell$ or $n_E = 0$ -- | |
981 | otherwise the encrypted counter values are too short. | |
982 | \end{definition} | |
983 | ||
984 | The following proposition relates the security of our artificial hybrid | |
985 | scheme to that of the practical schemes defined in | |
986 | definition~\ref{def:enc-scheme}. | |
987 | ||
988 | \begin{proposition} | |
989 | \label{prop:enc-hybrid} | |
990 | Let $F$ be a pseudorandom permutation on $\Bin^\ell$ (resp.\ a pseudorandom | |
991 | function from $\Bin^\ell$ to $\Bin^L$); let $m$ be a block cipher (resp.\ | |
992 | PRF) encryption mode. Then: | |
993 | \begin{enumerate} | |
994 | \def\ii#1{\item $\displaystyle#1$} | |
995 | \ii{\InSec{lor-cpa}(\Xid{E}{$m\$$}^F; t, q_E, \mu_E) \le | |
996 | \InSec{lor-cpa}(\Xid{E}{$m$H}^{F, V_0, c}_{0, 0, 0}; t, q_E, \mu_E)} | |
997 | \ii{\InSec{lor-cpa}(\Xid{E}{$m$C}^{F, c}; t, q_E, \mu_E) \le | |
998 | \InSec{lor-cpa} | |
999 | (\Xid{E}{$m$H}^{F, V_0, c}_{q_E, 0, 0}; t, q_E, \mu_E)} | |
1000 | \ii{\InSec{lor-cpa}(\Xid{E}{$m$E}^{F, c}; t, q_E, \mu_E) \le | |
1001 | \InSec{lor-cpa} | |
1002 | (\Xid{E}{$m$H}^{F, V_0, c}_{0, q_E, 0}; t, q_E, \mu_E)} | |
1003 | \ii{\InSec{lor-cpa}(\Xid{E}{$m$L}^{F, V_0}; t, q_E, \mu_E) \le | |
1004 | \InSec{lor-cpa} | |
1005 | (\Xid{E}{$m$H}^{F, V_0, c}_{0, 0, q_E}; t, q_E, \mu_E)} | |
1006 | \end{enumerate} | |
1007 | \end{proposition} | |
1008 | ||
1009 | \begin{proof} | |
1010 | For 1, it suffices to note that $\Xid{E}{$m\$$}^F \equiv \Xid{E}{$m$H}^{c, | |
1011 | V_0}_{0, 0, 0}$ for any $c$, $V_0$. For the others, we observe that, while | |
1012 | the IVs returned in the ciphertexts differ, it's very easy to simulate | |
1013 | encryption for the practical schemes given an encryption oracle for the | |
1014 | hybrid scheme: for the counter and encrypted-counter schemes, the counter | |
1015 | function is public knowledge; for the carry-over scheme, the correct IV for | |
1016 | the first message is known, and the IV any subsequent messages can be | |
1017 | computed from the previous IV and ciphertext according to condition~4 in | |
1018 | definition~\ref{def:enc-scheme}. | |
1019 | \end{proof} | |
1020 | ||
1021 | %%%-------------------------------------------------------------------------- | |
1022 | ||
1023 | \section{Ciphertext block chaining (CBC) encryption} | |
1024 | \label{sec:cbc} | |
1025 | ||
1026 | \subsection{Description} | |
1027 | \label{sec:cbc-desc} | |
1028 | ||
1029 | Suppose $E$ is an $\ell$-bit pseudorandom permutation. CBC mode works as | |
1030 | follows. Given a message $X$, we divide it into $\ell$-bit blocks $x_0$, | |
1031 | $x_1$, $\ldots$, $x_{n-1}$. Choose an initialization vector $v \in | |
1032 | \Bin^\ell$. Before passing each $x_i$ through $E$, we XOR it with the | |
1033 | previous ciphertext, with $v$ standing in for the first block: | |
1034 | \begin{equation} | |
1035 | y_0 = E_K(x_0 \xor v) \qquad | |
1036 | y_i = E_K(x_i \xor y_{i-1} \ \text{(for $1 \le i < n$)}. | |
1037 | \end{equation} | |
1038 | The ciphertext is then the concatenation of $v$ and the $y_i$. Decryption is | |
1039 | simple: | |
1040 | \begin{equation} | |
1041 | x_0 = E^{-1}_K(y_0) \xor v \qquad | |
1042 | x_i = E^{-1}_K(y_i) \xor y_{i-1} \ \text{(for $1 \le i < n$)} | |
1043 | \end{equation} | |
1044 | See figure~\ref{fig:cbc} for a diagram of CBC encryption. | |
1045 | ||
1046 | \begin{figure} | |
1047 | \begin{cgraph}{cbc-mode} | |
1048 | []!{0; <0.85cm, 0cm>: <0cm, 0.5cm>::} | |
1049 | *+=(1, 0)+[F]{\mathstrut x_0}="x" | |
1050 | :[dd] *{\xor}="xor" | |
1051 | [ll] *+=(1, 0)+[F]{\mathstrut v} :"xor" | |
1052 | :[dd] *+[F]{E}="e" :[ddd] *+=(1, 0)+[F]{\mathstrut y_0}="i" | |
1053 | "e" [l] {K} :"e" | |
1054 | [rrruuuu] *+=(1, 0)+[F]{\mathstrut x_1}="x" | |
1055 | :[dd] *{\xor}="xor" | |
1056 | "e" [d] :`r [ru] `u "xor" "xor" | |
1057 | :[dd] *+[F]{E}="e" :[ddd] | |
1058 | *+=(1, 0)+[F]{\mathstrut y_1}="i" | |
1059 | "e" [l] {K} :"e" | |
1060 | [rrruuuu] *+=(1, 0)+[F--]{\mathstrut x_i}="x" | |
1061 | :@{-->}[dd] *{\xor}="xor" | |
1062 | "e" [d] :@{-->}`r [ru] `u "xor" "xor" | |
1063 | :@{-->}[dd] *+[F]{E}="e" :@{-->}[ddd] | |
1064 | *+=(1, 0)+[F--]{\mathstrut y_i}="i" | |
1065 | "e" [l] {K} :@{-->}"e" | |
1066 | [rrruuuu] *+=(1, 0)+[F]{\mathstrut x_{n-1}}="x" | |
1067 | :[dd] *{\xor}="xor" | |
1068 | "e" [d] :@{-->}`r [ru] `u "xor" "xor" | |
1069 | :[dd] *+[F]{E}="e" :[ddd] | |
1070 | *+=(1, 0)+[F]{\mathstrut y_{n-1}}="i" | |
1071 | "e" [l] {K} :"e" | |
1072 | \end{cgraph} | |
1073 | ||
1074 | \caption{Encryption using CBC mode} | |
1075 | \label{fig:cbc} | |
1076 | \end{figure} | |
1077 | ||
1078 | \begin{definition}[CBC algorithms] | |
1079 | \label{def:cbc} | |
1080 | For any permutation $P\colon \Bin^\ell \to \Bin^\ell$, any initialization | |
1081 | vector $v \in \Bin^\ell$, any plaintext $x \in \Bin^{\ell\N}$ and any | |
1082 | ciphertext $y \in \Bin^*$, we define the encryption mode $\id{CBC} = | |
1083 | (\id{cbc-encrypt}, \id{cbc-decrypt})$ as follows: | |
1084 | \begin{program} | |
1085 | Algorithm $\id{cbc-encrypt}^{P(\cdot)}(v, x)$: \+ \\ | |
1086 | $y \gets \emptystring$; \\ | |
1087 | \FOR $i = 0$ \TO $|x|/\ell$ \DO \\ \ind | |
1088 | $x_i \gets x[\ell i \bitsto \ell (i + 1)]$; \\ | |
1089 | $y_i \gets P(x_i \xor v)$; \\ | |
1090 | $v \gets y_i$; \\ | |
1091 | $y \gets y \cat y_i$; \- \\ | |
1092 | \RETURN $(v, y)$; | |
1093 | \next | |
1094 | Algorithm $\id{cbc-decrypt}^{P(\cdot), P^{-1}(\cdot)}(v, y)$: \+ \\ | |
1095 | \IF $|y| \bmod \ell \ne 0$ \THEN \RETURN $\bot$; \\ | |
1096 | $x \gets \emptystring$; \\ | |
1097 | \FOR $1 = 0$ \TO $|y|/\ell$ \DO \\ \ind | |
1098 | $y_i \gets y[\ell i \bitsto \ell (i + 1)]$; \\ | |
1099 | $x_i \gets P^{-1}(y_i) \xor v$; \\ | |
1100 | $v \gets y_i$; \\ | |
1101 | $x \gets x \cat x_i$; \- \\ | |
1102 | \RETURN $(v, x)$; | |
1103 | \end{program} | |
1104 | Now, let $c$ be a generalized counter in $\Bin^\ell$. We define the | |
1105 | encryption schemes $\Xid{E}{CBC$\$$}^P$, $\Xid{E}{CBCE}^P$ and | |
1106 | $\Xid{E}{CBCH}^{P, c, \bot}_{0, 0, n_E}$, as described in | |
1107 | definition~\ref{def:enc-scheme}. | |
1108 | \end{definition} | |
1109 | ||
1110 | \subsection{Security of CBC mode} | |
1111 | ||
1112 | We now present our main theorem on CBC mode. | |
1113 | ||
1114 | \begin{theorem}[Security of hybrid CBC mode] | |
1115 | \label{thm:cbc} | |
1116 | Let $P\colon \keys P \times \Bin^\ell \to \Bin^\ell$ be a pseudorandom | |
1117 | permutation; let $v_0 \in \Bin^\ell$ be an initialization vector; let $n_L | |
1118 | \in \{\, 0, 1 \,\}$; let $c$ be a generalized counter in $\Bin^\ell$; and | |
1119 | let $n_C \in \N$ be a nonnegative integer; and suppose that at most one of | |
1120 | $n_L$ and $n_C$ is nonzero. Then, for any $t$, $q_E \ge n$ and $\mu_E$, | |
1121 | \[ \InSec{lor-cpa} | |
1122 | (\Xid{\E}{CBCH}^{P, c, v_0}_{n_L, 0, n_E}; t, q_E, \mu_E) \le | |
1123 | 2 \cdot \InSec{prp}(P; t + q t_P, q) + \frac{q (q - 1)}{2^\ell - q} | |
1124 | \] | |
1125 | where $q = n_L + n_E + \mu_E/\ell$ and $t_P$ is some small constant. | |
1126 | \end{theorem} | |
1127 | ||
1128 | The proof of this theorem we postpone until section~\ref{sec:cbc-proof}. As | |
1129 | promised, the security of our randomized and stateful schemes follow as | |
1130 | simple corollaries. | |
1131 | ||
1132 | \begin{corollary}[Security of practical CBC modes] | |
1133 | \label{cor:cbc} | |
1134 | Let $P$ and $c$ be as in theorem~\ref{thm:cbc}. Then for any $t$, $q_E$ | |
1135 | and $\mu_E$, and some small constant $t_P$, | |
1136 | \begin{eqnarray*}[rl] | |
1137 | \InSec{lor-cpa}(\Xid{\E}{CBC$\$$}^P; t, q_E, \mu_E) | |
1138 | & \le 2 \cdot \InSec{prp}(P; t + q t_P, q) + \frac{q (q - 1)}{2^\ell - q} | |
1139 | \\ | |
1140 | \InSec{lor-cpa}(\Xid{\E}{CBCE}^{P, c}; t, q_E, \mu_E) | |
1141 | & \le 2 \cdot \InSec{prp}(P; t + q' t_P, q') + | |
1142 | \frac{q' (q' - 1)}{2^\ell - q'} | |
1143 | \\ | |
1144 | \tabpause{and} | |
1145 | \InSec{lor-cpa}(\Xid{\E}{CBCL}^{P, v_0}; t, 1, \mu_E) | |
