| 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 | \def\hex#1{\texttt{#1}_{x}} |
| 85 | |
| 86 | \newslowboxenv{cgraph}{\par$$}{\begin{graph}}{\end{graph}}{$$\par} |
| 87 | \newslowboxenv{vgraph} |
| 88 | {\hfil$\vcenter\bgroup\hbox\bgroup} |
| 89 | {\begin{graph}} |
| 90 | {\end{graph}} |
| 91 | {\egroup\egroup$} |
| 92 | \newenvironment{vgraphs}{\hbox to\hsize\bgroup}{\hfil\egroup} |
| 93 | |
| 94 | \begin{document} |
| 95 | |
| 96 | %%%-------------------------------------------------------------------------- |
| 97 | |
| 98 | \maketitle |
| 99 | |
| 100 | \begin{abstract} |
| 101 | We study the standard block cipher modes of operation: CBC, CFB, OFB, and |
| 102 | CBCMAC and analyse their security. We don't look at ECB other than briefly |
| 103 | to note its insecurity, and we have no new results on counter mode. Our |
| 104 | results improve over those previously published in that (a) our bounds are |
| 105 | better, (b) our proofs are shorter and easier, (c) the proofs correct |
| 106 | errors we discovered in previous work, or some combination of these. We |
| 107 | provide a new security notion for symmetric encryption which turns out to |
| 108 | be rather useful when analysing block cipher modes. Finally, we define a |
| 109 | new condition for initialization vectors, introducing the concept of a |
| 110 | `generalized counter', and proving that generalized counters suffice for |
| 111 | security in (full-width) CFB and OFB modes and that generalized counters |
| 112 | encrypted using the block cipher (with the same key) suffice for all the |
| 113 | encryption modes we study. |
| 114 | \end{abstract} |
| 115 | |
| 116 | \iffancystyle |
| 117 | \newpage |
| 118 | \columnsep=2em \columnseprule=0pt |
| 119 | \tableofcontents[\begin{multicols}{2}\raggedright][\end{multicols}] |
| 120 | \listoffigures[\begin{multicols}{2}\raggedright][\end{multicols}] |
| 121 | \listoftables[\begin{multicols}{2}\raggedright][\end{multicols}] |
| 122 | \newpage |
| 123 | \fi |
| 124 | |
| 125 | %%%-------------------------------------------------------------------------- |
| 126 | |
| 127 | \section{Introduction} |
| 128 | \label{sec:intro} |
| 129 | |
| 130 | \subsection{Block cipher modes} |
| 131 | |
| 132 | Block ciphers -- keyed pseudorandom permutations -- are essential |
| 133 | cryptographic tools, widely used for bulk data encryption and to an |
| 134 | increasing extent for message authentication. Because the efficient block |
| 135 | ciphers we have operate on fixed and relatively small strings of bits -- 64 |
| 136 | or 128 bits at a time, one needs a `mode of operation' to explain how to |
| 137 | process longer messages. |
| 138 | |
| 139 | A collection of encryption modes, named ECB, CBC, CFB and OFB, were defined |
| 140 | in \cite{FIPS81}. Of these, ECB -- simply divide the message into blocks and |
| 141 | process them independently with the block cipher -- is just insecure and not |
| 142 | to be recommended for anything much. We describe the other three, and |
| 143 | analyse their security using the standard quantitative provable-security |
| 144 | approach. All three require an `initialization vector' or `IV' which |
| 145 | diversifies the output making it hard to correlate ciphertexts with |
| 146 | plaintexts. We investigate which conditions on these IVs suffice for secure |
| 147 | encryption. |
| 148 | |
| 149 | We also examine the CBC-MAC message-authentication scheme, because it's |
| 150 | intimately related to the CBC encryption scheme and the same techniques we |
| 151 | used in the analysis of the latter apply to the former. |
| 152 | |
| 153 | \subsection{Previous work} |
| 154 | |
| 155 | The first quantitative security proof for a block cipher mode is the analysis |
| 156 | of CBCMAC of \cite{Bellare:1994:SCB}. Security proofs for the encryption |
| 157 | modes CBC and CTR appeared in \cite{Bellare:2000:CST}, which also defines and |
| 158 | relates the standard security notions of symmetric encryption. The authors |
| 159 | of \cite{Alkassar:2001:OSS} offer a proof of CFB mode, though we believe it |
| 160 | to be flawed in a number of respects. |
| 161 | |
| 162 | \subsection{Our contribution} |
| 163 | |
| 164 | We introduce a new security notion for symmetric encryption, named |
| 165 | `result-or-garbage', or `ROG-CPA', which generalizes the `real-or-random' |
| 166 | notion of \cite{Bellare:2000:CST} and the `random-string' notion of |
| 167 | \cite{Rogaway:2001:OCB}. Put simply, it states that an encryption scheme is |
| 168 | secure if an adversary has difficulty distinguishing true ciphertexts from |
| 169 | strings chosen by an algorithm which is given only the \emph{length} of the |
| 170 | adversary's plaintext. This turns out to be just the right tool for |
| 171 | analysing our encryption modes. We relate this notion to the standard |
| 172 | `left-or-right' notion and, thereby, all the others. |
| 173 | |
| 174 | Our bound for CBC mode improves over the `tight' bound proven in |
| 175 | \cite{Bellare:2000:CST} by almost a factor of two. The difference comes |
| 176 | because they analyse the construction as if it were built from a PRF and add |
| 177 | in a `PRP-used-as-a-PRF' correction term: our analysis considers the effect |
| 178 | of a permutation directly. We prove that CBC mode is still secure if an |
| 179 | encrypted counter is used in place of a random string as the IV for each |
| 180 | message. Finally, we show that the `ciphertext stealing' technique is |
| 181 | secure. |
| 182 | |
| 183 | For CFB, we first discuss the work of \cite{Alkassar:2001:OSS}, who offer a |
| 184 | proof for both CFB mode and an optimized variant which enhances the |
| 185 | error-recovery properties of standard CFB. We believe that their proof is |
| 186 | defective in a number of ways. We then offer our own proof. Our bound is a |
| 187 | factor of two worse than theirs; however, we believe that fixing their proof |
| 188 | introduces this missing factor of two: that is, that our `poorer' bound |
| 189 | reflects the true security of CFB mode more accurately. We show that |
| 190 | full-width CFB is secure if the IV is any `generalized counter', and that |
| 191 | both full-width and truncated $t$-bit CFB are secure if the IV is an |
| 192 | encrypted counter. We also show that, unlike CBC mode, it is safe to `carry |
| 193 | over' the final shift-register value from the previous message as the IV for |
| 194 | the next message. |
| 195 | |
| 196 | OFB mode is in fact a simple modification to CFB mode, and we prove the |
| 197 | security of OFB by relating it to CFB. |
| 198 | |
| 199 | Finally, for CBCMAC, we analyse it using \emph{both} pseudorandom functions |
| 200 | \emph{and} pseudorandom permutations, showing that, in fact, using a block |
| 201 | cipher rather than a PRF reduces the security hardly at all. Also, we |
| 202 | improve on the (groundbreaking) work of \cite{Bellare:1994:SCB} firstly by |
| 203 | improving the security bound by a factor of almost four, and secondly by |
| 204 | extending the message space from a space of fixed-length messages to |
| 205 | \emph{any} prefix-free set of strings. |
| 206 | |
| 207 | As a convenient guide, our security bounds are summarized in |
| 208 | table~\ref{tab:summary}. |
| 209 | |
| 210 | \begin{table} |
| 211 | \def\lower#1{% |
| 212 | \vbox to\baselineskip{\vskip\baselineskip\vskip2pt\hbox{#1}\vss}} |
| 213 | \def\none{\multicolumn{1}{c|}{---}} |
| 214 | \let\hack=\relax |
| 215 | \begin{tabular}[C] |
| 216 | {| c | ?>{\hack}c | c | >{\displaystyle} Mc | >{\displaystyle} Mc |} |
| 217 | \hlx{hv[4]} |
| 218 | \multicolumn{1}{|c|}{\lower{\bfseries Mode}} & |
| 219 | \multicolumn{1}{c|}{\lower{\bfseries Section}} & |
| 220 | \multicolumn{1}{c|}{\lower{\bfseries Notion}} & |
| 221 | \multicolumn{2}{c|}{\bfseries Security with} \\ \hlx{v[4]zc{4-5}v} |
| 222 | & & & |
| 223 | \multicolumn{1}{c|}{\bfseries $(t, q, \epsilon)$-PRF} & |
| 224 | \multicolumn{1}{c|}{\bfseries $(t, q, \epsilon)$-PRP} |
| 225 | \\ \hlx{vhvv} |
| 226 | CBC & \ref{sec:cbc} & LOR-CPA & |
| 227 | \none & |
| 228 | 2\epsilon + \frac{q (q - 1)}{2^\ell - q} \\ \hlx{vvhvv} |
| 229 | CFB & \ref{sec:cfb} & LOR-CPA & |
| 230 | 2 \epsilon + \frac{q (q - 1)}{2^\ell} & |
| 231 | 2 \epsilon + \frac{q (q - 1)}{2^{\ell-1}} \\ \hlx{vvhvv} |
| 232 | OFB & \ref{sec:ofb} & LOR-CPA & |
| 233 | 2 \epsilon + \frac{q (q - 1)}{2^\ell} & |
| 234 | 2 \epsilon + \frac{q (q - 1)}{2^{\ell-1}} \\ \hlx{vvhvv} |
| 235 | CBCMAC & \ref{sec:cbcmac} & SUF-CMA & |
| 236 | \epsilon + \frac{q (q - 1) + 2 q_V}{2^{\ell+1}} & |
| 237 | \epsilon + |
| 238 | \frac{q (q - 1)}{2 \cdot (2^\ell - q)} + |
| 239 | \frac{q_V}{2^\ell - q_T} \\ \hlx{vvh} |
| 240 | \end{tabular} |
| 241 | |
| 242 | \caption[Summary of our results] |
| 243 | {Summary of our results. In all cases, $q$ is the number of block |
| 244 | cipher applications used in the game.} |
| 245 | \label{tab:summary} |
| 246 | \end{table} |
| 247 | |
| 248 | \subsection{The rest of the paper} |
| 249 | |
| 250 | In section~\ref{sec:prelim} we define the various bits of notation and |
| 251 | terminology we'll need in the rest of the paper. The formal definitions are |
| 252 | given for our new `result-or-garbage' security notion, and for our |
| 253 | generalized counters. In section~\ref{sec:cbc} we study CBC mode, and |
| 254 | ciphertext stealing. In section~\ref{sec:cfb} we study CFB mode. In |
| 255 | section~\ref{sec:ofb} we study OFB mode. In section~\ref{sec:cbcmac} we |
| 256 | study the CBCMAC message authentication scheme. |
| 257 | |
| 258 | %%%-------------------------------------------------------------------------- |
| 259 | |
| 260 | \section{Notation and definitions} |
| 261 | \label{sec:prelim} |
| 262 | |
| 263 | \subsection{Bit strings} |