1146 | & \le 2 \cdot \InSec{prp}(P; t + q t_P, q) + | |
1147 | \frac{q (q - 1)}{2^\ell - q} | |
1148 | \end{eqnarray*} | |
1149 | where $q = \mu_E/\ell$, and $q' = q + q_E$. | |
1150 | \end{corollary} | |
1151 | \begin{proof} | |
1152 | Follows from theorem~\ref{thm:cbc} and proposition~\ref{prop:enc-hybrid}. | |
1153 | \end{proof} | |
1154 | ||
1155 | \begin{remark} | |
1156 | The insecurity of CBC mode over that inherent in the underlying PRP is | |
1157 | essentially a birthday bound: note for $q \le 2^{\ell/2}$, our denominator | |
1158 | $2^\ell - q \approx 2^\ell$, and for larger $q$, the term $q (q - 1)/2^\ell | |
1159 | > 1$ anyway, so all security is lost (according to the above result). | |
1160 | Compared to \cite[theorem~17]{Bellare:2000:CST}, we gain the tiny extra | |
1161 | term in the denominator, but lose the PRP-as-a-PRF term $q^2 | |
1162 | 2^{-\ell-1}$.\footnote{% | |
1163 | In fact, they don't prove the stated bound of $q (3 q - 2)/2^{\ell+1}$ | |
1164 | but instead the larger $q (2 q - 1)/2^\ell$. The error is in the | |
1165 | application of their proposition~8: the PRF-insecurity term is doubled, | |
1166 | so the PRP-as-a-PRF term must be also.} % | |
1167 | \end{remark} | |
1168 | ||
1169 | \subsection{Ciphertext stealing} | |
1170 | ||
1171 | Ciphertext stealing \cite{Daemen:1995:CHF,Schneier:1996:ACP,RFC2040} allows | |
1172 | us to encrypt any message in $\Bin^*$ without the need for padding. The | |
1173 | trick is to fill in the `gap' at the end of the last block with the end bit | |
1174 | of the previous ciphertext, and then to put the remaining short penultimate | |
1175 | block at the end. Decryption proceeds by first decrypting the final block to | |
1176 | recover the remainder of the penultimate one. See | |
1177 | figure~\ref{fig:cbc-steal}. | |
1178 | ||
1179 | \begin{figure} | |
1180 | \begin{cgraph}{cbc-steal-enc} | |
1181 | []!{0; <0.85cm, 0cm>: <0cm, 0.5cm>::} | |
1182 | *+=(1, 0)+[F]{\mathstrut x_0}="x" | |
1183 | :[dd] *{\xor}="xor" | |
1184 | [ll] *+=(1, 0)+[F]{\mathstrut v} :"xor" | |
1185 | :[dd] *+[F]{E}="e" :[ddddd] *+=(1, 0)+[F]{\mathstrut y_0}="i" | |
1186 | "e" [l] {K} :"e" | |
1187 | [rrruuuu] *+=(1, 0)+[F]{\mathstrut x_1}="x" | |
1188 | :[dd] *{\xor}="xor" | |
1189 | "e" [d] :`r [ru] `u "xor" "xor" | |
1190 | :[dd] *+[F]{E}="e" :[ddddd] | |
1191 | *+=(1, 0)+[F]{\mathstrut y_1}="i" | |
1192 | "e" [l] {K} :"e" | |
1193 | [rrruuuu] *+=(1, 0)+[F--]{\mathstrut x_i}="x" | |
1194 | :@{-->}[dd] *{\xor}="xor" | |
1195 | "e" [d] :@{-->}`r [ru] `u "xor" "xor" | |
1196 | :@{-->}[dd] *+[F]{E}="e" :@{-->}[ddddd] | |
1197 | *+=(1, 0)+[F--]{\mathstrut y_i}="i" | |
1198 | "e" [l] {K} :@{-->}"e" | |
1199 | [rrruuuu] *+=(1, 0)+[F]{\mathstrut x_{n-2}}="x" | |
1200 | :[dd] *{\xor}="xor" | |
1201 | "e" [d] :@{-->}`r [ru] `u "xor" "xor" | |
1202 | :[dd] *+[F]{E}="e" | |
1203 | "e" [l] {K} :"e" | |
1204 | [rrruuuu] *+=(1, 0)+[F]{\mathstrut x_{n-1} \cat 0^{\ell-t}}="x" | |
1205 | :[dd] *{\xor}="xor" | |
1206 | "e" [d] :`r [ru] `u "xor" "xor" | |
1207 | "e" [dddddrrr] *+=(1, 0)+[F]{\mathstrut y_{n-1}[0 \bitsto t]}="i" | |
1208 | "e" [dd] ="x" | |
1209 | "i" [uu] ="y" | |
1210 | []!{"x"; "e" **{}, "x"+/4pt/ ="p", | |
1211 | "x"; "y" **{}, "x"+/4pt/ ="q", | |
1212 | "y"; "x" **{}, "y"+/4pt/ ="r", | |
1213 | "y"; "i" **{}, "y"+/4pt/ ="s", | |
1214 | "e"; | |
1215 | "p" **\dir{-}; | |
1216 | "q" **\crv{"x"}; | |
1217 | "r" **\dir{-}; | |
1218 | "s" **\crv{"y"}; | |
1219 | "i" **\dir{-}?>*\dir{>}} | |
1220 | "xor" :[dd] *+[F]{E}="e" | |
1221 | "e" [l] {K} :"e" | |
1222 | "e" [dddddlll] *+=(1, 0)+[F]{\mathstrut y_{n-2}}="i" | |
1223 | "e" [dd] ="x" | |
1224 | "i" [uu] ="y" | |
1225 | []!{"x"; "e" **{}, "x"+/4pt/ ="p", | |
1226 | "x"; "y" **{}, "x"+/4pt/ ="q", | |
1227 | "y"; "x" **{}, "y"+/4pt/ ="r", | |
1228 | "y"; "i" **{}, "y"+/4pt/ ="s", | |
1229 | "x"; "y" **{} ?="c" ?(0.5)/-4pt/ ="cx" ?(0.5)/4pt/ ="cy", | |
1230 | "e"; | |
1231 | "p" **\dir{-}; | |
1232 | "q" **\crv{"x"}; | |
1233 | "cx" **\dir{-}; | |
1234 | "c" *[@]\cir<4pt>{d^u}; | |
1235 | "cy"; | |
1236 | "r" **\dir{-}; | |
1237 | "s" **\crv{"y"}; | |
1238 | "i" **\dir{-}?>*\dir{>}} | |
1239 | \end{cgraph} | |
1240 | ||
1241 | \begin{cgraph}{cbc-steal-dec} | |
1242 | []!{0; <0.85cm, 0cm>: <0cm, 0.5cm>::} | |
1243 | *+=(1, 0)+[F]{\mathstrut y_0}="y" | |
1244 | :[ddddddd] *+[F]{D}="d" [l] {K} :"d" | |
1245 | [rrrdd] *{\xor} ="nx" "d" [u] :`r [rd] `d "nx" "nx" | |
1246 | "d" :[dd] *{\xor} ="xor" [ll] *+=(1, 0)+[F]{\mathstrut v} :"xor" | |
1247 | :[dd] *+=(1, 0)+[F]{\mathstrut x_0} "nx"="xor" | |
1248 | "y" [rrr] *+=(1, 0)+[F]{\mathstrut y_1}="y" | |
1249 | :[ddddddd] *+[F]{D}="d" [l] {K} :"d" | |
1250 | [rrrdd] *{\xor} ="nx" "d" [u] :@{-->}`r [rd] `d "nx" "nx" | |
1251 | "d" :"xor" | |
1252 | :[dd] *+=(1, 0)+[F]{\mathstrut x_1} "nx"="xor" | |
1253 | "y" [rrr] *+=(1, 0)+[F--]{\mathstrut y_i}="y" | |
1254 | :@{-->}[ddddddd] *+[F]{D}="d" [l] {K} :"d" | |
1255 | [rrrdd] *{\xor} ="nx" "d" [u] :@{-->}`r [rd] `d "nx" "nx" | |
1256 | "d" :"xor" | |
1257 | :[dd] *+=(1, 0)+[F--]{\mathstrut x_i} "nx"="xor" | |
1258 | "y" [rrr] *+=(1, 0)+[F]{\mathstrut y_{n-2}}="y" | |
1259 | [dddddrrr] *+[F]{D}="d" [r] {K} :"d" | |
1260 | "y" [dd] ="x" | |
1261 | "d" [uu] ="e" | |
1262 | []!{"x"; "y" **{}, "x"+/4pt/ ="p", | |
1263 | "x"; "e" **{}, "x"+/4pt/ ="q", | |
1264 | "e"; "x" **{}, "e"+/4pt/ ="r", | |
1265 | "e"; "d" **{}, "e"+/4pt/ ="s", | |
1266 | "y"; | |
1267 | "p" **\dir{-}; | |
1268 | "q" **\crv{"x"}; | |
1269 | "r" **\dir{-}; | |
1270 | "s" **\crv{"e"}; | |
1271 | "d" **\dir{-}?>*\dir{>}} | |
1272 | "d" :[dd] {z} ="z" | |
1273 | "z" [llluu] *{\xor} ="x1" | |
1274 | "z" :`l [lu] `u "x1" |*+{\scriptstyle 0^t \cat z[t \bitsto \ell]} "x1" | |
1275 | "z" :[dd] *{\xor} ="xor2" | |
1276 | :[dd] *+[F]{\mathstrut x_{n-1}[0 \bitsto t]} | |
1277 | "y" [rrr] *+=(1, 0)+[F]{\mathstrut y_{n-1} \cat 0^{\ell-t}}="y" | |
1278 | [dd] ="x" | |
1279 | "d" [llluu] ="e" | |
1280 | []!{"x"; "y" **{}, "x"+/4pt/ ="p", | |
1281 | "x"; "e" **{}, "x"+/4pt/ ="q", | |
1282 | "e"; "x" **{}, "e"+/4pt/ ="r", | |
1283 | "e"; "x1" **{}, "e"+/4pt/ ="s", | |
1284 | "x"; "e" **{} ?="c" ?(0.5)/-4pt/ ="cx" ?(0.5)/4pt/ ="cy", | |
1285 | "y"; | |
1286 | "p" **\dir{-}; | |
1287 | "q" **\crv{"x"}; | |
1288 | "cx" **\dir{-}; | |
1289 | "c" *[@]\cir<4pt>{d^u}; | |
1290 | "cy"; | |
1291 | "r" **\dir{-}; | |
1292 | "s" **\crv{"e"}; | |
1293 | "x1" **\dir{-}?>*\dir{>}} | |
1294 | "x1" [d] :`r [rd] `d "xor2" "xor2" | |
1295 | "x1" :[dd] *+[F]{D}="d" [l] {K} :"d" | |
1296 | "d" :"xor" | |
1297 | :[dd] *+[F]{\mathstrut x_{n-2}} | |
1298 | \end{cgraph} | |
1299 | ||
1300 | \caption{Encryption and decryption using CBC mode with ciphertext stealing} | |
1301 | \label{fig:cbc-steal} | |
1302 | \end{figure} | |
1303 | ||
1304 | Encrypting messages shorter than the block involves `IV stealing', which is a | |
1305 | grotty hack but works fine if IVs are random; if the IVs are encrypted | |
1306 | counters then there's nothing (modifiable) to steal from. | |
1307 | ||
1308 | We formally present a description of a randomized CBC stealing mode. | |
1309 | ||
1310 | \begin{definition}[CBC mode with ciphertext stealing] | |
1311 | \label{def:cbc-steal} | |
1312 | Let $P\colon \keys P \times \Bin^\ell \to \Bin^\ell$ be a pseudorandom | |
1313 | permutation. Let $c$ be a generalized counter on $\Bin^\ell$. We define | |
1314 | the randomized symmetric encryption scheme | |
1315 | $\Xid{\E}{CBC$\$$-steal}^P = (\Xid{G}{CBC$\$$-steal}^P, | |
1316 | \Xid{E}{CBC$\$$-steal}^P, \Xid{D}{CBC\$-steal}^P)$ for messages in $\Bin^*$ | |
1317 | as follows: | |
1318 | \begin{program} | |
1319 | Algorithm $\Xid{G}{CBC$\$$-steal}^P()$: \+ \\ | |
1320 | $K \getsr \keys P$; \\ | |
1321 | \RETURN $K$; | |
1322 | \- \\[\medskipamount] | |
1323 | Algorithm $\Xid{E}{CBC$\$$-steal}^P(K, x)$: \+ \\ | |
1324 | $t \gets |x| \bmod \ell$; \\ | |
1325 | \IF $t \ne 0$ \THEN $x \gets x \cat 0^{\ell-t}$; \\ | |
1326 | $v \getsr \Bin^\ell$; \\ | |
1327 | $y \gets v \cat \id{cbc-encrypt}(K, v, x)$; \\ | |
1328 | \IF $t \ne 0$ \THEN \\ \ind | |
1329 | $b \gets |y| - 2\ell$; \\ | |
1330 | $y \gets $\=$y[0 \bitsto b] \cat | |
1331 | y[b + \ell \bitsto |y|] \cat {}$ \\ | |
1332 | \>$y[b \bitsto b + t]$; \- \\ | |
1333 | \RETURN $y$; | |
1334 | \next | |
1335 | Algorithm $\Xid{D}{CBC$\$$-steal}^P(K, y)$: \+ \\ | |
1336 | \IF $|y| < \ell$ \THEN \RETURN $\bot$; \\ | |
1337 | $v \gets y[0 \bitsto \ell]$; \\ | |
1338 | $t = |y| \bmod \ell$; \\ | |
1339 | \IF $t \ne 0$ \THEN \\ \ind | |
1340 | $b \gets |y| - t - \ell$; \\ | |
1341 | $z \gets P^{-1}_K(y[b \bitsto b + \ell])$; \\ | |
1342 | $y \gets $\=$y[0 \bitsto b] \cat | |
1343 | y[b + \ell \bitsto |y|] \cat {}$ \\ | |
1344 | \>$z[t \bitsto \ell]$; \- \\ | |
1345 | $x \gets \id{cbc-decrypt}(K, v, y[\ell \bitsto |y|])$; \\ | |
1346 | \IF $t \ne 0$ \THEN \\ \ind | |
1347 | $x \gets x \cat z[0 \bitsto t] \xor y[b \bitsto b + t]$; \- \\ | |
1348 | \RETURN $x$; | |
1349 | \end{program} | |
1350 | \end{definition} | |
1351 | ||
1352 | The security of ciphertext stealing follows directly from the definition and | |
1353 | the security CBC mode. | |
1354 | ||
1355 | \begin{corollary}[Security of CBC with ciphertext stealing] | |
1356 | \label{cor:cbc-steal} | |
1357 | Let $P\colon \keys P \times \Bin^\ell \to \Bin^\ell$ be a pseudorandom | |
1358 | permutation. Then | |
1359 | \begin{eqnarray*}[rl] | |
1360 | \InSec{lor-cpa}(\Xid{\E}{CBC$\$$-steal}; t, q_E, \mu_E) | |
1361 | & \le \InSec{lor-cpa} | |
1362 | (\Xid{\E}{CBC$\$$}; t, q_E, \mu_E + q_E (\ell - 1)) \\ | |
1363 | & \le 2 \cdot \InSec{prp}(P; t + q t_P, q) + | |
1364 | \frac{q (q - 1)}{2^\ell - 2^{\ell/2}} | |
1365 | \end{eqnarray*} | |
1366 | where $q = \bigl\lfloor \bigl(\mu_E + q_E (\ell - 1)\bigr)/\ell | |
1367 | \bigr\rfloor$ and $t_P$ is some small constant. | |
1368 | \end{corollary} | |
1369 | ||
1370 | \begin{proof} | |
1371 | From the definition, we see that the encryption algorithm | |
1372 | $\Xid{E}{CBC-steal}$ simply pads a plaintext, encrypts it as for standard | |
1373 | CBC mode, and postprocesses the ciphertext. Hence, if $A$ is any adversary | |
1374 | attacking $\Xid{\E}{CBC-steal}$, we can construct an adversary | |
1375 | $A'$ which simply runs $A$ except that, on each query to the encryption | |
1376 | oracle, it pads both plaintexts, queries its CBC oracle, postprocesses the | |
1377 | ciphertext returned, and gives the result back to $A$. The fact that | |
1378 | plaintexts can now be up to $\ell - 1$ bits shorter than the next largest | |
1379 | whole number of blocks means that $B$ submits no more than $\mu_E + q_E | |
1380 | (\ell - 1)$ bits of plaintext to its oracle. The required result | |
1381 | follows then directly from theorem~\ref{thm:cbc}. | |
1382 | \end{proof} | |
1383 | ||
1384 | \subsection{Proof of theorem~\ref{thm:cbc}} | |
1385 | \label{sec:cbc-proof} | |
1386 | ||
1387 | The techniques and notation used in this proof will also be found in several | |
1388 | of the others. We recommend that readers try to follow this one carefully. | |
1389 | ||
1390 | We begin considering CBC mode using a completely random permutation. To | |
1391 | simplify notation slightly, we shall write $n = n_L + n_E$. Our main goal is | |
1392 | to prove the claim that there exists a garbage-emitter $W$ such that | |
1393 | \[ | |
1394 | \InSec{rog-cpa-$W$} | |
1395 | (\Xid{\E}{CBCH}^{\Perm{\ell}, c, v_0}_{n_L, 0, n_E}; | |
1396 | t, q_E, \mu_E) \le | |
1397 | \frac{q (q - 1)}{2 \cdot (2^\ell - n)}. | |
1398 | \] | |
1399 | From this, we can apply proposition~\ref{prop:rog-and-lor} to obtain | |
1400 | \[ | |
1401 | \InSec{lor-cpa} | |
1402 | (\Xid{\E}{CBCH}^{\Perm{\ell}, c, \bot}_{0, 0, n}; | |
1403 | t, q_E, \mu_E) \le | |
1404 | \frac{q (q - 1)}{2^\ell - n}. | |
1405 | \] | |
1406 | and, noting that there are precisely $q = \mu_E/\ell$ PRP-applications, we | |
1407 | apply proposition~\ref{prop:enc-info-to-real} to obtain the required result. | |
1408 | ||
1409 | Our garbage-emitter $W$ is a bit complicated. It chooses random but | |
1410 | \emph{distinct} blocks for the `ciphertext'; for the IVs, it uses $v_0$ for | |
1411 | the first message if $n_L = 1$, and otherwise it chooses random blocks | |
1412 | distinct from each other and the `ciphertext' blocks for the next $n_E$ | |
1413 | messages, and just random blocks for subsequent ones. The algorithm $W$ is | |
1414 | shown in figure~\ref{fig:cbc-garbage}. | |
1415 | ||
1416 | \begin{figure} | |
1417 | \begin{program} | |
1418 | Initialization: \+ \\ | |
1419 | $i \gets 0$; \\ | |
1420 | $\id{gone} \gets \emptyset$; | |
1421 | \- \\[\medskipamount] | |
1422 | Function $\id{fresh}()$ \+ \\ | |
1423 | $x \getsr \Bin^\ell \setminus \id{gone}$; \\ | |
1424 | $\id{gone} \gets \id{gone} \cup \{ x \}$; \\ | |
1425 | \RETURN $x$; | |
1426 | \next | |
1427 | Garbage emitter $W(m)$: \+ \\ | |
1428 | \IF $i \ge 2^\ell$ \THEN \ABORT; \\ | |
1429 | \IF $i < n_L$ \THEN $v \gets v_0$; \\ | |
1430 | \ELSE \IF $i < n$ \THEN $v \gets \id{fresh}()$; \\ | |
1431 | $i \gets i + 1$ \\ | |
1432 | \ELSE $v \getsr \Bin^\ell$; \\ | |
1433 | $y \gets \emptystring$; \\ | |
1434 | \FOR $j = 0$ \TO $m/\ell$; \\ \ind | |
1435 | $y_j \gets \id{fresh}()$; \\ | |
1436 | $y \gets y \cat y_j$; \- \\ | |
1437 | \RETURN $(v, y)$; | |
1438 | \end{program} | |
1439 | ||
1440 | \caption{Garbage emitter $W$ for CBC mode} | |
1441 | \label{fig:cbc-garbage} | |
1442 | \end{figure} | |
1443 | ||
1444 | Fortunately, it doesn't need to be efficient: the above simulations only need | |
1445 | to be able to do the LOR game, not the ROG one. The unpleasant-sounding | |
1446 | \ABORT\ only occurs after $2^\ell$ queries. If that happens we give up and | |
1447 | say the adversary won anyway: the claim is trivially true by this point, | |
1448 | since the adversary's maximum advantage is 1. | |
1449 | ||
1450 | Now we show that this lash-up is a good imitation of CBC encryption to | |
1451 | someone who doesn't know the key. The intuition works like this: every time | |
1452 | we query a random permutation at a new, fresh input value, we get a new, | |
1453 | different, random output value; conversely, if we repeat an input, we get the | |
1454 | same value out as last time. So, in the real `result' CBC game, if all the | |
1455 | permutation inputs are distinct, it looks just like the garbage emitted by | |
1456 | $W$. Unfortunately, that's not quite enough: the adversary can work out what | |
1457 | the permutation inputs ought to be and spot when there ought to have been a | |
1458 | collision but wasn't. So we'll show that, provided all the $P$-inputs -- | |
1459 | values which \emph{would} be input to the permutation if we're playing that | |
1460 | game -- are distinct, the two games look identical. | |
1461 | ||
1462 | We need some notation to describe the values in the game. Let $c_i = c(i)$ | |
1463 | be the $i$th counter value, for $0 \le i < n_E$, and let $v_i$ be the $i$th | |
1464 | initialization vector, where $v_0$ is as given if $n_L = 1$, $v_i = P(c_i - | |
1465 | n_L)$ if $n_L \le i < n$, and $v_i \inr \Bin^\ell$ if $n \le i < q_E$. Let | |
1466 | $q' = \mu_E/\ell = q - n$ be the total number of plaintext blocks in the | |
1467 | adversary's queries, let $x_i$ be the $i$th plaintext block queried, let | |
1468 | $y_i$ be the $i$th ciphertext block returned, let | |
1469 | \[ w_i = \begin{cases} | |
1470 | v_j & if block $i$ is the first block of the $j$th query, and \\ | |
1471 | y_{i-1} & otherwise | |
1472 | \end{cases} \] | |
1473 | and let $z_i = x_i \xor w_i$, all for $0 \le i < q'$. This is summarized | |
1474 | diagramatically in figure~\ref{fig:cbc-proof-notation}. The $P$-inputs are | |
1475 | now precisely the $c_i$ and the $z_i$. We'll denote probabilities in the | |
1476 | `result' game as $\Pr_R[\cdot]$ and in the `garbage' game as $\Pr_G[\cdot]$. | |