| 264 | \label{sec:bitstrings} |
| 265 | |
| 266 | Most of our notation for bit strings is standard. The main thing to note is |
| 267 | that everything is zero-indexed. |
| 268 | |
| 269 | \begin{itemize} |
| 270 | \item We write $\Bin = \{0, 1\}$ for the set of binary digits. Then $\Bin^n$ |
| 271 | is the set of $n$-bit strings, and $\Bin^*$ is the set of all (finite) bit |
| 272 | strings. |
| 273 | \item If $x$ is a bit string then $|x|$ is the length of $x$. If $x \in |
| 274 | \Bin^n$ then $|x| = n$. |
| 275 | \item If $x, y \in \Bin^n$ are strings of bits of the same length then $x |
| 276 | \xor y \in \Bin^n$ is their bitwise XOR. |
| 277 | \item If $x$ is a bit string and $i$ is an integer satisfying $0 \le i < |x|$ |
| 278 | then $x[i]$ is the $i$th bit of $x$. If $a$ and $b$ are integers |
| 279 | satisfying $0 \le a \le b \le |x|$ then $x[a \bitsto b]$ is the substring |
| 280 | of $x$ beginning with bit $a$ and ending just \emph{before} bit $b$. We |
| 281 | have $|x[i]| = 1$ and $|x[a \bitsto b]| = b - a$; if $y = x[a \bitsto b]$ |
| 282 | then $y[i] = x[a + i]$. |
| 283 | \item If $x$ and $y$ are bit strings then $x y$ is the result of |
| 284 | concatenating $y$ to $x$. If $z = x y$ then $|z| = |x| + |y|$; $z[i] = |
| 285 | x[i]$ if $0 \le i < |x|$ and $z[i] = y[i - |x|]$ if $|x| \le i < |x| + |
| 286 | |y|$. Sometimes, for clarity (e.g., to distinguish from integer |
| 287 | multiplication) we write $x \cat y$ instead of $x y$. |
| 288 | \item The empty string is denoted $\emptystring$. We have $|\emptystring| = |
| 289 | 0$, and $x = x \emptystring = \emptystring x$ for all strings $x |
| 290 | \in \Bin^*$. |
| 291 | \item If $x$ is a bit string and $n$ is a natural number then $x^n$ is the |
| 292 | result of concatenating $x$ to itself $n$ times. We have $x^0 = |
| 293 | \emptystring$ and if $n > 0$ then $x^n = x^{n-1} \cat x = x \cat x^{n-1}$. |
| 294 | \item If $x$ and $y$ are bit strings, $|x| = \ell$, and $|y| = t$, then we |
| 295 | define $x \shift{t} y$ as: |
| 296 | \[ x \shift{t} y = (x y)[t \bitsto t + \ell] = \begin{cases} |
| 297 | x[t \bitsto \ell] \cat y & if $t < \ell$, or \\ |
| 298 | y[t - \ell \bitsto t] & if $t \ge \ell$. |
| 299 | \end{cases} \] |
| 300 | Observe that, if $z = x \shift{t} y$ then $|z| = |x| = \ell$ and |
| 301 | \[ z[i] = (x y)[i + t] = \begin{cases} |
| 302 | x[i + t] & if $0 \le i < \ell - t$, or \\ |
| 303 | y[i + t - \ell] & if $\min(0, \ell - t) \le i < \ell$. |
| 304 | \end{cases} \] |
| 305 | Obviously $x \shift{0} \emptystring = x$, and if $|x| = |y| = t$ then $x |
| 306 | \shift{t} y = y$. Finally, if $|y| = t$ and $|z| = t'$ then $(x \shift{t} |
| 307 | y) \shift{t'} z = x \shift{t + t'} (y z)$. |
| 308 | \end{itemize} |
| 309 | |
| 310 | \subsection{Other notation} |
| 311 | \label{sec:miscnotation} |
| 312 | |
| 313 | \begin{itemize} |
| 314 | \iffalse |
| 315 | \item If $n$ is any natural number, then $\Nupto{n}$ is the set $\{\, i \in |
| 316 | \Z \mid 0 \le i < n \,\} = \{ 0, 1, \ldots, n \}$. |
| 317 | \fi |
| 318 | \item The symbol $\bot$ (`bottom') is a value different from every bit |
| 319 | string. |
| 320 | \item We write $\Func{l}{L}$ as the set of all functions from $\Bin^l$ to |
| 321 | $\Bin^L$, and $\Perm{l}$ as the set of all permutations on $\Bin^l$. |
| 322 | \end{itemize} |
| 323 | |
| 324 | \subsection{Algorithm descriptions} |
| 325 | \label{sec:algorithms} |
| 326 | |
| 327 | An \emph{adversary} is a probabilistic algorithm which attempts (possibly) to |
| 328 | `break' a cryptographic scheme. We will often provide adversaries with |
| 329 | oracles which compute values with secret data. The \emph{running time} of an |
| 330 | adversary conventionally includes the size of the adversary's description: |
| 331 | this is an attempt to `charge' the adversary for having large precomputed |
| 332 | tables. |
| 333 | |
| 334 | Most of the notation used in the algorithm descriptions should be obvious. |
| 335 | We briefly note a few features which may be unfamiliar. |
| 336 | \begin{itemize} |
| 337 | \item The notation $a \gets x$ denotes the action of assigning the value $x$ |
| 338 | to the variable $a$. |
| 339 | \item We write oracles as superscripts, with dots marking where inputs to |
| 340 | the oracle go, e.g., $A^{O(\cdot)}$. |
| 341 | \item The notation $a \getsr X$, where $X$ is a finite set, denotes the |
| 342 | action of assigning to $a$ a random value $x \in X$ according to the |
| 343 | uniform probability distribution on $X$; i.e., following $a \getsr X$, we |
| 344 | have $\Pr[a = x] = 1/|X|$ for any $x \in X$. |
| 345 | \end{itemize} |
| 346 | The notation is generally quite sloppy about types and scopes. We don't |
| 347 | think these informalities cause much confusion, and they greatly simplify the |
| 348 | presentation of the algorithms. |
| 349 | |
| 350 | \subsection{Pseudorandom functions and permutations} |
| 351 | \label{sec:prfs-and-prps} |
| 352 | |
| 353 | Our definitions of pseudorandom functions and permutations are standard. We |
| 354 | provide them for the sake of completeness. |
| 355 | |
| 356 | \begin{definition}[Pseudorandom function family] |
| 357 | \label{def:prf} |
| 358 | A \emph{pseudorandom function family (PRF)} $F = \{F_K\}_K$ is a collection |
| 359 | of functions $F_K\colon \Bin^\ell \to \Bin^L$ indexed by a \emph{key} $K |
| 360 | \in \keys F$. If $A$ is any adversary, we define $A$'s \emph{advantage in |
| 361 | distinguishing $F$ from a random function} to be |
| 362 | \[ \Adv{prf}{F}(A) = |
| 363 | \Pr[K \getsr \keys F: A^{F_K(\cdot)} = 1] - |
| 364 | \Pr[R \getsr \Func{\ell}{L}: A^{R(\cdot)} = 1] |
| 365 | \] |
| 366 | where the probability is taken over all choices of keys, random functions, |
| 367 | and the internal coin-tosses of $A$. The \emph{insecurity of $F$} is given |
| 368 | by |
| 369 | \[ \InSec{prf}(F; t, q) = \max_A \Adv{prf}{F}(A) \] |
| 370 | where the maximum is taken over all adversaries which run in time~$t$ and |
| 371 | issue at most $q$ oracle queries. If $\InSec{prf}(F; t, q) \le \epsilon$ |
| 372 | then we say that $F$ is a $(t, q, \epsilon)$-PRF. |
| 373 | \end{definition} |
| 374 | |
| 375 | \begin{definition}[Pseudorandom permutation family] |
| 376 | \label{def:prp} |
| 377 | A \emph{pseudorandom permutation family (PRP)} $E = \{E_K\}_K$ is a |
| 378 | collection of permutations $E_K\colon \Bin^\ell \to \Bin^\ell$ indexed by a |
| 379 | \emph{key} $K \in \keys E$. If $A$ is any adversary, we define $A$'s |
| 380 | \emph{advantage in distinguishing $E$ from a random permutation} to be |
| 381 | \[ \Adv{prp}{F}(A) = |
| 382 | \Pr[K \getsr \keys E: A^{E_K(\cdot)} = 1] - |
| 383 | \Pr[P \getsr \Perm{\ell}: A^{P(\cdot)} = 1] |
| 384 | \] |
| 385 | where the probability is taken over all choices of keys, random |
| 386 | permutations, and the internal coin-tosses of $A$. Note that the adversary |
| 387 | is not allowed to query the inverse permutation $E^{-1}_K(\cdot)$ or |
| 388 | $P^{-1}(\cdot)$. The \emph{insecurity of $E$} is given by |
| 389 | \[ \InSec{prp}(E; t, q) = \max_A \Adv{prf}{E}(A) \] |
| 390 | where the maximum is taken over all adversaries which run in time~$t$ and |
| 391 | issue at most $q$ oracle queries. If $\InSec{prp}(E; t, q) \le \epsilon$ |
| 392 | then we say that $E$ is a $(t, q, \epsilon)$-PRP. |
| 393 | \end{definition} |
| 394 | |
| 395 | The following result is standard; we shall require it for the security proofs |
| 396 | of CFB and OFB modes. The proof is given as an introduction to our general |
| 397 | approach. |
| 398 | |
| 399 | \begin{proposition} |
| 400 | \label{prop:prps-are-prfs} |
| 401 | Suppose $E$ is a PRP over $\Bin^\ell$. Then |
| 402 | \[ \InSec{prf}(E; t, q) |
| 403 | \le \InSec{prp}(E; t, q) + \frac{q (q - 1)}{2^{\ell+1}}. |
| 404 | \] |
| 405 | \end{proposition} |
| 406 | \begin{proof} |
| 407 | We claim |
| 408 | \[ \InSec{prf}(\Perm{\ell}; t, q) \le \frac{q (q - 1)}{2^{\ell+1}}, \] |
| 409 | i.e., that a \emph{perfect} $\ell$-bit random permutation is a PRF with the |
| 410 | stated bounds. The proposition follows immediately from this claim and the |
| 411 | definition of a PRP. |
| 412 | |
| 413 | We now prove the claim. Consider any adversary~$A$. Let $x_i$ be $A$'s |
| 414 | queries, and let $y_i$ be the responses, for $0 \le i < q$. Assume, |
| 415 | without loss of generality, that the $x_i$ are distinct. Let $C_n$ be the |
| 416 | event in the random-function game $\Expt{prf-$0$}{\Perm{\ell}}(A)$ that |
| 417 | $y_i = y_j$ for some $i$ and $j$ where $0 \le i < j < n$. Then |
| 418 | \begin{equation} |
| 419 | \Pr[C_n] \le \sum_{0\le i<n} \frac{i}{2^\ell} |
| 420 | = \frac{n (n - 1)}{2^{\ell+1}}. |
| 421 | \end{equation} |
| 422 | It's clear that the two games proceed identically if $C_q$ doesn't occur in |
| 423 | the random-function game. The claim follows. |
| 424 | \end{proof} |
| 425 | |
| 426 | \subsection{Symmetric encryption} |
| 427 | \label{sec:sym-enc} |
| 428 | |
| 429 | We begin with a purely syntactic description of a symmetric encryption |
| 430 | scheme, and then define our two notions of security. |
| 431 | |
| 432 | \begin{definition}[Symmetric encryption] |
| 433 | \label{def:symm-enc} |
| 434 | A \emph{symmetric encryption scheme} is a triple of algorithms $\E = (G, E, |
| 435 | D)$, with three (implicitly) associated sets: a keyspace, a plaintext |
| 436 | space, and a ciphertext space. |
| 437 | \begin{itemize} |
| 438 | \item $G$ is a probabilistic \emph{key-generation algorithm}. It is |
| 439 | invoked with no arguments, and returns a key $K$ which can be used with |
| 440 | the other two algorithms. We write $K \gets G()$. |
| 441 | \item $E$ is a probabilistic \emph{encryption algorithm}. It is invoked |
| 442 | with a key $K$ and a \emph{plaintext} $x$ in the plaintext space, and it |