1477 | ||
1478 | \begin{figure} | |
1479 | \begin{vgraphs} | |
1480 | \begin{vgraph}{cbc-notation-a} | |
1481 | [] !{<1.33cm, 0cm>: <0cm, 1cm>::} | |
1482 | {c_i} :[r] *+[F]{E}="e" [u] {K} :"e" :[r] {v_i} | |
1483 | \end{vgraph} | |
1484 | \begin{vgraph}{cbc-notation-b} | |
1485 | [] !{<1.33cm, 0cm>: <0cm, 1cm>::} | |
1486 | {x_i} :[r] *{\xor} ="xor" :[r] {z_i} | |
1487 | :[r] *+[F]{E}="e" [u] {K} :"e" :[r] {y_i} | |
1488 | "xor" [u] {w_i} ="w" :"xor" | |
1489 | "w" [lu] {v_j} ="v" :"w" | |
1490 | "w" [ru] {y_{i-1}} ="y" :"w" | |
1491 | "v" :@{.}|-*+\hbox{or} "y" | |
1492 | \end{vgraph} | |
1493 | \end{vgraphs} | |
1494 | ||
1495 | \caption{Notation for the proof of theorem~\ref{thm:cbc}.} | |
1496 | \label{fig:cbc-proof-notation} | |
1497 | \end{figure} | |
1498 | ||
1499 | Let $C_r$ be the event, in either game, that $z_i = z_j$ for some $0 \le i < | |
1500 | j < r$, or that $z_i = c_j$ for some $0 \le i < r$ and some $0 \le j < n_E$. | |
1501 | We need to bound the probability that $C_{q'}$ occurs in both the `result' | |
1502 | and `garbage' games. We'll do this inductively. By the definition, | |
1503 | $\Pr_R[C_0] = \Pr_G[C_0] = 0$. | |
1504 | ||
1505 | Firstly, tweak the games so that all of the IVs corresponding to counters are | |
1506 | chosen at the beginning, instead of as we go along. Obviously this doesn't | |
1507 | make any difference to the adversary's view of the proceedings, but it makes | |
1508 | our analysis easier. | |
1509 | ||
1510 | Let's assume that $C_r$ didn't happen; we want the probability that $C_{r+1}$ | |
1511 | did, which is obviously just the probability that $z_r$ collides with some | |
1512 | $z_i$ for $0 \le i < r$ or some $c_i$ for $0 \le i < n$. At this point, the | |
1513 | previous $z_i$ are fixed. So: | |
1514 | \begin{equation} | |
1515 | \label{eq:cbc-coll} | |
1516 | \Pr[C_{r+1} | \bar{C}_r] | |
1517 | = \sum_{z \in \Bin^\ell} \biggl( | |
1518 | \sum_{0\le i<n} \Pr[z = c_i] + | |
1519 | \sum_{0\le i<r} \Pr[z = z_i] | |
1520 | \biggr) \cdot \Pr[z_r = z] | |
1521 | \end{equation} | |
1522 | Now note that $z_r = w_r \xor x_r$. We've no idea how $x_r$ was chosen; but, | |
1523 | one of the following cases holds. | |
1524 | \begin{enumerate} | |
1525 | \item If $x_r$ is the first block of the first plaintext, i.e., $r = 0$, and | |
1526 | $n_L = 1$, then $w_r = v_0$. However, in this case we know that $n_E = 0$ | |
1527 | by hypothesis. There are no $z_i$ which $z_r$ might collide with, so the | |
1528 | probability of a collision is zero. | |
1529 | \item If $x_r$ is the first block of plaintext $i$, and $0 \le i < n$, then | |
1530 | $w_r = v_i$, and was chosen at random from a set of $2^\ell - i \le 2^\ell | |
1531 | - n \le 2^\ell - n - r$ possibilities, either by our random permutation or | |
1532 | by $W$. We know $x_r$ is independent of $w_r$ because none of the previous | |
1533 | $P$-inputs were equal to $c_i$, by our assumption of $\bar{C}_r$. | |
1534 | \item If $x_r$ is the first block of plaintext $i$, and $n \le i < q'$, then | |
1535 | $w_r = v_i$, and was chosen at random from all $2\ell$ possible $\ell$-bit | |
1536 | blocks. We know $x_r$ is independent of $w_r$ because we just chose $w_r$ | |
1537 | at random, after $x_r$ was chosen. | |
1538 | \item Otherwise, $x_r$ is a subsequent block in some message, and $w_r = | |
1539 | y_{r-1}$, and was chosen at random from a set of $2^\ell - n - r$ | |
1540 | possibilities, either by our random permutation or by $W$. We know $x_r$ | |
1541 | is independent of $w_r$ because $z_{r-1}$ is a new $P$-input, by our | |
1542 | assumption of $\bar{C}_r$. | |
1543 | \end{enumerate} | |
1544 | So, except in case~1, which isn't a problem anyway, $w_r$ is independent of | |
1545 | $x_r$, and chosen uniformly at random from a set of at least $2^\ell - r - n$ | |
1546 | elements, in either game -- so we can already see that $\Pr_R[C_i] = | |
1547 | \Pr_G[C_i]$ for any $i \ge 0$. Finally, the $z_i$ and $c_i$ are all | |
1548 | distinct, so the $z_i \xor x$ and $c_i \xor x$ must all be distinct, for any | |
1549 | fixed $x$. So: | |
1550 | \begin{eqnarray}[rl] | |
1551 | \Pr[C_{r+1} | \bar{C}_r] | |
1552 | & = \sum_{x \in \Bin^\ell} \biggl( | |
1553 | \sum_{0\le i<n} \Pr[w_r = x \xor c_i] + | |
1554 | \sum_{0\le i<r} \Pr[w_r = x \xor z_i] | |
1555 | \biggr) \cdot \Pr[x_r = x] \\ | |
1556 | & \le \sum_{x \in \Bin^\ell} \frac{r + n}{2^\ell - r - n} \Pr[x_r = x] | |
1557 | = \frac{r + n}{2^\ell - r - n} \sum_{x \in \Bin^\ell} \Pr[x_r = x] \\ | |
1558 | & = \frac{r + n}{2^\ell - r - n}. | |
1559 | \end{eqnarray} | |
1560 | Now we're almost home. All the $c_i$ and $z_i$ are distinct; all the $v_i$ | |
1561 | and $y_i$ are random, assuming $C_{q'}$. We can bound $\Pr[C_{q'}]$ thus: | |
1562 | \begin{equation} | |
1563 | \Pr[C_{q'}] | |
1564 | \le \sum_{0<i\le q'} \Pr[C_i | \bar{C}_{i-1}] | |
1565 | \le \sum_{0\le i\le q'} \frac{i + n - 1}{2^\ell - i - n + 1} | |
1566 | \end{equation} | |
1567 | Now, let $i' = i + n - 1$. Then | |
1568 | \begin{equation} | |
1569 | \Pr[C_{q'}] | |
1570 | \le \sum_{n-1\le i'\le q'+n-1} \frac{i'}{2^\ell - i'} | |
1571 | \le \sum_{0\le i'<q} \frac{i'}{2^\ell - q} | |
1572 | = \frac{q (q - 1)}{2 \cdot (2^\ell - q)} | |
1573 | \end{equation} | |
1574 | ||
1575 | Finally, let $R$ be the event that the adversary returned 1 at the end of the | |
1576 | game -- indicating a guess of `result'. Then, noting as we have, that | |
1577 | $\Pr_R[C_{q'}] = \Pr_G[C_{q'}]$, we get this: | |
1578 | \begin{eqnarray}[rl] | |
1579 | \Adv{rog-cpa-$W$}{\Xid{\E}{CBCH}^{P, c, n}}(A) | |
1580 | & = \Pr_R[R] - \Pr_G[R] \\ | |
1581 | & \begin{eqnalign}[rLl][b] | |
1582 | {} = & (\Pr_R[R | C_{q'}] \Pr_R[C_{q'}] + | |
1583 | \Pr_R[R | \bar{C}_{q'}] \Pr_R[\bar{C}_{q'}]) - {} \\ | |
1584 | & & (\Pr_G[R | C_{q'}] \Pr_R[C_{q'}] + | |
1585 | \Pr_G[R | \bar{C}_{q'}] \Pr_G[\bar{C}_{q'}]) | |
1586 | \end{eqnalign} \\ | |
1587 | & = \Pr_R[R | C_{q'}] \Pr_R[C_{q'}] - \Pr_G[R | C_{q'}] \Pr_G[C_{q'}] \\ | |
1588 | & \le \Pr[C_{q'}] \le \frac{q (q - 1)}{2 \cdot (2^\ell - q)} | |
1589 | \end{eqnarray} | |
1590 | And we're done! | |
1591 | \qed | |
1592 | ||
1593 | %%%-------------------------------------------------------------------------- | |
1594 | ||
1595 | \section{Ciphertext feedback (CFB) encryption} | |
1596 | \label{sec:cfb} | |
1597 | ||
1598 | \subsection{Description} | |
1599 | \label{sec:cfb-desc} | |
1600 | ||
1601 | Suppose $F$ is an $\ell$-bit-to-$L$-bit pseudorandom function, and let $t \le | |
1602 | L$. CFB mode works as follows. Given a message $X$, we divide it into | |
1603 | $t$-bit blocks $x_0$, $x_1$, $\ldots$, $x_{n-1}$. Choose an initialization | |
1604 | vector $v \in \Bin^\ell$. We maintain a \emph{shift register} $s_i$, whose | |
1605 | initial value is $v$. To encrypt a block $x_i$, we XOR it with the result of | |
1606 | passing the shift register through the PRF, forming $y_i$, and then update | |
1607 | the shift register by shifting in the ciphertext block $y_i$. That is, | |
1608 | \begin{equation} | |
1609 | s_0 = v \qquad | |
1610 | y_i = x_i \xor F_K(s_i) \qquad | |
1611 | s_{i+1} = s_i \shift{t} y_i \ \text{(for $0 \le i < n$)}. | |
1612 | \end{equation} | |
1613 | Decryption follows from noting that $x_i = y_i \xor F_K(s_i)$. See | |
1614 | figure~\ref{fig:cfb} for a diagrammatic representation. | |
1615 | ||
1616 | Also, we observe that the final plaintext block needn't be $t$ bits long: we | |
1617 | can pad it out to $t$ bits and truncate the result without affecting our | |
1618 | ability to decrypt. | |
1619 | ||
1620 | \begin{figure} | |
1621 | \begin{cgraph}{cfb-mode} | |
1622 | [] !{<0.425cm, 0cm>: <0cm, 0.5cm>::} | |
1623 | *+=(2, 0)+[F]{\mathstrut v} ="v" :|<>(0.35)@{/}_<>(0.35){\ell}[rrrrr] | |
1624 | *+[o][F]{\shift{t}} ="shift" | |
1625 | [ll] :|-@{/}^-{\ell}[dd] *+[F]{E} ="e" [ll] {K} :"e" | |
1626 | :|-@{/}^-{t}[dd] *{\xor} ="xor" | |