| 443 | returns a \emph{ciphertext} $y$ in the ciphertext space. We write $y |
| 444 | \gets E_K(x)$. |
| 445 | \item $D$ is a deterministic \emph{decryption algorithm}. It is invoked |
| 446 | with a key $K$ and a ciphertext $y$, and it returns either a plaintext |
| 447 | $x$ or the distinguished symbol $\bot$. We write $x \gets D_K(y)$. |
| 448 | \end{itemize} |
| 449 | For correctness, we require that whenever $y$ is a possible result of |
| 450 | computing $E_K(x)$, then $x = D_K(y)$. |
| 451 | \end{definition} |
| 452 | |
| 453 | Our primary notion of security is \emph{left-or-right indistinguishability |
| 454 | under chosen-plaintext attack} (LOR-CPA), since it offers the best reductions |
| 455 | to the other common notions. (We can't acheive security against chosen |
| 456 | ciphertext attack using any of our modes, so we don't even try.) See |
| 457 | \cite{Bellare:2000:CST} for a complete discussion of LOR-CPA, and how it |
| 458 | relates to other security notions for symmetric encryption. |
| 459 | |
| 460 | \begin{definition}[Left-or-right indistinguishability] |
| 461 | \label{def:lor-cpa} |
| 462 | Let $\E = (G, E, D)$ be a symmetric encryption scheme. Define the function |
| 463 | $\id{lr}(b, x_0, x_1) = x_b$. Then for any adversary $A$, we define $A$'s |
| 464 | \emph{advantage against the LOR-CPA security of $\E$} as |
| 465 | \[ \Adv{lor-cpa}{\E}(A) = |
| 466 | \Pr[K \gets G(): A^{E_K(\id{lr}(1, \cdot, \cdot))} = 1] - |
| 467 | \Pr[K \gets G(): A^{E_K(\id{lr}(0, \cdot, \cdot))} = 1]. |
| 468 | \] |
| 469 | We define the \emph{LOR-CPA insecurity of $\E$} to be |
| 470 | \[ \InSec{lor-cpa}(\E; t, q_E, \mu_E) = |
| 471 | \max_A \Adv{lor-cpa}{\E}(A) |
| 472 | \] |
| 473 | where the maximum is taken over all adversaries which run in time~$t$ and |
| 474 | issue at most $q_E$ encryption queries totalling at most $\mu_E$ bits. If |
| 475 | $\InSec{lor-cpa}(\E; t, q_E, \mu_E) \le \epsilon$ then we say that $\E$ is |
| 476 | $(t, q_E, \mu_E, \epsilon)$-LOR-CPA. |
| 477 | \end{definition} |
| 478 | |
| 479 | Our second notion is named \emph{result-or-garbage} and abbreviated ROG-CPA. |
| 480 | It is related to the notion used by \cite{Rogaway:2001:OCB}, though different |
| 481 | in important ways: for example, there are reductions both ways between |
| 482 | ROG-CPA and LOR-CPA (and hence the other standard notions of security for |
| 483 | symmetric encryption), whereas the notion of \cite{Rogaway:2001:OCB} is |
| 484 | strictly stronger than LOR-CPA. Our idea is that an encryption scheme is |
| 485 | secure if ciphertexts of given plaintexts -- \emph{results} -- hard to |
| 486 | distinguish from strings constructed independently of any plaintexts -- |
| 487 | \emph{garbage}. We formalize this notion in terms of a |
| 488 | \emph{garbage-emission algorithm} $W$ which is given only the length of the |
| 489 | plaintext. The algorithm $W$ will usually be probabilistic, and may maintain |
| 490 | state. Unlike \cite{Rogaway:2001:OCB}, we don't require that $W$'s output |
| 491 | `look random' in any way, just that it be chosen independently of the |
| 492 | adversary's plaintext selection. |
| 493 | |
| 494 | \begin{definition}[Result-or-garbage indistinguishability] |
| 495 | \label{def:rog-cpa} |
| 496 | Let $\E = (G, E, D)$ be a symmetric encryption scheme, and let $W$ be a |
| 497 | possibly-stateful, possibly-probabilistic \emph{garbage-emission} |
| 498 | algorithm. Then for any adversary $A$, we define $A$'s \emph{advantage |
| 499 | against the ROG-CPA-$W$ security of $\E$} as |
| 500 | \[ \Adv{rog-cpa-$W$}{\E}(A) = |
| 501 | \Pr[K \gets G(): A^{E_K(\cdot)} = 1] - \Pr[A^{W(|\cdot|)} = 1]. \] |
| 502 | We define the \emph{ROG-CPA insecurity of $\E$} to be |
| 503 | \[ \InSec{lor-cpa}(\E; t, q_E, \mu_E) = |
| 504 | \max_A \Adv{lor-cpa}{\E}(A) \] |
| 505 | where the maximum is taken over all adversaries which run in time~$t$ and |
| 506 | issue at most $q_E$ encryption queries totalling at most $\mu_E$ bits. If |
| 507 | $\InSec{rog-cpa-$W$}(\E; t, q_E, \mu_E) \le \epsilon$ for some $W$ then we |
| 508 | say that $\E$ is $(t, q_E, \mu_E, \epsilon)$-ROG-CPA. |
| 509 | \end{definition} |
| 510 | |
| 511 | The following proposition relates our new notion to the existing known |
| 512 | notions of security. |
| 513 | |
| 514 | \begingroup |
| 515 | \def\Wror{{W_{\text{ROR}}}} |
| 516 | \begin{proposition}[ROG $\Leftrightarrow$ LOR] |
| 517 | \label{prop:rog-and-lor} |
| 518 | Let $\E$ be a symmetric encryption scheme. Then, |
| 519 | \begin{enumerate} |
| 520 | \item for all garbage-emission algorithms $W$, |
| 521 | \[ \InSec{lor-cpa}(\E; t, q_E, \mu_E) |
| 522 | \le 2 \cdot |
| 523 | \InSec{rog-cpa-$W$}(\E; t + t_E \mu_E, q_E, \mu_E) |
| 524 | \] |
| 525 | and |
| 526 | \item there exists a garbage-emission algorithm $\Wror$ for which |
| 527 | \[ \InSec{rog-cpa-$\Wror$}(\E; t, q_E, \mu_E) |
| 528 | \le \InSec{lor-cpa}(\E; t + t_E \mu_E, q_E, \mu_E) |
| 529 | \] |
| 530 | \end{enumerate} |
| 531 | for some fairly small constant $t_E$. |
| 532 | \end{proposition} |
| 533 | \begin{remark} |
| 534 | Note the asymmetry between these two statements. ROG-CPA-$W$ implies |
| 535 | LOR-CPA for \emph{any} garbage emitter $W$, but LOR-CPA implies |
| 536 | ROG-CPA-$\Wror$ for the specific emitter $\Wror$ only. |
| 537 | \end{remark} |
| 538 | \begin{proof}[Proof of proposition \ref{prop:rog-and-lor}] |
| 539 | \begin{enumerate} |
| 540 | \item Let $W$ and $\E$ be given, and let $A$ be an adversary attacking the |
| 541 | LOR-CPA security of $\E$. Consider adversary $B$ attacking $\E$'s |
| 542 | ROG-CPA-$W$ security. |
| 543 | \begin{program} |
| 544 | Adversary $B^E(\cdot)$: \+ \\ |
| 545 | $b^* \getsr \Bin$; \\ |
| 546 | $b \gets A^{E(\id{lr}(b^*, \cdot, \cdot))}$; \\ |
| 547 | \IF $b = b^*$ \THEN \RETURN $1$ \ELSE \RETURN $0$; |
| 548 | \next |
| 549 | Function $\id{lr}(b, x_0, x_1)$: \+ \\ |
| 550 | \IF $b = 0$ \THEN \RETURN $x_0$ \ELSE \RETURN $x_1$; |
| 551 | \end{program} |
| 552 | If $E(\cdot)$ is the `result' encryption oracle, then $B$ simulates the |
| 553 | left-or-right game for the benefit of $A$, and therefore returns $1$ with |
| 554 | probability $(\Adv{lor-cpa}{\E}(A) + 1)/2$. On the other hand, if |
| 555 | $E(\cdot)$ returns `garbage' then the oracle responses are entirely |
| 556 | independent of $b^*$. This follows because $A$ is constrained to query |
| 557 | only on pairs of plaintexts with equal lengths, and the responses are |
| 558 | dependent only on these (common) lengths and any internal state and coin |
| 559 | tosses of $W$. So $b$ is independent of $b^*$ and $\Pr[b = b^*] = |
| 560 | \frac{1}{2}$. The result follows. |
| 561 | \item Let $\E = (G, E, D)$ be given. Our garbage emitter simulates the |
| 562 | real-or-random game of \cite{Bellare:2000:CST}. Let $K_W = \bot$ |
| 563 | initially: we define our emitter $\Wror$ thus: |
| 564 | \begin{program} |
| 565 | Garbage emitter $\Wror(n)$: \+ \\ |
| 566 | \IF $K_W = \bot$ \THEN $K_W \gets G()$; \\ |
| 567 | $x \getsr \Bin^n$; \\ |
| 568 | \RETURN $E_{K_W}(x)$; |
| 569 | \end{program} |
| 570 | We now show that $\InSec{rog-cpa-$\Wror$}(\E; t, q_E, \mu_E) \le |
| 571 | \InSec{lor-cpa}(\E; t + t_E \mu_E, q_E, \mu_E)$ for our $\Wror$. Let $A$ |
| 572 | be an adversary attacking the ROG-CPA-$\Wror$ security of $\E$. Consider |
| 573 | adversary $B$ attacking $\E$'s LOR-CPA security: |
| 574 | \begin{program} |
| 575 | Adversary $B^{E(\cdot, \cdot)}$: \+ \\ |
| 576 | $b \gets A^{\id{lorify}(\cdot)}$; \\ |
| 577 | \RETURN $b$; |
| 578 | \next |
| 579 | Function $\id{lorify}(x)$: \+ \\ |
| 580 | $x' \getsr \Bin^{|x|}$; \\ |
| 581 | \RETURN $E(x', x)$; |
| 582 | \end{program} |
| 583 | The adversary simulates the ROG-CPA-$\Wror$ games perfectly, since the |
| 584 | game has chosen the random $K_W$ for us already: the `left' game returns |
| 585 | only the results of encrypting random `garbage' plaintexts $x'$, while |
| 586 | the right game returns correct results of encrypting the given plaintexts |
| 587 | $x$. The result follows. \qed |
| 588 | \end{enumerate} |
| 589 | \end{proof} |
| 590 | \endgroup |
| 591 | |
| 592 | |
| 593 | \subsection{Message authentication} |
| 594 | \label{sec:mac} |
| 595 | |
| 596 | Our definitions for message authentication are standard; little needs to be |
| 597 | said of them. As with symmetric encryption, we begin with a syntactic |
| 598 | definition, and then describe our notion of security. |
| 599 | |
| 600 | \begin{definition}[Message authentication code] |
| 601 | \label{def:mac} |
| 602 | A \emph{message authentication code (MAC)} is a triple of algorithms $\M = |
| 603 | (G, T, V)$ with three (implicitly) associated sets: a keyspace, a message |
| 604 | space, and a tag space. |
| 605 | \begin{itemize} |
| 606 | \item $G$ is a probabilistic \emph{key-generation algorithm}. It is |
| 607 | invoked with no arguments, and returns a key $K$ which can be used with |
| 608 | the other two algorithms. We write $K \gets G()$. |
| 609 | \item $T$ is a probabilistic \emph{tagging algorithm}. It is invoked with |
| 610 | a key $K$ and a \emph{message} $x$ in the message space, and it returns a |
| 611 | \emph{tag} $\tau$ in the tag space. We write $\tau \gets T_K(x)$. |
| 612 | \item $V$ is a deterministic \emph{verification algorithm}. It is invoked |
| 613 | with a key $K$, a message $x$ and a tag $\tau$, and returns a bit $b \in |
| 614 | \Bin$. We write $b \gets V_K(x, \tau)$. |
| 615 | \end{itemize} |
| 616 | For correctness, we require that whenever $\tau$ is a possible result of |
| 617 | computing $T_K(x)$, then $V_K(x, \tau) = 1$. |
| 618 | \end{definition} |
| 619 | |
| 620 | Our notion of security is the strong unforgeability of |
| 621 | \cite{Abdalla:2001:DHIES,Bellare:2000:AER}. |