1627 | [lll] *+=(2, 0)+[F]{\mathstrut x_0} :|-@{/}_-{t} "xor" | |
1628 | :|-@{/}^-{t}[ddd] *+=(2, 0)+[F]{\mathstrut y_0} | |
1629 | "xor" [d] :`r "shift" "shift"|-@{/}_-{t} | |
1630 | :|-@{/}_-{\ell}[rrrrrrr] *+[o][F]{\shift{t}} ="shift" | |
1631 | [ll] :|-@{/}^-{\ell}[dd] *+[F]{E} ="e" [ll] {K} :"e" | |
1632 | :|-@{/}^-{t}[dd] *{\xor} ="xor" | |
1633 | [lll] *+=(2, 0)+[F]{\mathstrut x_1} :|-@{/}_-{t} "xor" | |
1634 | :|-@{/}^-{t}[ddd] *+=(2, 0)+[F]{\mathstrut y_1} | |
1635 | "xor" [d] :`r "shift" "shift"|-@{/}_-{t} | |
1636 | :@{-->}|-@{/}_-{\ell}[rrrrrrr] *+[o][F]{\shift{t}} ="shift" | |
1637 | [ll] :@{-->}|-@{/}^-{\ell}[dd] *+[F]{E} ="e" [ll] {K} :"e" | |
1638 | :@{-->}|-@{/}^-{t}[dd] *{\xor} ="xor" | |
1639 | [lll] *+=(2, 0)+[F--]{\mathstrut x_i} :@{-->}|-@{/}_-{t} "xor" | |
1640 | :@{-->}|-@{/}^-{t}[ddd] *+=(2, 0)+[F--]{\mathstrut y_i} | |
1641 | "xor" [d] :@{-->} `r "shift" "shift"|-@{/}_-{t} | |
1642 | [rrrrrdd] *+[F]{E} ="e" | |
1643 | "shift" :@{-->}`r "e" |-@{/}_-{\ell} "e" | |
1644 | [ll] {K} :"e" | |
1645 | :|-@{/}^-{t}[dd] *{\xor} ="xor" | |
1646 | [lll] *+=(2, 0)+[F]{\mathstrut x_{n-1}} :|-@{/}_-{t} "xor" | |
1647 | :|-@{/}^-{t}[ddd] *+=(2, 0)+[F]{\mathstrut y_{n-1}} | |
1648 | \end{cgraph} | |
1649 | ||
1650 | \caption{Encryption using CFB mode} | |
1651 | \label{fig:cfb} | |
1652 | \end{figure} | |
1653 | ||
1654 | \begin{definition}[CFB algorithms] | |
1655 | For any function $F\colon \Bin^\ell \to \Bin^t$, any initialization vector | |
1656 | $v \in \Bin^\ell$, any plaintext $x \in \Bin^*$ and any ciphertext $y \in | |
1657 | \Bin^*$, we define PRF encryption mode $\id{CFB} = (\id{cfb-encrypt}, | |
1658 | \id{cfb-decrypt})$ as follows: | |
1659 | \begin{program} | |
1660 | Algorithm $\id{cfb-encrypt}(F, v, x)$: \+ \\ | |
1661 | $s \gets v$; \\ | |
1662 | $L \gets |x|$; \\ | |
1663 | $x \gets x \cat 0^{t\lceil L/t \rceil - L}$; \\ | |
1664 | $y \gets \emptystring$; \\ | |
1665 | \FOR $i = 0$ \TO $(|x| - t')/t$ \DO \\ \ind | |
1666 | $x_i \gets x[ti \bitsto t(i + 1)]$; \\ | |
1667 | $y_i \gets x_i \xor F(s)$; \\ | |
1668 | $s \gets s \shift{t} y_i$; \\ | |
1669 | $y \gets y \cat y_i$; \- \\ | |
1670 | \RETURN $(s, y[0 \bitsto L])$; | |
1671 | \next | |
1672 | Algorithm $\id{cfb-decrypt}(F, v, y)$: \+ \\ | |
1673 | $s \gets v$; \\ | |
1674 | $L \gets |y|$; \\ | |
1675 | $y \gets y \cat 0^{t\lceil L/t \rceil - L}$; \\ | |
1676 | $x \gets \emptystring$; \\ | |
1677 | \FOR $i = 0$ \TO $(|x| - t')/t$ \DO \\ \ind | |
1678 | $y_i \gets y[ti \bitsto t(i + 1)]$; \\ | |
1679 | $x_i \gets x_i \xor F(s)$; \\ | |
1680 | $s \gets s \shift{t} y_i$; \\ | |
1681 | $x \gets x \cat x_i$; \- \\ | |
1682 | \RETURN $x[0 \bitsto L]$; | |
1683 | \end{program} | |
1684 | We now define the schemes $\Xid{\E}{CFB$\$$}^F$, | |
1685 | $\Xid{\E}{CFBC}^{F, c}$, $\Xid{\E}{CFBE}^{F, c}$, and | |
1686 | $\Xid{\E}{CFBL}^{F, V_0}$ according to | |
1687 | definition~\ref{def:enc-scheme}; and we define the hybrid scheme | |
1688 | $\Xid{\E}{CFBH}^{F, V_0, c}_{n_L, n_C, n_E}$ according to | |
1689 | definition~\ref{def:enc-hybrid}. | |
1690 | \end{definition} | |
1691 | ||
1692 | \subsection{Security of CFB mode} | |
1693 | ||
1694 | %% I suspect David will want to put some negative results here, and complain | |
1695 | %% about Alkassar et al.'s alleged proof. I'll press on with the positive | |
1696 | %% stuff. | |
1697 | %% | |
1698 | %% The problems come when $t < \ell$. Then C-mode isn't necessarily secure | |
1699 | %% (well, we get a similar bound with $t$ instead of $\ell$, which isn't very | |
1700 | %% impressive). The L-mode needs careful selection of the initial IV. | |
1701 | ||
1702 | \begin{definition}[Sliding strings] | |
1703 | \label{def:slide} | |
1704 | We say that an $\ell$-bit string $x$ \emph{$t$-slides} if there exist | |
1705 | integers $i$ and $j$ such that $0 \le j < i < \ell/t$ and $x[i t \bitsto | |
1706 | \ell] = x[j t \bitsto \ell - (i - j) t]$. | |
1707 | \end{definition} | |
1708 | \begin{remark} | |
1709 | For all $\ell > 0$ and $t < \ell$, the string $0^{\ell-1} 1$ does not | |
1710 | $t$-slide. | |
1711 | \end{remark} | |
1712 | ||
1713 | \begin{theorem}[Security of CFB mode] | |
1714 | \label{thm:cfb} | |
1715 | Let $F$ be a pseudorandom function from $\Bin^\ell$ to $\Bin^t$; let $V_0 | |
1716 | \in \Bin^\ell$ be a non-$t$-sliding string; let $c$ be a generalized | |
1717 | counter in $\Bin^\ell$; and let $n_L$, $n_C$, $n_E$ and $q_E$ be | |
1718 | nonnegative integers; and furthermore suppose that | |
1719 | \begin{itemize} | |
1720 | \item $n_L + n_C + n_E \le q_E$, | |
1721 | \item $n_L = 0$, or $n_C = n_E = 0$, or $\ell \le t$ and $V_0 \ne c(i)$ | |
1722 | for any $n_L \le i < n_L + n_C + n_E$, and | |
1723 | \item either $n_C = 0$ or $\ell \le t$. | |
1724 | \end{itemize} | |
1725 | Then, for any $t_E$ and $\mu_E$, and whenever | |
1726 | we have | |
1727 | \[ \InSec{lor-cpa}(\Xid{\E}{CFBH}^{F, V_0, c}_{n_L, n_C, n_E}; | |
1728 | t_E, q_E, \mu_E) \le | |
1729 | 2 \cdot \InSec{prf}(F; t_E + q t_F, q) + \frac{q (q - 1)}{2^\ell} | |
1730 | \] | |
1731 | where $q = \bigl\lfloor \bigl(\mu_E + q_E (t - 1)\bigr)/t \bigr\rfloor + | |
1732 | n_E$, and $t_F$ is some small constant. | |
1733 | \end{theorem} | |
1734 | ||
1735 | The proof is a bit involved; we postpone it until | |
1736 | section~\ref{sec:cfb-proof}. | |
1737 | ||
1738 | \begin{corollary} | |
1739 | \label{cor:cfb-prf} | |
1740 | Let $F$, $c$ and $V_0$ be as in theorem~\ref{thm:cfb}. Then for any $t_E$, | |
1741 | $q_E$ and $\mu_E$, | |
1742 | \begin{eqnarray*}[rl] | |
1743 | \InSec{lor-cpa}(\Xid{\E}{CFB$\$$}^F; t_E, q_E, \mu_E) | |
1744 | & \le 2 \cdot \InSec{prf}(F; t_E + q t_F, q) + \frac{q (q - 1)}{2^\ell} | |
1745 | \\ | |
1746 | \InSec{lor-cpa}(\Xid{\E}{CFBE}^{F, c}; t_E, q_E, \mu_E) | |
1747 | & \le 2 \cdot \InSec{prf}(F; t_E + q' t_F, q') + | |
1748 | \frac{q' (q' - 1)}{2^\ell} | |
1749 | \\ | |
1750 | \InSec{lor-cpa}(\Xid{\E}{CFBL}^{F, V_0}; t_E, q_E, \mu_E) | |
1751 | & \le 2 \cdot \InSec{prf}(F; t_E + q t_F, q) + \frac{q (q - 1)}{2^\ell} | |
1752 | \\ | |
1753 | \tabpause{and, if $\ell \le t$,} | |
1754 | \InSec{lor-cpa}(\Xid{\E}{CFBC}^{F, c}; t_E, q_E, \mu_E) | |
1755 | & \le 2 \cdot \InSec{prf}(F; t_E + q t_F, q) + \frac{q (q - 1)}{2^\ell} | |
1756 | \end{eqnarray*} | |
1757 | where $q = \bigl\lfloor \bigl(\mu_E + q_E (t - 1)\bigr)/t \bigr\rfloor + | |
1758 | n_E$, $q' = q + q_E$, and $t_F$ is some small constant. | |
1759 | \end{corollary} | |
1760 | \begin{proof} | |
1761 | Follows from theorem~\ref{thm:cfb} and proposition~\ref{prop:enc-hybrid}. | |
1762 | \end{proof} | |
1763 | ||
1764 | \begin{corollary} | |
1765 | \label{cor:cfb-prp} | |
1766 | Let $P$ be a pseudorandom permutation on $\Bin^\ell$, and let $c$ and $V_0$ | |
1767 | be as in theorem~\ref{thm:cfb}. Then for any $t_E$, $q_E$ and $\mu_E$, | |
1768 | \begin{eqnarray*}[rl] | |
1769 | \InSec{lor-cpa}(\Xid{\E}{CFB$\$$}^P; t_E, q_E, \mu_E) | |
1770 | & \le 2 \cdot \InSec{prp}(P; t_E + q t_F, q) + \frac{q (q - 1)}{2^\ell} | |
1771 | \\ | |
1772 | \InSec{lor-cpa}(\Xid{\E}{CFBE}^{P, c}; t_E, q_E, \mu_E) | |
1773 | & \le 2 \cdot \InSec{prp}(P; t_E + q' t_F, q') + | |
1774 | \frac{q' (q' - 1)}{2^\ell} | |
1775 | \\ | |
1776 | \InSec{lor-cpa}(\Xid{\E}{CFBL}^{P, V_0}; t_E, q_E, \mu_E) | |
1777 | & \le 2 \cdot \InSec{prp}(P; t_E + q t_F, q) + \frac{q (q - 1)}{2^\ell} | |
1778 | \\ | |
1779 | \tabpause{and, if $\ell \le t$,} | |
1780 | \InSec{lor-cpa}(\Xid{\E}{CFBC}^{P, c}; t_E, q_E, \mu_E) | |
1781 | & \le 2 \cdot \InSec{prp}(P; t_E + q t_F, q) + \frac{q (q - 1)}{2^\ell} | |
1782 | \end{eqnarray*} | |
1783 | where $q = \bigl\lfloor \bigl(\mu_E + q_E (t - 1)\bigr)/t \bigr\rfloor + | |
1784 | n_E$, $q' = q + q_E$, and $t_F$ is some small constant. | |
1785 | \end{corollary} | |
1786 | \begin{proof} | |
1787 | Follows from corollary~\ref{cor:cfb-prf} and | |
1788 | proposition~\ref{prop:prps-are-prfs}. | |
1789 | \end{proof} | |
1790 | ||