| 622 | |
| 623 | \begin{definition}[Strong unforgeability] |
| 624 | Let $\M = (G, T, V)$ be a message authentication code, and let $A$ |
| 625 | be an adversary. We perform the following experiment. |
| 626 | \begin{program} |
| 627 | Experiment $\Expt{suf-cma}{\M}(A)$: \+ \\ |
| 628 | $K \gets G()$; \\ |
| 629 | $\Xid{T}{list} \gets \emptyset$; \\ |
| 630 | $\id{good} \gets 0$; \\ |
| 631 | $A^{\id{tag}(\cdot), \id{verify}(\cdot, \cdot)}$; \\ |
| 632 | \RETURN $\id{good}$; |
| 633 | \newline |
| 634 | Oracle $\id{tag}(x)$: \+ \\ |
| 635 | $\tau \gets T_K(x)$; \\ |
| 636 | $\Xid{T}{list} \gets \Xid{T}{list} \cup \{(x, \tau)\}$; \\ |
| 637 | \RETURN $\tau$; |
| 638 | \next |
| 639 | Oracle $\id{verify}(x, \tau)$: \+ \\ |
| 640 | $b \gets V_K(x, \tau)$; \\ |
| 641 | \IF $b = 1 \land (x, \tau) \notin \Xid{T}{list}$ \THEN |
| 642 | $\id{good} \gets 1$; \\ |
| 643 | \RETURN $b$; |
| 644 | \end{program} |
| 645 | That is, the adversary `wins' if it submits a query to its verification |
| 646 | oracle which is \emph{new} -- doesn't match any message/tag pair from the |
| 647 | tagging oracle -- and \emph{valid} -- the verification oracle returned |
| 648 | success. We define the adversary's \emph{success probability} as |
| 649 | \[ \Succ{suf-cma}{\M}(A) = |
| 650 | \Pr[\Expt{suf-cma}{\M}(A) = 1]. \] |
| 651 | We define the \emph{SUF-CMA insecurity of $\M$} to be |
| 652 | \[ \InSec{suf-cma}(\M; t, q_T, \mu_T, q_V, \mu_V) = |
| 653 | \max_A \Adv{suf-cma}{\M}(A) \] |
| 654 | where the maximum is taken over all adversaries which run in time~$t$, |
| 655 | issue at most $q_T$ tagging queries totalling at most $\mu_T$ bits, and |
| 656 | issue at most $q_V$ verification queries totalling at most $\mu_V$ bits. |
| 657 | If $\InSec{suf-cma}(\M; t, q_T, \mu_T, q_V, \mu_V) \le \epsilon$ |
| 658 | then we say that $\E$ is $(t, q_T, \mu_T, q_V, \mu_V)$-SUF-CMA. |
| 659 | \end{definition} |
| 660 | |
| 661 | \subsection{Initialization vectors and encryption modes} |
| 662 | \label{sec:iv} |
| 663 | |
| 664 | In order to reduce the number of definitions in this paper to a tractable |
| 665 | level, we will describe the basic modes independently of how initialization |
| 666 | vectors (IVs) are chosen, and then construct the actual encryption schemes by |
| 667 | applying various IV selection methods from the modes. |
| 668 | |
| 669 | We consider the following IV selection methods. |
| 670 | \begin{description} |
| 671 | \item[Random selection] An IV is chosen uniformly at random just before |
| 672 | encrypting each message. |
| 673 | \item[Counter] The IV for each message is a \emph{generalized counter} (see |
| 674 | discussion below, and definition~\ref{def:genctr}). |
| 675 | \item[Encrypted counter] The IV for a message is the result of applying the |
| 676 | block cipher to a generalized counter, using the same key as for message |
| 677 | encryption. |
| 678 | \item[Carry-over] The IV for the first message is fixed; the IV for |
| 679 | subsequent messages is some function of the previous plaintexts or |
| 680 | ciphertexts (e.g., the last ciphertext block of the previous message). |
| 681 | \end{description} |
| 682 | Not all of these methods are secure for all of the modes we consider. |
| 683 | |
| 684 | \begin{definition}[Generalized counters] |
| 685 | \label{def:genctr} |
| 686 | If $S$ is a finite set, then a \emph{generalized counter in $S$} is an |
| 687 | bijection $c\colon \Nupto{|S|} \leftrightarrow S$. For brevity, we shall |
| 688 | refer simply to `counters', leaving implicit the generalization. |
| 689 | \end{definition} |
| 690 | |
| 691 | \begin{remark}[Examples of generalized counters] \leavevmode |
| 692 | \begin{itemize} |
| 693 | \item There is a `natural' binary representation of the natural numbers |
| 694 | $\Nupto{2^\ell}$ as $\ell$-bit strings: for any $n \in \Nupto{2^\ell}$, |
| 695 | let $R(n)$ be the unique $r \in \Bin^\ell$ such that $\smash{n = |
| 696 | \sum_{0\le i<\ell} 2^i r[i]}$. Then $R(\cdot)$ is a generalized counter |
| 697 | in $\Bin^\ell$. |
| 698 | \item We can represent elements of the finite field $\gf{2^\ell}$ as |
| 699 | $\ell$-bit strings. Let $p(x) \in \gf{2}[x]$ be a primitive polynomial |
| 700 | of degree $\ell$; then represent $\gf{2^\ell}$ by $\gf{2}[x]/(p(x))$. |
| 701 | Now for any $a \in \gf{2^\ell}$, let $R(a)$ be the unique $r \in |
| 702 | \Bin^\ell$ such that $\smash{a = \sum_{0\le i<\ell} x^i r[i]}$. Because |
| 703 | $p(x)$ is primitive, $x$ generates the multiplicative group |
| 704 | $\gf{2^\ell}^{\,*}$, so define $c(n) = R(x^n)$ for $0 \le n < 2^\ell - 1$ |
| 705 | and $c(2^{\ell - 1}) = 0^\ell$; then $c(\cdot)$ is a generalized counter |
| 706 | in $\Bin^\ell$. This counter can be implemented efficiently in hardware |
| 707 | using a linear feedback shift register. |
| 708 | \end{itemize} |
| 709 | \end{remark} |
| 710 | |
| 711 | \begin{definition}[Encryption modes] |
| 712 | \label{def:enc-modes} |
| 713 | |
| 714 | A \emph{block cipher encryption mode} $m_P = (\id{encrypt}, \id{decrypt})$ |
| 715 | is a pair of deterministic oracle algorithms (and implicitly-defined |
| 716 | plaintext and ciphertext spaces) which satisfy the following conditions: |
| 717 | \begin{enumerate} |
| 718 | \item The algorithm $\id{encrypt}$ runs with oracle access to a permutation |
| 719 | $P\colon \Bin^\ell \leftrightarrow \Bin^\ell$; on input a plaintext $x$ |
| 720 | and an initialization vector $v \in \Bin^\ell$, it returns a ciphertext |
| 721 | $y$ and a \emph{chaining value} $v' \in \Bin^\ell$. We write $(v', y) = |
| 722 | \id{encrypt}^{P(\cdot)}(v, x)$. |
| 723 | \item The algorithm $\id{decrypt}$ runs with oracle access to a permutation |
| 724 | $P\colon \Bin^\ell \leftrightarrow \Bin^\ell$ and its inverse |
| 725 | $P^{-1}(\cdot)$; on input a ciphertext $y$ and an initialization vector |
| 726 | $v \in \Bin^\ell$, it returns a plaintext $x$. We write that $x = |
| 727 | \id{decrypt}^{P(\cdot), P^{-1}(\cdot)}(v, y)$. |
| 728 | \item For all permutations $P\colon \Bin^\ell \leftrightarrow \Bin^\ell$, |
| 729 | all plaintexts $x$ and all initialization vectors $v$, if $(v', y) = |
| 730 | \id{encrypt}^{P(\cdot)}(v, x)$ then $x = \id{decrypt}^{P(\cdot), |
| 731 | P^{-1}(\cdot)}(v, y)$. |
| 732 | \end{enumerate} |
| 733 | Similarly, a \emph{PRF encryption mode} $m_F = (\id{encrypt}, |
| 734 | \id{decrypt})$ is a pair of deterministic oracle algorithms (and |
| 735 | implicitly-defined plaintext and ciphertext spaces) which satisfy the |
| 736 | following conditions: |
| 737 | \begin{enumerate} |
| 738 | \item The algorithm $\id{encrypt}$ runs with oracle access to a function |
| 739 | $F\colon \Bin^\ell \to \Bin^L$; on input a plaintext $x$ and an |
| 740 | initialization vector $v \in \Bin^\ell$, it returns a ciphertext $y$ and |
| 741 | a \emph{chaining value} $v' \in \Bin^\ell$. We write $(v', y) = |
| 742 | \id{encrypt}^{F(\cdot)}(v, x)$. |
| 743 | \item The algorithm $\id{decrypt}$ runs with oracle access to a function |
| 744 | $F\colon \Bin^\ell \to \Bin^L$; on input a ciphertext $y$ and an |
| 745 | initialization vector $v \in \Bin^\ell$, it returns a plaintext $x$. We |
| 746 | write that $(v', x) = \id{decrypt}^{F(\cdot)}(v, y)$. |
| 747 | \item For all functions $F\colon \Bin^\ell \to \Bin^L$, all plaintexts $x$ |
| 748 | and all initialization vectors $v$, if $(v', y) = |
| 749 | \id{encrypt}^{F(\cdot)}(v, x)$ then $x = \id{decrypt}^{F(\cdot)}(v, y)$. |
| 750 | \qed |
| 751 | \end{enumerate} |
| 752 | \end{definition} |
| 753 | |
| 754 | \begin{definition}[Symmetric encryption schemes from modes] |
| 755 | \label{def:enc-scheme} |
| 756 | Let $F$ be a pseudorandom permutation on $\Bin^\ell$ (resp.\ a |
| 757 | pseudorandom function from $\Bin^\ell$ to $\Bin^L$); let $m = |
| 758 | (\id{encrypt}, \id{decrypt})$ be a block cipher (resp.\ PRF) encryption |
| 759 | mode. (To save on repetition, if $F$ is a PRF then define $F_K^{-1}(x) = |
| 760 | \bot$ for all keys $K$ and inputs $x$.) We define the following symmetric |
| 761 | encryption schemes according to how IVs are selected. |
| 762 | |
| 763 | \begin{itemize} |
| 764 | \def\Enc{\Xid{\E}{$m$\what}^{\super}} |
| 765 | \def\GG{\Xid{G}{$m$\what}^{\super}} |
| 766 | \def\EE{\Xid{E}{$m$\what}^{\super}} |
| 767 | \def\DD{\Xid{D}{$m$\what}^{\super}} |
| 768 | |
| 769 | \def\what{$\$$} |
| 770 | \def\super{F} |
| 771 | \item Randomized selection: define $\Enc = (\GG, \EE, \DD)$, where |
| 772 | \begin{program} |
| 773 | Algorithm $\GG()$: \+ \\ |
| 774 | $K \getsr \keys F$; \\ |
| 775 | \RETURN $K$; |
| 776 | \next |
| 777 | Algorithm $\EE(K, x)$: \+ \\ |
| 778 | $v \getsr \Bin^\ell$; \\ |
| 779 | $(v', x) \gets \id{encrypt}^{F_K(\cdot)}(v, x)$; \\ |
| 780 | \RETURN $(v, y)$; |
| 781 | \next |
| 782 | Algorithm $\DD(K, y')$; \+ \\ |
| 783 | $(v, y) \gets y'$; \\ |
| 784 | $(v', x) \gets {}$ \\ |
| 785 | \qquad $\id{decrypt}^{F_K(\cdot), F^{-1}_K(\cdot)}(v, y)$; \\ |
| 786 | \RETURN $x$; |
| 787 | \end{program} |
| 788 | |
| 789 | \def\what{C} |
| 790 | \def\super{F, c} |
| 791 | \def\imsg{\Xid{i}{msg}} |
| 792 | \item Generalized counters: define $\Enc = (\GG, \EE, \DD)$, where $c$ is a |
| 793 | generalized counter in $\Bin^\ell$, and |
| 794 | \begin{program} |
| 795 | Algorithm $\GG()$: \+ \\ |
| 796 | $K \getsr \keys F$; \\ |
| 797 | $\imsg \gets 0$; \\ |
| 798 | \RETURN $K$; |
| 799 | \next |
| 800 | Algorithm $\EE(K, x)$: \+ \\ |
| 801 | $i \gets c(\imsg)$; \\ |
| 802 | $(v', x) \gets \id{encrypt}^{F_K(\cdot)}(i, x)$; \\ |
| 803 | $\imsg \gets \imsg + 1$; \\ |
| 804 | \RETURN $(i, y)$; |
| 805 | \next |
| 806 | Algorithm $\DD(K, y')$; \+ \\ |
| 807 | $(i, y) \gets y'$; \\ |
| 808 | $(v', x) \gets {}$ \\ |
| 809 | \qquad $\id{decrypt}^{F_K(\cdot), F^{-1}_K(\cdot)}(i, y)$; \\ |
| 810 | \RETURN $x$; |
| 811 | \end{program} |
| 812 | |
| 813 | \def\what{E} |
| 814 | \def\super{F, c} |
| 815 | \def\imsg{\Xid{i}{msg}} |
| 816 | \item Encrypted counters: if $L \ge \ell$, then define $\Enc = (\GG, \EE, |
| 817 | \DD)$, where $c$ is a generalized counter in $\Bin^\ell$, and |
| 818 | \begin{program} |
| 819 | Algorithm $\GG()$: \+ \\ |
| 820 | $K \getsr \keys F$; \\ |
| 821 | $\imsg \gets 0$; \\ |
| 822 | \RETURN $K$; |
| 823 | \next |
| 824 | Algorithm $\EE(K, x)$: \+ \\ |
| 825 | $i \gets c(\imsg)$; \\ |
| 826 | $v \gets F_K(i)[0 \bitsto \ell]$; \\ |
| 827 | $(v', x) \gets \id{encrypt}^{F_K(\cdot)}(v, x)$; \\ |
| 828 | $\imsg \gets \imsg + 1$; \\ |
| 829 | \RETURN $(i, y)$; |
| 830 | \next |
| 831 | Algorithm $\DD(K, y')$; \+ \\ |
| 832 | $(i, y) \gets y'$; \\ |
| 833 | $v \gets F_K(i)[0 \bitsto \ell]$; \\ |
| 834 | $(v', x) \gets {}$ \\ |
| 835 | \qquad $\id{decrypt}^{F_K(\cdot), F^{-1}_K(\cdot)}(v, y)$; \\ |
| 836 | \RETURN $x$; |
| 837 | \end{program} |
| 838 | (We require $L \ge \ell$ for this to be well-defined; otherwise the |
| 839 | encrypted counter value is too short.) |
| 840 | |
| 841 | \def\what{L} |
| 842 | \def\super{F, V_0} |
| 843 | \def\vnext{\Xid{v}{next}} |
| 844 | \item Carry-over: define $\Enc = (\GG, \EE, \DD)$ where $V_0 \in \Bin^\ell$ |
| 845 | is the initialization vector for the first message, and |
| 846 | \begin{program} |
| 847 | Algorithm $\GG()$: \+ \\ |
| 848 | $K \getsr \keys F$; \\ |
| 849 | $\vnext \gets V_0$; \\ |
| 850 | \RETURN $K$; |
| 851 | \next |
| 852 | Algorithm $\EE(K, x)$: \+ \\ |
| 853 | $v \gets \vnext$; \\ |
| 854 | $(v', x) \gets \id{encrypt}^{F_K(\cdot)}(v, x)$; \\ |
| 855 | $\vnext \gets v'$; \\ |
| 856 | \RETURN $(v, y)$; |
| 857 | \next |
| 858 | Algorithm $\DD(K, y')$; \+ \\ |
| 859 | $(v, y) \gets y'$; \\ |
| 860 | $(v', x) \gets {}$ \\ |
| 861 | \qquad $\id{decrypt}^{F_K(\cdot), F^{-1}_K(\cdot)}(v, y)$; \\ |
| 862 | \RETURN $x$; |
| 863 | \end{program} |
| 864 | \end{itemize} |
| 865 | |
| 866 | Note that, while the encryption algorithms of the above schemes are either |
| 867 | randomized or stateful, the decryption algorithms are simple and |
| 868 | deterministic. |
| 869 | \end{definition} |
| 870 | |
| 871 | The following simple and standard result will be very useful in our proofs. |
| 872 | |
| 873 | \begin{proposition} |
| 874 | \label{prop:enc-info-to-real} |
| 875 | \leavevmode |
| 876 | \begin{enumerate} |
| 877 | \item Suppose that $\E^P = (G^P, E^P, D^P)$ is one of the symmetric |
| 878 | encryption schemes of definition~\ref{def:enc-scheme}, constructed from a |
| 879 | pseudorandom permutation $P\colon \Bin^\ell \leftrightarrow \Bin^\ell$. |
| 880 | If $q$ is an upper bound on the number of PRP applications required for |
| 881 | the encryption $q_E$ messages totalling $\mu_E$ bits, then there is some |
| 882 | small constant $t'$ such that |
| 883 | \[ \InSec{lor-cpa}(\E^P; t, q_E, \mu_E) \le |
| 884 | \InSec{lor-cpa}(\E^{\Perm{\ell}}; t, q_E, \mu_E) + |
| 885 | 2 \cdot \InSec{prp}(P; t + q t', q) . |
| 886 | \] |
| 887 | \item Similarly, suppose that $\E^F = (G^F, E^F, D^F)$ is one of the |
| 888 | symmetric encryption schemes of definition~\ref{def:enc-scheme}, |
| 889 | constructed from a pseudorandom function $F\colon \Bin^\ell \to \Bin^L$. |
| 890 | If $q$ is an upper bound on the number of PRP applications required for |
| 891 | the encryption $q_E$ messages totalling $\mu_E$ bits, then there is some |
| 892 | small constant $t'$ such that |
| 893 | \[ \InSec{lor-cpa}(\E^F; t, q_E, \mu_E) \le |
| 894 | \InSec{lor-cpa}(\E^{\Func{\ell}{L}}; t, q_E, \mu_E) + |
| 895 | 2 \cdot \InSec{prf}(F; t + q t', q) . |
| 896 | \] |
| 897 | \end{enumerate} |
| 898 | \end{proposition} |
| 899 | \begin{proof} |
| 900 | \begin{enumerate} |
| 901 | \item Let $A$ be an adversary attacking the LOR-CPA security of $\E^P$, |
| 902 | which takes time $t$ and issues $q_E$ encryption queries totalling |
| 903 | $\mu_E$ bits. We construct an adversary $B$ attacking the security of |
| 904 | the PRP $P$ as follows. $B$ selects a random $b^* \inr \Bin$. It then |
| 905 | runs $A$, simulating the LOR-CPA game by using $b^*$ to decide whether to |
| 906 | encrypt the left or right plaintext, and using its oracle access to $P$ |
| 907 | to do the encryption. Eventually, $A$ returns a bit $b$. If $b = b^*$, |
| 908 | $B$ returns $1$ (indicating `pseudorandom'); otherwise it returns $0$. |
| 909 | |
| 910 | If $B$'s oracle is selected from the PRP $P$, then $B$ correctly |
| 911 | simulates the LOR-CPA game for $\E^P$, and $B$ returns $1$ with |
| 912 | probability precisely $(\Adv{lor-cpa}{\E^P}(A) + 1)/2$. Conversely, if |
| 913 | $B$'s oracle is a random permutation, then $B$ correctly simulates the |
| 914 | LOR-CPA game for $\E^{\Perm{\ell}}$, so $B$ returns $1$ with probability |
| 915 | $(\Adv{lor-cpa}{\E^P}(A) + 1)/2$. Thus, we have |
| 916 | \begin{eqnarray}[rl] |
| 917 | \Adv{prp}{P}(B) |
| 918 | & = (\Adv{lor-cpa}{\E^P}(A) + 1)/2 |
| 919 | - (\Adv{lor-cpa}{\E^{\Perm{\ell}}}(A) + 1)/2 \\ |
| 920 | & = (\Adv{lor-cpa}{\E^P}(A) |
| 921 | - \Adv{lor-cpa}{\E^{\Perm{\ell}}}(A))/2 . |
| 922 | \end{eqnarray} |
| 923 | Note that the extra work which $B$ does over $A$ -- initialization, |
| 924 | tidying up and encrypting messages -- is bounded by some small constant |
| 925 | $t_P$ multiplied by the number of oracle queries~$q$ made by~$B$, and the |
| 926 | required result follows by multiplying through by~$2$ and rearranging. |
| 927 | \item The proof for this case is almost identical: merely substitute $F$ |
| 928 | for $P$, `PRF' for `PRP' and $\Func{\ell}{L}$ for $\Perm{\ell}$ in the |
| 929 | above. \qed |
| 930 | \end{enumerate} |
| 931 | \end{proof} |
| 932 | |
| 933 | Of course, proving theorems about each of the above schemes individually will |
| 934 | be very tedious. We therefore define a `hybrid' scheme which switches |
| 935 | between the above selection methods. This isn't a practical encryption |
| 936 | scheme -- just a `trick' to reduce the number of complicated proofs we need |
| 937 | to give. |
| 938 | |
| 939 | \begin{definition}[Hybrid encryption modes] |
| 940 | \label{def:enc-hybrid} |
| 941 | \def\Enc{\Xid{\E}{$m$\what}^{\super}_{\sub}} |
| 942 | \def\GG{\Xid{G}{$m$\what}^{\super}_{\sub}} |
| 943 | \def\EE{\Xid{E}{$m$\what}^{\super}_{\sub}} |
| 944 | \def\DD{\Xid{D}{$m$\what}^{\super}_{\sub}} |
| 945 | \def\what{H} |
| 946 | \def\super{F, V_0, c} |
| 947 | \def\sub{n_L, n_C, n_E} |
| 948 | \def\imsg{\Xid{i}{msg}} |
| 949 | \def\vnext{\Xid{v}{next}} |
| 950 | Let $n_L$, $n_C$ and $n_E$ be nonnegative integers, with $n_L + n_C + n_E |
| 951 | \le 2^{\ell}$; let $F$ be a pseudorandom permutation on $\Bin^\ell$ (resp.\ |
| 952 | a pseudorandom function from $\Bin^\ell$ to $\Bin^L$); let $m = |
| 953 | (\id{encrypt}, \id{decrypt})$ be a block cipher (resp.\ PRF) encryption |
| 954 | mode let $V_0 \in \Bin^\ell$ be an initialization vector; and let $c\colon |
| 955 | \Nupto{2^\ell} \to \Bin^\ell$ be a generalized counter. (Again, if $F$ is |
| 956 | a PRF, we set $F_K(x) = \bot$ for all $K$ and $x$.) We define the scheme |
| 957 | $\Enc = (\GG, \EE, \DD)$ as follows. |
| 958 | \begin{program} |
| 959 | Algorithm $\GG()$: \+ \\ |
| 960 | $K \getsr \keys F$; \\ |
| 961 | $\imsg \gets 0$; \\ |
| 962 | $\vnext \gets V_0$; \\ |
| 963 | \RETURN $K$; |
| 964 | \next |
| 965 | Algorithm $\DD(K, y')$; \+ \\ |
| 966 | $(v, y) \gets y'$; \\ |
| 967 | $(v', x) \gets \id{decrypt}^{F_K(\cdot), F^{-1}_K(\cdot)}(v, y)$; \\ |
| 968 | \RETURN $x$; |
| 969 | \newline |
| 970 | Algorithm $\EE(K, x)$: \+ \\ |
| 971 | \IF $\imsg < n_L$ \THEN $v \gets \vnext$; \\ |
| 972 | \ELSE\IF $\imsg < n_L + n_C$ \THEN $v \gets c(\imsg)$; \\ |
| 973 | \ELSE\IF $\imsg < n_L + n_C + n_E$ \THEN |
| 974 | $v \gets F_K(c(\imsg)[0 \bitsto \ell])$; \\ |
| 975 | \ELSE $v \getsr \Bin^\ell$; \\ |
| 976 | $(v', x) \gets \id{encrypt}^{F_K(\cdot)}(v, x)$; \\ |
| 977 | $\vnext \gets v'$; \\ |
| 978 | $\imsg \gets \imsg + 1$; \\ |
| 979 | \RETURN $(v, y)$; |
| 980 | \end{program} |
| 981 | For this to be well-defined, we require that $L \ge \ell$ or $n_E = 0$ -- |
| 982 | otherwise the encrypted counter values are too short. |
| 983 | \end{definition} |
| 984 | |
| 985 | The following proposition relates the security of our artificial hybrid |
| 986 | scheme to that of the practical schemes defined in |
| 987 | definition~\ref{def:enc-scheme}. |
| 988 | |
| 989 | \begin{proposition} |
| 990 | \label{prop:enc-hybrid} |
| 991 | Let $F$ be a pseudorandom permutation on $\Bin^\ell$ (resp.\ a pseudorandom |
| 992 | function from $\Bin^\ell$ to $\Bin^L$); let $m$ be a block cipher (resp.\ |
| 993 | PRF) encryption mode. Then: |
| 994 | \begin{enumerate} |
| 995 | \def\ii#1{\item $\displaystyle#1$} |
| 996 | \ii{\InSec{lor-cpa}(\Xid{E}{$m\$$}^F; t, q_E, \mu_E) \le |
| 997 | \InSec{lor-cpa}(\Xid{E}{$m$H}^{F, V_0, c}_{0, 0, 0}; t, q_E, \mu_E)} |
| 998 | \ii{\InSec{lor-cpa}(\Xid{E}{$m$C}^{F, c}; t, q_E, \mu_E) \le |
| 999 | \InSec{lor-cpa} |
| 1000 | (\Xid{E}{$m$H}^{F, V_0, c}_{q_E, 0, 0}; t, q_E, \mu_E)} |
| 1001 | \ii{\InSec{lor-cpa}(\Xid{E}{$m$E}^{F, c}; t, q_E, \mu_E) \le |
| 1002 | \InSec{lor-cpa} |
| 1003 | (\Xid{E}{$m$H}^{F, V_0, c}_{0, q_E, 0}; t, q_E, \mu_E)} |
| 1004 | \ii{\InSec{lor-cpa}(\Xid{E}{$m$L}^{F, V_0}; t, q_E, \mu_E) \le |
| 1005 | \InSec{lor-cpa} |
| 1006 | (\Xid{E}{$m$H}^{F, V_0, c}_{0, 0, q_E}; t, q_E, \mu_E)} |
| 1007 | \end{enumerate} |
| 1008 | \end{proposition} |
| 1009 | |
| 1010 | \begin{proof} |
| 1011 | For 1, it suffices to observe that $\Xid{E}{$m\$$}^F \equiv |
| 1012 | \Xid{E}{$m$H}^{F, V_0, c}_{0, 0, 0}$ for any $c$, $V_0$. For 2, note that |
| 1013 | $\Xid{E}{$m$C}^{F, c}$ behaves identically to $\Xid{E}{$m$H}^{F, V_0, |
| 1014 | c}_{q_E, 0, 0}$ for any $c$, $V_0$ for the first $q_E$ encryption |