1791 | \subsection{Proof of theorem~\ref{thm:cfb}} | |
1792 | \label{sec:cfb-proof} | |
1793 | ||
1794 | Our proof follows the same lines as for CBC mode: we show the ROG-CPA | |
1795 | security of hybrid-CFB mode using an ideal random function, and then apply | |
1796 | our earlier results to complete the proof. However, the ROG-CPA result will | |
1797 | be useful later when we consider the security of OFB mode, so we shall be a | |
1798 | little more formal about defining it. | |
1799 | ||
1800 | The garbage emitter is in some sense the `perfect' one: it emits a `correct' | |
1801 | IV followed by a uniform random string of the correct length. | |
1802 | ||
1803 | \begin{definition}[The $W_\$$ garbage emitter] | |
1804 | Let natural numbers $n_L$, $n_C$, and $V_0 \in \Bin^\ell$ be given; then we | |
1805 | define the garbage emitter $W_\$$ as follows. | |
1806 | \begin{program} | |
1807 | Initialization: \+ \\ | |
1808 | $i \gets 0$; \\ | |
1809 | $v \gets V_0$; | |
1810 | \- \\[\medskipamount] | |
1811 | Garbage emitter $W_\$(m)$: \+ \\ | |
1812 | \IF $i < n_L$ \THEN $v' \gets v$; \\ | |
1813 | \ELSE \IF $n_L \le i < n_L + n_C$ \THEN $v' \gets c(i)$; \\ | |
1814 | \ELSE \IF $n_L + n_C \le i$ \THEN $v' \getsr \Bin^\ell$; \\ | |
1815 | $i \gets i + 1$; \\ | |
1816 | $m' \gets t \lfloor (m + t - 1)/t\rfloor$; \\ | |
1817 | $y \getsr \Bin^{m'}$; \\ | |
1818 | $v \gets v' \shift{m'} y$; \\ | |
1819 | \RETURN $(v', y[0 \bitsto m])$ | |
1820 | \end{program} | |
1821 | \end{definition} | |
1822 | ||
1823 | We now show that CFB mode with a random function is hard to distinguish from | |
1824 | $W_\$$. | |
1825 | \begin{lemma}[Pseudorandomness of CFB mode] | |
1826 | \label{lem:cfb-rog} | |
1827 | Let $\ell$, $t$, $n_L$, $n_C$, $n_E$, $q_E$, $c$, $V_0$, and $q$ be as in | |
1828 | theorem~\ref{thm:cfb}. Then, for any $t_E$ and $\mu_E$, | |
1829 | \[ \InSec{rog-cpa-$W_\$$} | |
1830 | (\Xid{\E}{CFBH}^{\Func{\ell}{t}, V_0, c}_{n_L, n_C, n_E}; | |
1831 | t, q_E, \mu_E) \le | |
1832 | \frac{q (q - 1)}{2^{\ell+1}}. | |
1833 | \] | |
1834 | \end{lemma} | |
1835 | Theorem~\ref{thm:cfb} follows from this result by application of | |
1836 | propositions \ref{prop:rog-and-lor} and~\ref{prop:enc-info-to-real}. It | |
1837 | remains therefore for us to prove lemma~\ref{lem:cfb-rog}. | |
1838 | ||
1839 | To reduce the weight of notation, let us agree to suppress the adornments on | |
1840 | $\Adv{}{}$ and $\InSec{}$ symbols. Also, let $m_L = n_L$; let $m_C$ = $n_L + | |
1841 | n_C$; and let $m_E = n_L + n_C + n_E$. (Remember: the $m$s are | |
1842 | cu\textit{m}ulative.) | |
1843 | ||
1844 | The truncation of ciphertext blocks makes matters complicated. Let us say | |
1845 | that an adversary is \emph{block-respecting} if all of its plaintext queries | |
1846 | are a multiple of $t$ bits in length; obviously all of the oracle responses | |
1847 | for a block-respecting adversary are also a multiple of $t$ bits in length. | |
1848 | \begin{claim*} | |
1849 | If $A'$ be a block-respecting adversary querying a total of $\mu_E$ bits of | |
1850 | plaintext queries; then | |
1851 | \[ \Adv{}{}(A') \le \frac{q (q - 1)}{2^{\ell+1}} \] | |
1852 | where $q = \mu_E/t$. | |
1853 | \end{claim*} | |
1854 | Lemma~\ref{lem:cfb-rog} follows from this claim: if $A$ is any adversary, | |
1855 | then we construct a block-respecting adversary $A'$ by padding $A$'s | |
1856 | plaintext queries and truncating the oracle responses; and if $A$ makes $q_E$ | |
1857 | queries totalling $\mu_E$ bits, then the total bits queried by $A'$ is no | |
1858 | more than $\bigl\lfloor\bigl( \mu_E + q_E (t - 1) \bigr)\bigr\rfloor$ bits. | |
1859 | We now proceed to the proof of the above claim. | |
1860 | ||
1861 | Suppose, then, that we are given a block-respecting adversary $A$ which makes | |
1862 | $q$ queries to its encryption oracle. Let $F(\cdot)$ denote the application | |
1863 | of the random function. We want to show that, provided all of the $F$-inputs | |
1864 | are distinct, the $F$-outputs are uniformly random, and hence the CFB | |
1865 | ciphertexts are uniformly random. As for the CBC case, life isn't that good | |
1866 | to us: we have to deal with the case where the adversary can see that two | |
1867 | $F$-inputs would have collided, and therefore that a garbage string couldn't | |
1868 | have been generated by CFB encryption of his plaintext. | |
1869 | ||
1870 | Our notation will be similar to, yet slightly different from, that of | |
1871 | section~\ref{sec:cbc-proof}. | |
1872 | ||
1873 | Let $q' = q - n_E$ be the number of $t$-bit plaintext blocks the adversary | |
1874 | submits, and for $0 \le i < q'$, let $x_i$ be the $i$th plaintext block | |
1875 | queried, and let $y_i$ be the $i$th ciphertext block returned. | |
1876 | ||
1877 | For $m_L \le i < m_E$, let $c_i = c(i)$ be the $i$th counter value. For $0 | |
1878 | \le i < q_E$ let $v_i$ be the $i$th initialization vector, i.e., | |
1879 | \[ v_i = \begin{cases} | |
1880 | V_0 & if $i = 0$ and $n_L > 0$; \\ | |
1881 | v_{i-1} \shift{t} Y_{i-1} | |
1882 | & if $1 \le i < m_L$ and $Y_{i-1}$ was the ciphertext | |
1883 | from query $i - 1$; \\ | |
1884 | c_i & if $m_L \le i < m_C$; \\ | |
1885 | F(c_i) & if the oracle is `result', and $m_C \le i < m_E$; | |
1886 | or \\ | |
1887 | R_i & for some $R_i \inr \Bin^\ell$, otherwise. | |
1888 | \end{cases} | |
1889 | \] | |
1890 | Note that the only difference in the $v_i$ between the `result' and `garbage' | |
1891 | games occurs in the encrypted-counters phase. Furthermore, if no other | |
1892 | $F$-input is equal to any $c_i$ for $m_C \le i < m_E$ then the IVs are | |
1893 | identically distributed. | |
1894 | ||
1895 | Now, for $0 \le i < q'$, define | |
1896 | \[ z_i = \begin{cases} | |
1897 | v_j & if block $i$ is the first block of the | |
1898 | $j$th query, or \\ | |
1899 | z_{i-1} \shift{t} y_{i-1} & otherwise | |
1900 | \end{cases} | |
1901 | \] | |
1902 | and let $w_i = x_i \xor y_i$. In the `result' game, we have $w_i = F(z_i)$, | |
1903 | of course. All of this notation is summarized diagrammatically in | |
1904 | figure~\ref{fig:cfb-proof-notation}. The $F$-inputs are precisely the $z_i$ | |
1905 | and $c_i$ for $m_C \le i < m_E$. | |
1906 | ||
1907 | We'll denote probabilities in the `result' game as $\Pr_R[\cdot]$ and in the | |
1908 | `garbage' game as $\Pr_G[\cdot]$. | |
1909 | ||
1910 | \begin{figure} | |
1911 | \begin{vgraphs} | |
1912 | \begin{vgraph}{cfb-notation-a} | |
1913 | [] !{<1.333cm, 0cm>: <0cm, 1cm>::} | |
1914 | {z_i} ="z" :|-@{/}_-{\ell}[r] *+[F]{F} ="F" | |
1915 | :|-@{/}_-{t}[r] {w_i} ="w" :|-@{/}_-{t}[r] *{\xor} ="xor" | |
1916 | "xor" [u] {x_i} ="x" :|-@{/}^-{t}"xor" :|-@{/}^-{t}[d] {y_i} ="y" | |
1917 | "z" [lu] {v_j} ="v" :"z" | |
1918 | "z" [ru] {z_{i-1} \shift{t} y_{i-1}} ="y" :"z" | |
1919 | "v" :@{.}|-*+\hbox{or} "y" | |
1920 | \end{vgraph} | |
1921 | \end{vgraphs} | |
1922 | ||
1923 | \caption{Notation for the proof of lemma~\ref{lem:cfb-rog}.} | |
1924 | \label{fig:cfb-proof-notation} | |
1925 | \end{figure} | |
1926 | ||
1927 | Let $C_r$ be the event, in either game, that $z_i = z_j$ for some $0 \le i < | |
1928 | j < r$, or that $z_i = c_j$ for some $0 \le i < r$ and some $m_C \le j < | |
1929 | m_E$. | |
1930 | ||
1931 | Let's assume that $C_r$ didn't happen; we want the probability that $C_{r+1}$ | |
1932 | did, which is just the probability that $z_r$ collides with some $z_i$ where | |
1933 | $0 \le i < r$, or some $c_i$ for $m_C \le i < m_E$. Observe that, under this | |
1934 | assumption, all the $w_i$, and hence the $y_i$, are uniformly distributed, | |
1935 | and that therefore the two games are indistinguishable. | |
1936 | ||
1937 | One of the following cases holds. | |
1938 | \begin{enumerate} | |
1939 | \item If $r = 0$ and $m_L > 0$ then $z_r = V_0$. There is no other $z_i$ yet | |
1940 | for $z_r$ to collide with, though it might collide with some encrypted | |