| 1015 | queries; but no adversary is permitted to exceed this limit, and hence no |
| 1016 | adversary can distinguish the two. Similarly, for 4, note that |
| 1017 | $\Xid{E}{$m$L}^{F, V_0}$ behaves identically to $\Xid{E}{$m$H}^{F, V_0, |
| 1018 | c}_{0, 0, q_E}$ for any $c$, $V_0$ for the first $q_E$ encryption |
| 1019 | queries. |
| 1020 | |
| 1021 | The case of 3 is slightly more complicated: $\Xid{E}{$m$E}^{F, c}$ behaves |
| 1022 | identically to $\Xid{E}{$m$H}^{F, V_0, c}_{0, q_E, 0}$ for the first $q_E$ |
| 1023 | encryption queries \emph{except} that the latter returns different |
| 1024 | initialization vectors from its encryption oracle. However, since the |
| 1025 | counter $c$ is fixed public knowledge, it is trivial to construct a fully |
| 1026 | faithful replica of the $m$E game given the hybrid oracle, such that no |
| 1027 | adversary can distinguish the two. |
| 1028 | \end{proof} |
| 1029 | |
| 1030 | %%%-------------------------------------------------------------------------- |
| 1031 | |
| 1032 | \section{Ciphertext block chaining (CBC) encryption} |
| 1033 | \label{sec:cbc} |
| 1034 | |
| 1035 | \subsection{Description} |
| 1036 | \label{sec:cbc-desc} |
| 1037 | |
| 1038 | Suppose $E$ is an $\ell$-bit pseudorandom permutation. CBC mode works as |
| 1039 | follows. Given a message $X$, we divide it into $\ell$-bit blocks $x_0$, |
| 1040 | $x_1$, $\ldots$, $x_{n-1}$. Choose an initialization vector $v \in |
| 1041 | \Bin^\ell$. Before passing each $x_i$ through $E$, we XOR it with the |
| 1042 | previous ciphertext, with $v$ standing in for the first block: |
| 1043 | \begin{equation} |
| 1044 | y_0 = E_K(x_0 \xor v) \qquad |
| 1045 | y_i = E_K(x_i \xor y_{i-1} \ \text{(for $1 \le i < n$)}. |
| 1046 | \end{equation} |
| 1047 | The ciphertext is then the concatenation of $v$ and the $y_i$. Decryption is |
| 1048 | simple: |
| 1049 | \begin{equation} |
| 1050 | x_0 = E^{-1}_K(y_0) \xor v \qquad |
| 1051 | x_i = E^{-1}_K(y_i) \xor y_{i-1} \ \text{(for $1 \le i < n$)} |
| 1052 | \end{equation} |
| 1053 | See figure~\ref{fig:cbc} for a diagram of CBC encryption. |
| 1054 | |
| 1055 | \begin{figure} |
| 1056 | \begin{cgraph}{cbc-mode} |
| 1057 | []!{0; <0.85cm, 0cm>: <0cm, 0.5cm>::} |
| 1058 | *+=(1, 0)+[F]{\mathstrut x_0}="x" |
| 1059 | :[dd] *{\xor}="xor" |
| 1060 | [ll] *+=(1, 0)+[F]{\mathstrut v} :"xor" |
| 1061 | :[dd] *+[F]{E}="e" :[ddd] *+=(1, 0)+[F]{\mathstrut y_0}="i" |
| 1062 | "e" [l] {K} :"e" |
| 1063 | [rrruuuu] *+=(1, 0)+[F]{\mathstrut x_1}="x" |
| 1064 | :[dd] *{\xor}="xor" |
| 1065 | "e" [d] :`r [ru] `u "xor" "xor" |
| 1066 | :[dd] *+[F]{E}="e" :[ddd] |
| 1067 | *+=(1, 0)+[F]{\mathstrut y_1}="i" |
| 1068 | "e" [l] {K} :"e" |
| 1069 | [rrruuuu] *+=(1, 0)+[F--]{\mathstrut x_i}="x" |
| 1070 | :@{-->}[dd] *{\xor}="xor" |
| 1071 | "e" [d] :@{-->}`r [ru] `u "xor" "xor" |
| 1072 | :@{-->}[dd] *+[F]{E}="e" :@{-->}[ddd] |
| 1073 | *+=(1, 0)+[F--]{\mathstrut y_i}="i" |
| 1074 | "e" [l] {K} :@{-->}"e" |
| 1075 | [rrruuuu] *+=(1, 0)+[F]{\mathstrut x_{n-1}}="x" |
| 1076 | :[dd] *{\xor}="xor" |
| 1077 | "e" [d] :@{-->}`r [ru] `u "xor" "xor" |
| 1078 | :[dd] *+[F]{E}="e" :[ddd] |
| 1079 | *+=(1, 0)+[F]{\mathstrut y_{n-1}}="i" |
| 1080 | "e" [l] {K} :"e" |
| 1081 | \end{cgraph} |
| 1082 | |
| 1083 | \caption{Encryption using CBC mode} |
| 1084 | \label{fig:cbc} |
| 1085 | \end{figure} |
| 1086 | |
| 1087 | \begin{definition}[CBC algorithms] |
| 1088 | \label{def:cbc} |
| 1089 | For any permutation $P\colon \Bin^\ell \to \Bin^\ell$, any initialization |
| 1090 | vector $v \in \Bin^\ell$, any plaintext $x \in \Bin^{\ell\N}$ and any |
| 1091 | ciphertext $y \in \Bin^*$, we define the encryption mode $\id{CBC} = |
| 1092 | (\id{cbc-encrypt}, \id{cbc-decrypt})$ as follows: |
| 1093 | \begin{program} |
| 1094 | Algorithm $\id{cbc-encrypt}^{P(\cdot)}(v, x)$: \+ \\ |
| 1095 | $y \gets \emptystring$; \\ |
| 1096 | \FOR $i = 0$ \TO $|x|/\ell$ \DO \\ \ind |
| 1097 | $x_i \gets x[\ell i \bitsto \ell (i + 1)]$; \\ |
| 1098 | $y_i \gets P(x_i \xor v)$; \\ |
| 1099 | $v \gets y_i$; \\ |
| 1100 | $y \gets y \cat y_i$; \- \\ |
| 1101 | \RETURN $(v, y)$; |
| 1102 | \next |
| 1103 | Algorithm $\id{cbc-decrypt}^{P(\cdot), P^{-1}(\cdot)}(v, y)$: \+ \\ |
| 1104 | \IF $|y| \bmod \ell \ne 0$ \THEN \RETURN $\bot$; \\ |
| 1105 | $x \gets \emptystring$; \\ |
| 1106 | \FOR $1 = 0$ \TO $|y|/\ell$ \DO \\ \ind |
| 1107 | $y_i \gets y[\ell i \bitsto \ell (i + 1)]$; \\ |
| 1108 | $x_i \gets P^{-1}(y_i) \xor v$; \\ |
| 1109 | $v \gets y_i$; \\ |
| 1110 | $x \gets x \cat x_i$; \- \\ |
| 1111 | \RETURN $(v, x)$; |
| 1112 | \end{program} |
| 1113 | Now, let $c$ be a generalized counter in $\Bin^\ell$. We define the |
| 1114 | encryption schemes $\Xid{E}{CBC$\$$}^P$, $\Xid{E}{CBCE}^{P, c}$ and |
| 1115 | $\Xid{E}{CBCH}^{P, c, \bot}_{0, 0, n_E}$, as described in |
| 1116 | definition~\ref{def:enc-scheme}. |
| 1117 | \end{definition} |
| 1118 | |
| 1119 | \subsection{Security of CBC mode} |
| 1120 | |
| 1121 | We now present our main theorem on CBC mode. |
| 1122 | |
| 1123 | \begin{theorem}[Security of hybrid CBC mode] |
| 1124 | \label{thm:cbc} |
| 1125 | Let $P\colon \keys P \times \Bin^\ell \to \Bin^\ell$ be a pseudorandom |
| 1126 | permutation; let $V_0 \in \Bin^\ell$ be an initialization vector; let $n_L |
| 1127 | \in \{ 0, 1 \}$; let $c$ be a generalized counter in $\Bin^\ell$; and |
| 1128 | let $n_C \in \N$ be a nonnegative integer; and suppose that at most one of |
| 1129 | $n_L$ and $n_C$ is nonzero. Then, for any $t$, $q_E \ge n$ and $\mu_E$, |
| 1130 | \[ \InSec{lor-cpa} |
| 1131 | (\Xid{\E}{CBCH}^{P, c, V_0}_{n_L, 0, n_E}; t, q_E, \mu_E) \le |
| 1132 | 2 \cdot \InSec{prp}(P; t + q t_P, q) + \frac{q (q - 1)}{2^\ell - q} |
| 1133 | \] |
| 1134 | where $q = n_L + n_E + \mu_E/\ell$ and $t_P$ is some small constant. |
| 1135 | \end{theorem} |
| 1136 | |
| 1137 | The proof of this theorem we postpone until section~\ref{sec:cbc-proof}. As |
| 1138 | promised, the security of our randomized and stateful schemes follow as |
| 1139 | simple corollaries. |
| 1140 | |
| 1141 | \begin{corollary}[Security of practical CBC modes] |
| 1142 | \label{cor:cbc} |
| 1143 | Let $P$ and $c$ be as in theorem~\ref{thm:cbc}. Then for any $t$, $q_E$ |
| 1144 | and $\mu_E$, and some small constant $t_P$, |
| 1145 | \begin{eqnarray*}[rl] |
| 1146 | \InSec{lor-cpa}(\Xid{\E}{CBC$\$$}^P; t, q_E, \mu_E) |
| 1147 | & \le 2 \cdot \InSec{prp}(P; t + q t_P, q) + \frac{q (q - 1)}{2^\ell - q} |
| 1148 | \\ |
| 1149 | \InSec{lor-cpa}(\Xid{\E}{CBCE}^{P, c}; t, q_E, \mu_E) |
| 1150 | & \le 2 \cdot \InSec{prp}(P; t + q' t_P, q') + |
| 1151 | \frac{q' (q' - 1)}{2^\ell - q'} |
| 1152 | \\ |
| 1153 | \tabpause{and} |
| 1154 | \InSec{lor-cpa}(\Xid{\E}{CBCL}^{P, V_0}; t, 1, \mu_E) |
| 1155 | & \le 2 \cdot \InSec{prp}(P; t + q t_P, q) + |
| 1156 | \frac{q (q - 1)}{2^\ell - q} |
| 1157 | \end{eqnarray*} |
| 1158 | where $q = \mu_E/\ell$, and $q' = q + q_E$. |
| 1159 | \end{corollary} |
| 1160 | \begin{proof} |
| 1161 | Follows from theorem~\ref{thm:cbc} and proposition~\ref{prop:enc-hybrid}. |
| 1162 | \end{proof} |
| 1163 | |
| 1164 | \begin{remark} |
| 1165 | The insecurity of CBC mode over that inherent in the underlying PRP is |
| 1166 | essentially a birthday bound: note for $q \le 2^{\ell/2}$, our denominator |
| 1167 | $2^\ell - q \approx 2^\ell$, and for larger $q$, the term $q (q - 1)/2^\ell |
| 1168 | > 1$ anyway, so all security is lost (according to the above result). |
| 1169 | Compared to \cite[theorem~17]{Bellare:2000:CST}, we gain the tiny extra |
| 1170 | term in the denominator, but lose the PRP-as-a-PRF term |
| 1171 | $q^2/2^{\ell-1}$.\footnote{% |
| 1172 | In fact, they don't prove the stated bound of $q (3 q - 2)/2^{\ell+1}$ |
| 1173 | but instead the larger $q (2 q - 1)/2^\ell$. The error is in the |
| 1174 | application of their proposition~8: the PRF-insecurity term is doubled, |
| 1175 | so the PRP-as-a-PRF term must be also.} % |
| 1176 | \end{remark} |
| 1177 | |
| 1178 | \subsection{Ciphertext stealing} |
| 1179 | |
| 1180 | Ciphertext stealing \cite{Daemen:1995:CHF,Schneier:1996:ACP,RFC2040} allows |
| 1181 | us to encrypt any message in $\Bin^*$ without the need for padding. The |
| 1182 | trick is to fill in the `gap' at the end of the last block with the end bit |
| 1183 | of the previous ciphertext, and then to put the remaining short penultimate |
| 1184 | block at the end. Decryption proceeds by first decrypting the final block to |
| 1185 | recover the remainder of the penultimate one. See |
| 1186 | figure~\ref{fig:cbc-steal}. |
| 1187 | |
| 1188 | \begin{figure} |
| 1189 | \begin{cgraph}{cbc-steal-enc} |
| 1190 | []!{0; <0.85cm, 0cm>: <0cm, 0.5cm>::} |
| 1191 | *+=(1, 0)+[F]{\mathstrut x_0}="x" |
| 1192 | :[dd] *{\xor}="xor" |
| 1193 | [ll] *+=(1, 0)+[F]{\mathstrut v} :"xor" |
| 1194 | :[dd] *+[F]{E}="e" :[ddddd] *+=(1, 0)+[F]{\mathstrut y_0}="i" |
| 1195 | "e" [l] {K} :"e" |
| 1196 | [rrruuuu] *+=(1, 0)+[F]{\mathstrut x_1}="x" |
| 1197 | :[dd] *{\xor}="xor" |
| 1198 | "e" [d] :`r [ru] `u "xor" "xor" |
| 1199 | :[dd] *+[F]{E}="e" :[ddddd] |
| 1200 | *+=(1, 0)+[F]{\mathstrut y_1}="i" |
| 1201 | "e" [l] {K} :"e" |
| 1202 | [rrruuuu] *+=(1, 0)+[F--]{\mathstrut x_i}="x" |
| 1203 | :@{-->}[dd] *{\xor}="xor" |
| 1204 | "e" [d] :@{-->}`r [ru] `u "xor" "xor" |
| 1205 | :@{-->}[dd] *+[F]{E}="e" :@{-->}[ddddd] |
| 1206 | *+=(1, 0)+[F--]{\mathstrut y_i}="i" |
| 1207 | "e" [l] {K} :@{-->}"e" |
| 1208 | [rrruuuu] *+=(1, 0)+[F]{\mathstrut x_{n-2}}="x" |
| 1209 | :[dd] *{\xor}="xor" |
| 1210 | "e" [d] :@{-->}`r [ru] `u "xor" "xor" |
| 1211 | :[dd] *+[F]{E}="e" |
| 1212 | "e" [l] {K} :"e" |
| 1213 | [rrruuuu] *+=(1, 0)+[F]{\mathstrut x_{n-1} \cat 0^{\ell-t}}="x" |
| 1214 | :[dd] *{\xor}="xor" |
| 1215 | "e" [d] :`r [ru] `u "xor" "xor" |
| 1216 | "e" [dddddrrr] *+=(1, 0)+[F]{\mathstrut y_{n-1}[0 \bitsto t]}="i" |
| 1217 | "e" [dd] ="x" |
| 1218 | "i" [uu] ="y" |
| 1219 | []!{"x"; "e" **{}, "x"+/4pt/ ="p", |
| 1220 | "x"; "y" **{}, "x"+/4pt/ ="q", |
| 1221 | "y"; "x" **{}, "y"+/4pt/ ="r", |
| 1222 | "y"; "i" **{}, "y"+/4pt/ ="s", |
| 1223 | "e"; |
| 1224 | "p" **\dir{-}; |
| 1225 | "q" **\crv{"x"}; |
| 1226 | "r" **\dir{-}; |
| 1227 | "s" **\crv{"y"}; |
| 1228 | "i" **\dir{-}?>*\dir{>}} |
| 1229 | "xor" :[dd] *+[F]{E}="e" |
| 1230 | "e" [l] {K} :"e" |
| 1231 | "e" [dddddlll] *+=(1, 0)+[F]{\mathstrut y_{n-2}}="i" |
| 1232 | "e" [dd] ="x" |
| 1233 | "i" [uu] ="y" |
| 1234 | []!{"x"; "e" **{}, "x"+/4pt/ ="p", |
| 1235 | "x"; "y" **{}, "x"+/4pt/ ="q", |
| 1236 | "y"; "x" **{}, "y"+/4pt/ ="r", |
| 1237 | "y"; "i" **{}, "y"+/4pt/ ="s", |
| 1238 | "x"; "y" **{} ?="c" ?(0.5)/-4pt/ ="cx" ?(0.5)/4pt/ ="cy", |
| 1239 | "e"; |
| 1240 | "p" **\dir{-}; |
| 1241 | "q" **\crv{"x"}; |
| 1242 | "cx" **\dir{-}; |
| 1243 | "c" *[@]\cir<4pt>{d^u}; |
| 1244 | "cy"; |
| 1245 | "r" **\dir{-}; |
| 1246 | "s" **\crv{"y"}; |
| 1247 | "i" **\dir{-}?>*\dir{>}} |
| 1248 | \end{cgraph} |
| 1249 | |
| 1250 | \begin{cgraph}{cbc-steal-dec} |
| 1251 | []!{0; <0.85cm, 0cm>: <0cm, 0.5cm>::} |
| 1252 | *+=(1, 0)+[F]{\mathstrut y_0}="y" |
| 1253 | :[ddddddd] *+[F]{D}="d" [l] {K} :"d" |
| 1254 | [rrrdd] *{\xor} ="nx" "d" [u] :`r [rd] `d "nx" "nx" |
| 1255 | "d" :[dd] *{\xor} ="xor" [ll] *+=(1, 0)+[F]{\mathstrut v} :"xor" |
| 1256 | :[dd] *+=(1, 0)+[F]{\mathstrut x_0} "nx"="xor" |
| 1257 | "y" [rrr] *+=(1, 0)+[F]{\mathstrut y_1}="y" |
| 1258 | :[ddddddd] *+[F]{D}="d" [l] {K} :"d" |
| 1259 | [rrrdd] *{\xor} ="nx" "d" [u] :@{-->}`r [rd] `d "nx" "nx" |
| 1260 | "d" :"xor" |
| 1261 | :[dd] *+=(1, 0)+[F]{\mathstrut x_1} "nx"="xor" |
| 1262 | "y" [rrr] *+=(1, 0)+[F--]{\mathstrut y_i}="y" |
| 1263 | :@{-->}[ddddddd] *+[F]{D}="d" [l] {K} :"d" |
| 1264 | [rrrdd] *{\xor} ="nx" "d" [u] :@{-->}`r [rd] `d "nx" "nx" |
| 1265 | "d" :"xor" |
| 1266 | :[dd] *+=(1, 0)+[F--]{\mathstrut x_i} "nx"="xor" |
| 1267 | "y" [rrr] *+=(1, 0)+[F]{\mathstrut y_{n-2}}="y" |
| 1268 | [dddddrrr] *+[F]{D}="d" [r] {K} :"d" |
| 1269 | "y" [dd] ="x" |
| 1270 | "d" [uu] ="e" |
| 1271 | []!{"x"; "y" **{}, "x"+/4pt/ ="p", |
| 1272 | "x"; "e" **{}, "x"+/4pt/ ="q", |
| 1273 | "e"; "x" **{}, "e"+/4pt/ ="r", |
| 1274 | "e"; "d" **{}, "e"+/4pt/ ="s", |
| 1275 | "y"; |
| 1276 | "p" **\dir{-}; |
| 1277 | "q" **\crv{"x"}; |
| 1278 | "r" **\dir{-}; |
| 1279 | "s" **\crv{"e"}; |
| 1280 | "d" **\dir{-}?>*\dir{>}} |
| 1281 | "d" :[dd] {z} ="z" |
| 1282 | "z" [llluu] *{\xor} ="x1" |
| 1283 | "z" :`l [lu] `u "x1" |*+{\scriptstyle 0^t \cat z[t \bitsto \ell]} "x1" |
| 1284 | "z" :[dd] *{\xor} ="xor2" |
| 1285 | :[dd] *+[F]{\mathstrut x_{n-1}[0 \bitsto t]} |
| 1286 | "y" [rrr] *+=(1, 0)+[F]{\mathstrut y_{n-1} \cat 0^{\ell-t}}="y" |
| 1287 | [dd] ="x" |
| 1288 | "d" [llluu] ="e" |
| 1289 | []!{"x"; "y" **{}, "x"+/4pt/ ="p", |
| 1290 | "x"; "e" **{}, "x"+/4pt/ ="q", |
| 1291 | "e"; "x" **{}, "e"+/4pt/ ="r", |
| 1292 | "e"; "x1" **{}, "e"+/4pt/ ="s", |
| 1293 | "x"; "e" **{} ?="c" ?(0.5)/-4pt/ ="cx" ?(0.5)/4pt/ ="cy", |
| 1294 | "y"; |
| 1295 | "p" **\dir{-}; |
| 1296 | "q" **\crv{"x"}; |
| 1297 | "cx" **\dir{-}; |
| 1298 | "c" *[@]\cir<4pt>{d^u}; |
| 1299 | "cy"; |
| 1300 | "r" **\dir{-}; |
| 1301 | "s" **\crv{"e"}; |
| 1302 | "x1" **\dir{-}?>*\dir{>}} |
| 1303 | "x1" [d] :`r [rd] `d "xor2" "xor2" |
| 1304 | "x1" :[dd] *+[F]{D}="d" [l] {K} :"d" |
| 1305 | "d" :"xor" |
| 1306 | :[dd] *+[F]{\mathstrut x_{n-2}} |
| 1307 | \end{cgraph} |
| 1308 | |
| 1309 | \caption{Encryption and decryption using CBC mode with ciphertext stealing} |
| 1310 | \label{fig:cbc-steal} |
| 1311 | \end{figure} |
| 1312 | |
| 1313 | Encrypting messages shorter than the block involves `IV stealing' -- using |
| 1314 | the IV instead of the ciphertext from the last full-length block -- which is |
| 1315 | a grotty hack but works fine if IVs are random; if the IVs are encrypted |
| 1316 | counters then there's nothing (modifiable) to steal from. |
| 1317 | |
| 1318 | We formally present a description of a randomized CBC stealing mode. |
| 1319 | |
| 1320 | \begin{definition}[CBC mode with ciphertext stealing] |
| 1321 | \label{def:cbc-steal} |
| 1322 | Let $P\colon \keys P \times \Bin^\ell \to \Bin^\ell$ be a pseudorandom |
| 1323 | permutation. Let $c$ be a generalized counter on $\Bin^\ell$. We define |
| 1324 | the randomized symmetric encryption scheme |
| 1325 | $\Xid{\E}{CBC$\$$-steal}^P = (\Xid{G}{CBC$\$$-steal}^P, |
| 1326 | \Xid{E}{CBC$\$$-steal}^P, \Xid{D}{CBC\$-steal}^P)$ for messages in $\Bin^*$ |
| 1327 | as follows: |
| 1328 | \begin{program} |
| 1329 | Algorithm $\Xid{G}{CBC$\$$-steal}^P()$: \+ \\ |
| 1330 | $K \getsr \keys P$; \\ |
| 1331 | \RETURN $K$; |
| 1332 | \- \\[\medskipamount] |
| 1333 | Algorithm $\Xid{E}{CBC$\$$-steal}^P(K, x)$: \+ \\ |
| 1334 | $t \gets |x| \bmod \ell$; \\ |
| 1335 | \IF $t \ne 0$ \THEN $x \gets x \cat 0^{\ell-t}$; \\ |
| 1336 | $v \getsr \Bin^\ell$; \\ |
| 1337 | $y \gets v \cat \id{cbc-encrypt}(K, v, x)$; \\ |
| 1338 | \IF $t \ne 0$ \THEN \\ \ind |
| 1339 | $b \gets |y| - 2\ell$; \\ |
| 1340 | $y \gets $\=$y[0 \bitsto b] \cat |
| 1341 | y[b + \ell \bitsto |y|] \cat {}$ \\ |
| 1342 | \>$y[b \bitsto b + t]$; \- \\ |
| 1343 | \RETURN $y$; |
| 1344 | \next |
| 1345 | Algorithm $\Xid{D}{CBC$\$$-steal}^P(K, y)$: \+ \\ |
| 1346 | \IF $|y| < \ell$ \THEN \RETURN $\bot$; \\ |
| 1347 | $v \gets y[0 \bitsto \ell]$; \\ |
| 1348 | $t = |y| \bmod \ell$; \\ |
| 1349 | \IF $t \ne 0$ \THEN \\ \ind |
| 1350 | $b \gets |y| - t - \ell$; \\ |
| 1351 | $z \gets P^{-1}_K(y[b \bitsto b + \ell])$; \\ |
| 1352 | $y \gets $\=$y[0 \bitsto b] \cat |
| 1353 | y[b + \ell \bitsto |y|] \cat {}$ \\ |
| 1354 | \>$z[t \bitsto \ell]$; \- \\ |
| 1355 | $x \gets \id{cbc-decrypt}(K, v, y[\ell \bitsto |y|])$; \\ |
| 1356 | \IF $t \ne 0$ \THEN \\ \ind |
| 1357 | $x \gets x \cat z[0 \bitsto t] \xor y[b \bitsto b + t]$; \- \\ |
| 1358 | \RETURN $x$; |
| 1359 | \end{program} |
| 1360 | \end{definition} |
| 1361 | |
| 1362 | The security of ciphertext stealing follows directly from the definition and |
| 1363 | the security CBC mode. |
| 1364 | |
| 1365 | \begin{corollary}[Security of CBC with ciphertext stealing] |
| 1366 | \label{cor:cbc-steal} |
| 1367 | Let $P\colon \keys P \times \Bin^\ell \to \Bin^\ell$ be a pseudorandom |
| 1368 | permutation. Then |
| 1369 | \begin{eqnarray*}[rl] |
| 1370 | \InSec{lor-cpa}(\Xid{\E}{CBC$\$$-steal}; t, q_E, \mu_E) |
| 1371 | & \le \InSec{lor-cpa} |
| 1372 | (\Xid{\E}{CBC$\$$}; t, q_E, \mu_E + q_E (\ell - 1)) \\ |
| 1373 | & \le 2 \cdot \InSec{prp}(P; t + q t_P, q) + |
| 1374 | \frac{q (q - 1)}{2^\ell - 2^{\ell/2}} |
| 1375 | \end{eqnarray*} |
| 1376 | where $q = \bigl\lfloor \bigl(\mu_E + q_E (\ell - 1)\bigr)/\ell |
| 1377 | \bigr\rfloor$ and $t_P$ is some small constant. |
| 1378 | \end{corollary} |
| 1379 | |
| 1380 | \begin{proof} |
| 1381 | From the definition, we see that the encryption algorithm |
| 1382 | $\Xid{E}{CBC-steal}$ simply pads a plaintext, encrypts it as for standard |
| 1383 | CBC mode, and postprocesses the ciphertext. Hence, if $A$ is any adversary |
| 1384 | attacking $\Xid{\E}{CBC-steal}$, we can construct an adversary |
| 1385 | $A'$ which simply runs $A$ except that, on each query to the encryption |
| 1386 | oracle, it pads both plaintexts, queries its CBC oracle, postprocesses the |
| 1387 | ciphertext returned, and gives the result back to $A$. The fact that |
| 1388 | plaintexts can now be up to $\ell - 1$ bits shorter than the next largest |
| 1389 | whole number of blocks means that $B$ submits no more than $\mu_E + q_E |
| 1390 | (\ell - 1)$ bits of plaintext to its oracle. The required result |
| 1391 | follows then directly from theorem~\ref{thm:cbc}. |
| 1392 | \end{proof} |
| 1393 | |
| 1394 | \subsection{Proof of theorem~\ref{thm:cbc}} |
| 1395 | \label{sec:cbc-proof} |
| 1396 | |
| 1397 | The techniques and notation used in this proof will also be found in several |
| 1398 | of the others. We recommend that readers try to follow this one carefully. |
| 1399 | |
| 1400 | We begin considering CBC mode using a completely random permutation. To |
| 1401 | simplify notation slightly, we shall write $n = n_L + n_E$. Our main goal is |
| 1402 | to prove the claim that there exists a garbage-emitter $W$ such that |
| 1403 | \[ |
| 1404 | \InSec{rog-cpa-$W$} |
| 1405 | (\Xid{\E}{CBCH}^{\Perm{\ell}, c, V_0}_{n_L, 0, n_E}; |
| 1406 | t, q_E, \mu_E) \le |
| 1407 | \frac{q (q - 1)}{2 \cdot (2^\ell - n)}. |
| 1408 | \] |
| 1409 | From this, we can apply proposition~\ref{prop:rog-and-lor} to obtain |
| 1410 | \[ |
| 1411 | \InSec{lor-cpa} |
| 1412 | (\Xid{\E}{CBCH}^{\Perm{\ell}, c, \bot}_{0, 0, n}; |
| 1413 | t, q_E, \mu_E) \le |
| 1414 | \frac{q (q - 1)}{2^\ell - n}. |
| 1415 | \] |
| 1416 | and, noting that there are precisely $q = \mu_E/\ell$ PRP-applications, we |
| 1417 | apply proposition~\ref{prop:enc-info-to-real} to obtain the required result. |
| 1418 | |
| 1419 | Our garbage-emitter $W$ is a bit complicated. It chooses random but |
| 1420 | \emph{distinct} blocks for the `ciphertext'; for the IVs, it uses $V_0$ for |
| 1421 | the first message if $n_L = 1$, and otherwise it chooses random blocks |
| 1422 | distinct from each other and the `ciphertext' blocks for the next $n_E$ |
| 1423 | messages, and just random blocks for subsequent ones. The algorithm $W$ is |
| 1424 | shown in figure~\ref{fig:cbc-garbage}. |
| 1425 | |
| 1426 | \begin{figure} |
| 1427 | \begin{program} |
| 1428 | Initialization: \+ \\ |
| 1429 | $i \gets 0$; \\ |
| 1430 | $\id{gone} \gets \emptyset$; |
| 1431 | \- \\[\medskipamount] |
| 1432 | Function $\id{fresh}()$ \+ \\ |
| 1433 | $x \getsr \Bin^\ell \setminus \id{gone}$; \\ |
| 1434 | $\id{gone} \gets \id{gone} \cup \{ x \}$; \\ |