1941 | counter $F(c_i)$, with probability $n_E/2^\ell$. | |
1942 | \item If $z_r = c_i$ is the IV for some message $i$ where $m_L \le i < m_C$, | |
1943 | life is a bit complicated. It can't collide with $V_0$ or other $c_i$ by | |
1944 | assumption; the encrypted counters and random IVs haven't been chosen yet; | |
1945 | and either $n_C = 0$ (in which case there's nothing to do here anyway) or | |
1946 | $\ell \le t$, so there are no $z_i$ containing partial copies of $V_0$ to | |
1947 | worry about. This leaves non-IV $z_i$: again, $\ell \le t$, so $z_i = | |
1948 | y_i[t - \ell \bitsto t]$, which is random by our assumption of $\bar{C}_r$; | |
1949 | hence a collision with one of these $z_i$ occurs with probability at most | |
1950 | $r/2^\ell$. | |
1951 | \item If $z_r$ is the IV for some message $i$ where $m_C \le i < m_E$, then | |
1952 | it can collide with previous $z_i$ or either previous or future $c_i$. We | |
1953 | know, however, that no $F$-input has collided with $c_i$, so in the | |
1954 | `result' game, $z_r = F(c_r)$ is uniformly distributed; in the `garbage' | |
1955 | game, $W_\$$ generates $z_r$ at random anyway. It collides, therefore, | |
1956 | with probability at most $(r + n_E)/2^\ell$. | |
1957 | \item If $z_r$ is the IV for some message $i$ where $m_E \le i < q'$ then | |
1958 | $z_r$ was chosen uniformly at random. Hence it collides with probability | |
1959 | at most $(r + n_E)/2^\ell$. | |
1960 | \item Finally, either $z_r$ is not the IV for a message, or it is, but the | |
1961 | message number $i < n_L$, so in either case, $z_r = z_{r-1} \shift{t} | |
1962 | y_{r-1}$. We have two subcases to consider. | |
1963 | \begin{enumerate} | |
1964 | \item If $1 \le r < \ell/t$ (we dealt with the case $r = 0$ above) then | |
1965 | some of $V_0$ remains in the shift register. If $z_r$ collides with some | |
1966 | $z_i$, for $0 \le i < r$, then we must have $z_r[0 \bitsto \ell - t r] = | |
1967 | z_i[0 \bitsto \ell - t r]$; but $z_r[0 \bitsto \ell - t r] = V_0[t r | |
1968 | \bitsto \ell]$, and $z_i[0 \bitsto \ell - t r] = V_0[t i \bitsto \ell - t | |
1969 | (r - i)]$, i.e., we have found a $t$-sliding of $V_0$, which is | |
1970 | impossible by hypothesis. Hence, $z_r$ cannot collide with any earlier | |
1971 | $z_i$. Also by hypothesis, $n_C = n_E = 0$ if $\ell > t$, so $z_r$ | |
1972 | cannot collide with any counters $c_i$. | |
1973 | \item Suppose, then, that $r \ge \ell/t$. For $0 \le j < \ell/t$, let $H_j | |
1974 | = \ell - t j$, $L_j = \max(0, H_j - t)$, and $N_j = H_j - L_j$. (Note | |
1975 | that $\sum_{0\le j<\ell/t} N_j = \ell$.) Then $z_r[L_j \bitsto H_j] = | |
1976 | y_{r-j-1}[t - N_j \bitsto t]$; but the $y_i$ for $i < r$ are uniformly | |
1977 | distributed. Thus, $z_r$ collides with some specific other value $z'$ | |
1978 | only with probability $1/2^{\sum_j N_j} = 1/2^\ell$. The overall | |
1979 | collision probablity for $z_r$ is then at most $(r + n_E)/2^\ell$. | |
1980 | \end{enumerate} | |
1981 | \end{enumerate} | |
1982 | In all these cases, it's clear that the collision probability is no more than | |
1983 | $(r + n_E)/2^\ell$. | |
1984 | ||
1985 | The probability that there is a collision during the course of the game is | |
1986 | $\Pr[C_{q'}]$, which we can now bound thus: | |
1987 | \begin{equation} | |
1988 | \Pr[C_q'] \le \sum_{0<i\le q'} \Pr[C_i | \bar{C}_{i-1}] | |
1989 | \le \sum_{0<i\le q'} \frac{i + n_E}{2^\ell}. | |
1990 | \end{equation} | |
1991 | If we set $i' = i + n_E$, then we get | |
1992 | \begin{equation} | |
1993 | \Pr[C_q'] \le \sum_{0\le i'\le q} \frac{i'}{2^\ell} | |
1994 | = \frac{q (q - 1)}{2^{\ell+1}}. | |
1995 | \end{equation} | |
1996 | Finally, then, we can apply the same argument as we used at the end of | |
1997 | section~\ref{sec:cbc-proof} to show that | |
1998 | \begin{equation} | |
1999 | \Adv{}{}(A') \le \frac{q (q - 1)}{2^{\ell+1}} | |
2000 | \end{equation} | |
2001 | as claimed. This completes the proof. | |
2002 | ||
2003 | %%%-------------------------------------------------------------------------- | |
2004 | ||
2005 | \section{OFB mode encryption} | |
2006 | \label{sec:ofb} | |
2007 | ||
2008 | \subsection{Description} | |
2009 | \label{sec:ofb-desc} | |
2010 | ||
2011 | Suppose $F$ is an $\ell$-bit-to-$L$-bit pseudorandom function, and let $t \le | |
2012 | L$. OFB mode works as follows. Given a message $X$, we divide it into | |
2013 | $t$-bit blocks $x_0$, $x_1$, $\ldots$, $x_{n-1}$. Choose an initialization | |
2014 | vector $v \in \Bin^\ell$. We maintain a \emph{shift register} $s_i$, whose | |
2015 | initial value is $v$. To encrypt a block $x_i$, we XOR it with the result | |
2016 | $z_i$ of passing the shift register through the PRF, forming $y_i$, and then | |
2017 | update the shift register by shifting in the PRF output $z_i$. That | |
2018 | is, | |
2019 | \begin{equation} | |
2020 | s_0 = v \qquad | |
2021 | z_i = F_K(s_i) \qquad | |
2022 | y_i = x_i \xor z_i \qquad | |
2023 | s_{i+1} = s_i \shift{t} z_i \ \text{(for $0 \le i < n$)}. | |
2024 | \end{equation} | |
2025 | Decryption is precisely the same operation. | |
2026 | ||
2027 | Also, we observe that the final plaintext block needn't be $t$ bits long: we | |
2028 | can pad it out to $t$ bits and truncate the result without affecting our | |
2029 | ability to decrypt. | |
2030 | ||
2031 | \begin{figure} | |
2032 | \begin{cgraph}{ofb-mode} | |
2033 | [] !{<0.425cm, 0cm>: <0cm, 0.5cm>::} | |
2034 | *+=(2, 0)+[F]{\mathstrut v} ="v" :|<>(0.35)@{/}_<>(0.35){\ell}[rrrrr] | |
2035 | *+[o][F]{\shift{t}} ="shift" | |
2036 | [ll] :|-@{/}^-{\ell}[dd] *+[F]{E} ="e" [ll] {K} :"e" | |
2037 | :|-@{/}^-{t}[ddd] *{\xor} ="xor" | |
2038 | [lll] *+=(2, 0)+[F]{\mathstrut x_0} :|-@{/}_-{t} "xor" | |
2039 | :|-@{/}^-{t}[dd] *+=(2, 0)+[F]{\mathstrut y_0} | |
2040 | "xor" [u] :`r "shift" "shift"|-@{/}_-{t} | |
2041 | :|-@{/}_-{\ell}[rrrrrrr] *+[o][F]{\shift{t}} ="shift" | |
2042 | [ll] :|-@{/}^-{\ell}[dd] *+[F]{E} ="e" [ll] {K} :"e" | |
2043 | :|-@{/}^-{t}[ddd] *{\xor} ="xor" | |
2044 | [lll] *+=(2, 0)+[F]{\mathstrut x_1} :|-@{/}_-{t} "xor" | |
2045 | :|-@{/}^-{t}[dd] *+=(2, 0)+[F]{\mathstrut y_1} | |
2046 | "xor" [u] :`r "shift" "shift"|-@{/}_-{t} | |
2047 | :@{-->}|-@{/}_-{\ell}[rrrrrrr] *+[o][F]{\shift{t}} ="shift" | |
2048 | [ll] :@{-->}|-@{/}^-{\ell}[dd] *+[F]{E} ="e" [ll] {K} :"e" | |
2049 | :@{-->}|-@{/}^-{t}[ddd] *{\xor} ="xor" | |
2050 | [lll] *+=(2, 0)+[F--]{\mathstrut x_i} :@{-->}|-@{/}_-{t} "xor" | |
2051 | :@{-->}|-@{/}^-{t}[dd] *+=(2, 0)+[F--]{\mathstrut y_i} | |
2052 | "xor" [u] :@{-->} `r "shift" "shift"|-@{/}_-{t} | |
2053 | [rrrrrdd] *+[F]{E} ="e" | |
2054 | "shift" :@{-->}`r "e" |-@{/}_-{\ell} "e" | |
2055 | [ll] {K} :"e" | |
2056 | :|-@{/}^-{t}[ddd] *{\xor} ="xor" | |
2057 | [lll] *+=(2, 0)+[F]{\mathstrut x_{n-1}} :|-@{/}_-{t} "xor" | |
2058 | :|-@{/}^-{t}[dd] *+=(2, 0)+[F]{\mathstrut y_{n-1}} | |
2059 | \end{cgraph} | |
2060 | ||
2061 | \caption{Encryption using OFB mode} | |
2062 | \label{fig:ofb} | |
2063 | \end{figure} | |
2064 | ||
2065 | \begin{definition}[OFB algorithms] | |
2066 | For any function $F\colon \Bin^\ell \to \Bin^t$, any initialization vector | |
2067 | $v \in \Bin^\ell$, any plaintext $x \in \Bin^*$ and any ciphertext $y \in | |
2068 | \Bin^*$, we define PRF encryption mode $\id{OFB} = (\id{ofb-encrypt}, | |
2069 | \id{ofb-decrypt})$ as follows: | |
2070 | \begin{program} | |
2071 | Algorithm $\id{ofb-encrypt}(F, v, x)$: \+ \\ | |
2072 | $s \gets v$; \\ | |
2073 | $L \gets |x|$; \\ | |
2074 | $x \gets x \cat 0^{t\lceil L/t \rceil - L}$; \\ | |
2075 | $y \gets \emptystring$; \\ | |
2076 | \FOR $i = 0$ \TO $(|x| - t')/t$ \DO \\ \ind | |
2077 | $x_i \gets x[ti \bitsto t(i + 1)]$; \\ | |
2078 | $z_i \gets F(s)$; \\ | |
2079 | $y_i \gets x_i \xor z_i$; \\ | |
2080 | $s \gets s \shift{t} z_i$; \\ | |
2081 | $y \gets y \cat y_i$; \- \\ | |
2082 | \RETURN $(s, y[0 \bitsto L])$; | |
2083 | \next | |
2084 | Algorithm $\id{ofb-decrypt}(F, v, y)$: \+ \\ | |
2085 | \RETURN $\id{ofb-encrypt}(F, v, y)$; | |
2086 | \end{program} | |
2087 | We now define the schemes $\Xid{\E}{OFB$\$$}^F$, $\Xid{\E}{OFBC}^{F, c}$, | |
2088 | $\Xid{\E}{OFBE}^{F, c}$, and $\Xid{\E}{OFBL}^{F, V_0}$ according to | |
2089 | definition~\ref{def:enc-scheme}; and we define the hybrid scheme | |
2090 | $\Xid{\E}{OFBH}^{F, V_0, c}_{n_L, n_C, n_E}$ according to | |
2091 | definition~\ref{def:enc-hybrid}. | |
2092 | \end{definition} | |
2093 | ||
2094 | \begin{remark}[Similarity to CFB mode] | |
2095 | \label{rem:ofb-like-cfb} | |
2096 | OFB mode is strongly related to CFB mode: we can OFB encrypt a message $x$ | |
2097 | by \emph{CFB-encrypting} the all-zero string $0^{|x|}$ with the same key | |
2098 | and IV. That is, we could have written $\id{ofb-encrypt}$ and | |
2099 | $\id{ofb-decrypt}$ like this: | |
2100 | \begin{program} | |
2101 | Algorithm $\id{ofb-encrypt}(F, v, x)$: \+ \\ | |
2102 | $(s, z) \gets \id{cfb-encrypt}(F, v, 0^{|x|})$; \\ | |
2103 | \RETURN $(s, x \xor z)$; | |
2104 | \next | |
2105 | Algorithm $\id{ofb-decrypt}(F, v, y)$: \+ \\ | |
2106 | \RETURN $\id{ofb-encrypt}(F, v, y)$; | |
2107 | \end{program} | |
2108 | We shall use this fact to prove the security of OFB mode in the next | |
2109 | section. | |
2110 | \end{remark} | |
2111 | ||
2112 | \subsection{Security of OFB mode} | |
2113 | ||
2114 | \begin{theorem}[Security of OFB mode] | |
2115 | \label{thm:ofb} | |
2116 | Let $F$ be a pseudorandom function from $\Bin^\ell$ to $\Bin^t$; let $V_0 | |
2117 | \in \Bin^\ell$ be a non-$t$-sliding string; let $c$ be a generalized | |
2118 | counter in $\Bin^\ell$; and let $n_L$, $n_C$, $n_E$ and $q_E$ be | |
2119 | nonnegative integers; and furthermore suppose that | |
2120 | \begin{itemize} | |
2121 | \item $n_L + n_C + n_E \le q_E$, | |
2122 | \item $n_L = 0$, or $n_C = n_E = 0$, or $\ell \le t$ and $V_0 \ne c(i)$ | |
2123 | for any $n_L \le i < n_L + n_C + n_E$, and | |
2124 | \item either $n_C = 0$ or $\ell \le t$. | |
2125 | \end{itemize} | |
2126 | Then, for any $t_E$ and $\mu_E$, and whenever | |
2127 | we have | |
2128 | \[ \InSec{lor-cpa}(\Xid{\E}{OFBH}^{F, V_0, c}_{n_L, n_C, n_E}; | |
2129 | t_E, q_E, \mu_E) \le | |
2130 | 2 \cdot \InSec{prf}(F; t_E + q t_F, q) + \frac{q (q - 1)}{2^\ell} | |
2131 | \] | |
2132 | where $q = \bigl\lfloor \bigl(\mu_E + q_E (t - 1)\bigr)/t \bigr\rfloor + | |
2133 | n_E$, and $t_F$ is some small constant. | |
2134 | \end{theorem} | |
2135 | \begin{proof} | |
2136 | We claim that | |
2137 | \[ \InSec{rog-cpa-$W_\$$} | |
2138 | (\Xid{\E}{OFBH}^{\Func{\ell}{t}, V_0, c}_{n_L, n_C, n_E}; | |
2139 | t, q_E, \mu_E) \le | |
2140 | \frac{q (q - 1)}{2^{\ell+1}}. | |
2141 | \] | |
2142 | This follows from lemma~\ref{lem:cfb-rog}, which makes the same statement | |
2143 | about CFB mode, and the observation in remark~\ref{rem:ofb-like-cfb}. | |
2144 | Suppose $A$ attempts to distinguish OFBH encryption from $W_\$$. We define | |
2145 | the adversary $B$ which uses $A$ to attack CFBH encryption, as follows: | |
2146 | \begin{program} | |
2147 | Adversary $B^{E(\cdot)}$: \+ \\ | |
2148 | \RETURN $A^{\id{ofb}(\cdot)}$; \- | |
2149 | \next | |
2150 | Function $\id{ofb}(x)$: \+ \\ | |
2151 | $(v, z) \gets E(0^{|x|})$; \\ | |
2152 | \RETURN $(v, x \xor z)$; | |
2153 | \end{program} | |
2154 | Now we apply proposition~\ref{prop:rog-and-lor}; the theorem follows. | |
2155 | \end{proof} | |
2156 | ||
2157 | \begin{corollary} | |
2158 | \label{cor:ofb-prf} | |
2159 | Let $F$, $c$ and $V_0$ be as in theorem~\ref{thm:ofb}. Then for any $t_E$, | |
2160 | $q_E$ and $\mu_E$, | |
2161 | \begin{eqnarray*}[rl] | |
2162 | \InSec{lor-cpa}(\Xid{\E}{OFB$\$$}^F; t_E, q_E, \mu_E) | |
2163 | & \le 2 \cdot \InSec{prf}(F; t_E + q t_F, q) + \frac{q (q - 1)}{2^\ell} | |
2164 | \\ | |
2165 | \InSec{lor-cpa}(\Xid{\E}{OFBE}^{F, c}; t_E, q_E, \mu_E) | |
2166 | & \le 2 \cdot \InSec{prf}(F; t_E + q' t_F, q') + | |
2167 | \frac{q' (q' - 1)}{2^\ell} | |
2168 | \\ | |
2169 | \InSec{lor-cpa}(\Xid{\E}{OFBL}^{F, V_0}; t_E, q_E, \mu_E) | |
2170 | & \le 2 \cdot \InSec{prf}(F; t_E + q t_F, q) + \frac{q (q - 1)}{2^\ell} | |
2171 | \\ | |
2172 | \tabpause{and, if $\ell \le t$,} | |
2173 | \InSec{lor-cpa}(\Xid{\E}{OFBC}^{F, c}; t_E, q_E, \mu_E) | |
2174 | & \le 2 \cdot \InSec{prf}(F; t_E + q t_F, q) + \frac{q (q - 1)}{2^\ell} | |
2175 | \end{eqnarray*} | |
2176 | where $q = \bigl\lfloor \bigl(\mu_E + q_E (t - 1)\bigr)/t \bigr\rfloor + | |
2177 | n_E$, $q' = q + q_E$, and $t_F$ is some small constant. | |
2178 | \end{corollary} | |
2179 | \begin{proof} | |
2180 | Follows from theorem~\ref{thm:ofb} and proposition~\ref{prop:enc-hybrid}. | |
2181 | \end{proof} | |
2182 | ||
2183 | \begin{corollary} | |
2184 | \label{cor:ofb-prp} | |
2185 | Let $P$ be a pseudorandom permutation on $\Bin^\ell$, and let $c$ and $V_0$ | |
2186 | be as in theorem~\ref{thm:ofb}. Then for any $t_E$, $q_E$ and $\mu_E$, | |
2187 | \begin{eqnarray*}[rl] | |
2188 | \InSec{lor-cpa}(\Xid{\E}{OFB$\$$}^P; t_E, q_E, \mu_E) | |
2189 | & \le 2 \cdot \InSec{prp}(P; t_E + q t_F, q) + \frac{q (q - 1)}{2^\ell} | |
2190 | \\ | |
2191 | \InSec{lor-cpa}(\Xid{\E}{OFBE}^{P, c}; t_E, q_E, \mu_E) | |
2192 | & \le 2 \cdot \InSec{prp}(P; t_E + q' t_F, q') + | |
2193 | \frac{q' (q' - 1)}{2^\ell} | |
2194 | \\ | |
2195 | \InSec{lor-cpa}(\Xid{\E}{OFBL}^{P, V_0}; t_E, q_E, \mu_E) | |
2196 | & \le 2 \cdot \InSec{prp}(P; t_E + q t_F, q) + \frac{q (q - 1)}{2^\ell} | |
2197 | \\ | |
2198 | \tabpause{and, if $\ell \le t$,} | |
2199 | \InSec{lor-cpa}(\Xid{\E}{OFBC}^{P, c}; t_E, q_E, \mu_E) | |
2200 | & \le 2 \cdot \InSec{prp}(P; t_E + q t_F, q) + \frac{q (q - 1)}{2^\ell} | |
2201 | \end{eqnarray*} | |
2202 | where $q = \bigl\lfloor \bigl(\mu_E + q_E (t - 1)\bigr)/t \bigr\rfloor + | |
2203 | n_E$, $q' = q + q_E$, and $t_F$ is some small constant. | |
2204 | \end{corollary} | |
2205 | \begin{proof} | |
2206 | Follows from corollary~\ref{cor:ofb-prf} and | |
2207 | proposition~\ref{prop:prps-are-prfs}. | |
2208 | \end{proof} | |
2209 | ||
2210 | %%%-------------------------------------------------------------------------- | |
2211 | ||
2212 | \section{CBCMAC mode message authentication} | |
2213 | \label{sec:cbcmac} | |
2214 | ||
2215 | ||
2216 | ||
2217 | \begin{figure} | |
2218 | \begin{cgraph}{cbc-mac} | |
2219 | []!{<0.425cm, 0cm>: <0cm, 0.75cm>::} | |
2220 | *+=(2, 0)+[F]{\mathstrut x_0} | |
2221 | :`d [dr] [rrr] *+[F]{E} ="e" [d] {K} :"e" | |
2222 | :[rrr] *{\xor} ="xor" | |
2223 | [u] *+=(2, 0)+[F]{\mathstrut x_1} :"xor" | |
2224 | :[rrr] *+[F]{E} ="e" [d] {K} :"e" | |
2225 | :@{-->}[rrr] *{\xor} ="xor" | |
2226 | [u] *+=(2, 0)+[F--]{\mathstrut x_i} :@{-->}"xor" | |
2227 | :@{-->}[rrr] *+[F]{E} ="e" [d] {K} :@{-->}"e" | |
2228 | :@{-->}[rrr] *{\xor} ="xor" | |
2229 | [u] *+=(2, 0)+[F]{\mathstrut x_{n-1}} :"xor" | |
2230 | :[rrr] *+[F]{E} ="e" [d] {K} :"e" | |
2231 | :[rrr] *+=(2, 0)+[F]{\mathstrut \tau} | |
2232 | \end{cgraph} | |
2233 | ||
2234 | \caption{Message authentication using CBCMAC mode} | |
2235 | \label{fig:cbcmac} | |
2236 | \end{figure} | |
2237 | ||
2238 | \fixme | |
2239 | Alas, it's been so long since I've picked this up that I've (a) forgotten how | |
2240 | the proof for this mode went, and (b) lost my notes. You'll either have to | |
2241 | wait, or I'll have to drop this bit. | |
2242 | ||
2243 | %%%-------------------------------------------------------------------------- | |
2244 | ||
2245 | \section{Ackonowledgements} | |
2246 | ||
2247 | Thanks to Clive Jones for his suggestions on notation, and his help in | |
2248 | structuring the proofs. | |
2249 | ||
2250 | %%%----- That's all, folks -------------------------------------------------- | |
2251 | ||
2252 | \bibliography{mdw-crypto,cryptography2000,cryptography,rfc} | |
2253 | ||
2254 | \end{document} | |
2255 | ||
2256 | %%% Local Variables: | |
2257 | %%% mode: latex | |
2258 | %%% TeX-master: t | |
2259 | %%% End: |