| 1435 | \RETURN $x$; |
| 1436 | \next |
| 1437 | Garbage emitter $W(m)$: \+ \\ |
| 1438 | \IF $i \ge 2^\ell$ \THEN \ABORT; \\ |
| 1439 | \IF $i < n_L$ \THEN $v \gets V_0$; \\ |
| 1440 | \ELSE \IF $i < n$ \THEN $v \gets \id{fresh}()$; \\ |
| 1441 | $i \gets i + 1$ \\ |
| 1442 | \ELSE $v \getsr \Bin^\ell$; \\ |
| 1443 | $y \gets \emptystring$; \\ |
| 1444 | \FOR $j = 0$ \TO $m/\ell$; \\ \ind |
| 1445 | $y_j \gets \id{fresh}()$; \\ |
| 1446 | $y \gets y \cat y_j$; \- \\ |
| 1447 | \RETURN $(v, y)$; |
| 1448 | \end{program} |
| 1449 | |
| 1450 | \caption{Garbage emitter $W$ for CBC mode} |
| 1451 | \label{fig:cbc-garbage} |
| 1452 | \end{figure} |
| 1453 | |
| 1454 | Fortunately, it doesn't need to be efficient: the above simulations only need |
| 1455 | to be able to do the LOR game, not the ROG one. The unpleasant-sounding |
| 1456 | \ABORT\ only occurs after $2^\ell$ queries. If that happens we give up and |
| 1457 | say the adversary won anyway: the claim is trivially true by this point, |
| 1458 | since the adversary's maximum advantage is 1. |
| 1459 | |
| 1460 | Now we show that this lash-up is a good imitation of CBC encryption to |
| 1461 | someone who doesn't know the key. The intuition works like this: every time |
| 1462 | we query a random permutation at a new, fresh input value, we get a new, |
| 1463 | different, random output value; conversely, if we repeat an input, we get the |
| 1464 | same value out as last time. So, in the real `result' CBC game, if all the |
| 1465 | permutation inputs are distinct, it looks just like the garbage emitted by |
| 1466 | $W$. Unfortunately, that's not quite enough: the adversary can work out what |
| 1467 | the permutation inputs ought to be and spot when there ought to have been a |
| 1468 | collision but wasn't. So we'll show that, provided all the $P$-inputs -- |
| 1469 | values which \emph{would} be input to the permutation if we're playing that |
| 1470 | game -- are distinct, the two games look identical. |
| 1471 | |
| 1472 | We need some notation to describe the values in the game. Let $c_i = c(i)$ |
| 1473 | be the $i$th counter value, for $0 \le i < n_E$, and let $v_i$ be the $i$th |
| 1474 | initialization vector, where $v_0 = V_0$ is as given if $n_L = 1$, $v_i = |
| 1475 | P(c_i - n_L)$ if $n_L \le i < n$, and $v_i \inr \Bin^\ell$ if $n \le i < |
| 1476 | q_E$. Let $q' = \mu_E/\ell = q - n$ be the total number of plaintext blocks |
| 1477 | in the adversary's queries, let $x_i$ be the $i$th plaintext block queried, |
| 1478 | let $y_i$ be the $i$th ciphertext block returned, let |
| 1479 | \[ w_i = \begin{cases} |
| 1480 | v_j & if block $i$ is the first block of the $j$th query, and \\ |
| 1481 | y_{i-1} & otherwise |
| 1482 | \end{cases} \] |
| 1483 | and let $z_i = x_i \xor w_i$, all for $0 \le i < q'$. This is summarized |
| 1484 | diagramatically in figure~\ref{fig:cbc-proof-notation}. The $P$-inputs are |
| 1485 | now precisely the $c_i$ and the $z_i$. We'll denote probabilities in the |
| 1486 | `result' game as $\Pr_R[\cdot]$ and in the `garbage' game as $\Pr_G[\cdot]$. |
| 1487 | |
| 1488 | \begin{figure} |
| 1489 | \begin{vgraphs} |
| 1490 | \begin{vgraph}{cbc-notation-a} |
| 1491 | [] !{<1.33cm, 0cm>: <0cm, 1cm>::} |
| 1492 | {c_i} :[r] *+[F]{E}="e" [u] {K} :"e" :[r] {v_i} |
| 1493 | \end{vgraph} |
| 1494 | \begin{vgraph}{cbc-notation-b} |
| 1495 | [] !{<1.33cm, 0cm>: <0cm, 1cm>::} |
| 1496 | {x_i} :[r] *{\xor} ="xor" :[r] {z_i} |
| 1497 | :[r] *+[F]{E}="e" [u] {K} :"e" :[r] {y_i} |
| 1498 | "xor" [u] {w_i} ="w" :"xor" |
| 1499 | "w" [lu] {v_j} ="v" :"w" |
| 1500 | "w" [ru] {y_{i-1}} ="y" :"w" |
| 1501 | "v" :@{.}|-*+\hbox{or} "y" |
| 1502 | \end{vgraph} |
| 1503 | \end{vgraphs} |
| 1504 | |
| 1505 | \caption{Notation for the proof of theorem~\ref{thm:cbc}.} |
| 1506 | \label{fig:cbc-proof-notation} |
| 1507 | \end{figure} |
| 1508 | |
| 1509 | Let $C_r$ be the event, in either game, that $z_i = z_j$ for some $0 \le i < |
| 1510 | j < r$, or that $z_i = c_j$ for some $0 \le i < r$ and some $0 \le j < n_E$. |
| 1511 | We need to bound the probability that $C_{q'}$ occurs in both the `result' |
| 1512 | and `garbage' games. We'll do this inductively. By the definition, |
| 1513 | $\Pr_R[C_0] = \Pr_G[C_0] = 0$. |
| 1514 | |
| 1515 | Firstly, tweak the games so that all of the IVs corresponding to counters are |
| 1516 | chosen at the beginning, instead of as we go along. Obviously this doesn't |
| 1517 | make any difference to the adversary's view of the proceedings, but it makes |
| 1518 | our analysis easier. |
| 1519 | |
| 1520 | Let's assume that $C_r$ didn't happen; we want the probability that $C_{r+1}$ |
| 1521 | did, which is obviously just the probability that $z_r$ collides with some |
| 1522 | $z_i$ for $0 \le i < r$ or some $c_i$ for $0 \le i < n$. At this point, the |
| 1523 | previous $z_i$ are fixed. So: |
| 1524 | \begin{equation} |
| 1525 | \label{eq:cbc-coll} |
| 1526 | \Pr[C_{r+1} | \bar{C}_r] |
| 1527 | = \sum_{z \in \Bin^\ell} \biggl( |
| 1528 | \sum_{0\le i<n} \Pr[z = c_i] + |
| 1529 | \sum_{0\le i<r} \Pr[z = z_i] |
| 1530 | \biggr) \cdot \Pr[z_r = z] |
| 1531 | \end{equation} |
| 1532 | Now note that $z_r = w_r \xor x_r$. We've no idea how $x_r$ was chosen; but, |
| 1533 | one of the following cases holds. |
| 1534 | \begin{enumerate} |
| 1535 | \item If $x_r$ is the first block of the first plaintext, i.e., $r = 0$, and |
| 1536 | $n_L = 1$, then $w_r = v_0$. However, in this case we know that $n_E = 0$ |
| 1537 | by hypothesis. There are no $z_i$ which $z_r$ might collide with, so the |
| 1538 | probability of a collision is zero. |
| 1539 | \item If $x_r$ is the first block of plaintext $i$, and $0 \le i < n$, then |
| 1540 | $w_r = v_i$, and was chosen at random from a set of $2^\ell - i \le 2^\ell |
| 1541 | - n \le 2^\ell - n - r$ possibilities, either by our random permutation or |
| 1542 | by $W$. We know $x_r$ is independent of $w_r$ because none of the previous |
| 1543 | $P$-inputs were equal to $c_i$, by our assumption of $\bar{C}_r$. |
| 1544 | \item If $x_r$ is the first block of plaintext $i$, and $n \le i < q'$, then |
| 1545 | $w_r = v_i$, and was chosen at random from all $2\ell$ possible $\ell$-bit |
| 1546 | blocks. We know $x_r$ is independent of $w_r$ because we just chose $w_r$ |
| 1547 | at random, after $x_r$ was chosen. |
| 1548 | \item Otherwise, $x_r$ is a subsequent block in some message, and $w_r = |
| 1549 | y_{r-1}$, and was chosen at random from a set of $2^\ell - n - r$ |
| 1550 | possibilities, either by our random permutation or by $W$. We know $x_r$ |
| 1551 | is independent of $w_r$ because $z_{r-1}$ is a new $P$-input, by our |
| 1552 | assumption of $\bar{C}_r$. |
| 1553 | \end{enumerate} |
| 1554 | So, except in case~1, which isn't a problem anyway, $w_r$ is independent of |
| 1555 | $x_r$, and chosen uniformly at random from a set of at least $2^\ell - r - n$ |
| 1556 | elements, in either game -- so we can already see that $\Pr_R[C_i] = |
| 1557 | \Pr_G[C_i]$ for any $i \ge 0$. Finally, the $z_i$ and $c_i$ are all |
| 1558 | distinct, so the $z_i \xor x$ and $c_i \xor x$ must all be distinct, for any |
| 1559 | fixed $x$. So: |
| 1560 | \begin{eqnarray}[rl] |
| 1561 | \Pr[C_{r+1} | \bar{C}_r] |
| 1562 | & = \sum_{x \in \Bin^\ell} \biggl( |
| 1563 | \sum_{0\le i<n} \Pr[w_r = x \xor c_i] + |
| 1564 | \sum_{0\le i<r} \Pr[w_r = x \xor z_i] |
| 1565 | \biggr) \cdot \Pr[x_r = x] \\ |
| 1566 | & \le \sum_{x \in \Bin^\ell} \frac{r + n}{2^\ell - r - n} \Pr[x_r = x] |
| 1567 | = \frac{r + n}{2^\ell - r - n} \sum_{x \in \Bin^\ell} \Pr[x_r = x] \\ |
| 1568 | & = \frac{r + n}{2^\ell - r - n}. |
| 1569 | \end{eqnarray} |
| 1570 | Now we're almost home. All the $c_i$ and $z_i$ are distinct; all the $v_i$ |
| 1571 | and $y_i$ are random, assuming $C_{q'}$. We can bound $\Pr[C_{q'}]$ thus: |
| 1572 | \begin{equation} |
| 1573 | \Pr[C_{q'}] |
| 1574 | \le \sum_{0<i\le q'} \Pr[C_i | \bar{C}_{i-1}] |
| 1575 | \le \sum_{0\le i\le q'} \frac{i + n - 1}{2^\ell - i - n + 1} |
| 1576 | \end{equation} |
| 1577 | Now, let $i' = i + n - 1$. Then |
| 1578 | \begin{equation} |
| 1579 | \Pr[C_{q'}] |
| 1580 | \le \sum_{n-1\le i'\le q'+n-1} \frac{i'}{2^\ell - i'} |
| 1581 | \le \sum_{0\le i'<q} \frac{i'}{2^\ell - q} |
| 1582 | = \frac{q (q - 1)}{2 \cdot (2^\ell - q)} |
| 1583 | \end{equation} |
| 1584 | |
| 1585 | Finally, let $R$ be the event that the adversary returned 1 at the end of the |
| 1586 | game -- indicating a guess of `result'. Then, noting as we have, that |
| 1587 | $\Pr_R[C_{q'}] = \Pr_G[C_{q'}]$, we get this: |
| 1588 | \begin{eqnarray}[rl] |
| 1589 | \Adv{rog-cpa-$W$}{\Xid{\E}{CBCH}^{P, c, n}}(A) |
| 1590 | & = \Pr_R[R] - \Pr_G[R] \\ |
| 1591 | & \begin{eqnalign}[rLl][b] |
| 1592 | {} = & (\Pr_R[R | C_{q'}] \Pr_R[C_{q'}] + |
| 1593 | \Pr_R[R | \bar{C}_{q'}] \Pr_R[\bar{C}_{q'}]) - {} \\ |
| 1594 | & & (\Pr_G[R | C_{q'}] \Pr_R[C_{q'}] + |
| 1595 | \Pr_G[R | \bar{C}_{q'}] \Pr_G[\bar{C}_{q'}]) |
| 1596 | \end{eqnalign} \\ |
| 1597 | & = \Pr_R[R | C_{q'}] \Pr_R[C_{q'}] - \Pr_G[R | C_{q'}] \Pr_G[C_{q'}] \\ |
| 1598 | & \le \Pr[C_{q'}] \le \frac{q (q - 1)}{2 \cdot (2^\ell - q)} |
| 1599 | \end{eqnarray} |
| 1600 | And we're done! |
| 1601 | \qed |
| 1602 | |
| 1603 | %%%-------------------------------------------------------------------------- |
| 1604 | |
| 1605 | \section{Ciphertext feedback (CFB) encryption} |
| 1606 | \label{sec:cfb} |
| 1607 | |
| 1608 | \subsection{Description} |
| 1609 | \label{sec:cfb-desc} |
| 1610 | |
| 1611 | Suppose $F$ is an $\ell$-bit-to-$L$-bit pseudorandom function, and let $t \le |
| 1612 | L$. CFB mode works as follows. Given a message $X$, we divide it into |
| 1613 | $t$-bit blocks $x_0$, $x_1$, $\ldots$, $x_{n-1}$. Choose an initialization |
| 1614 | vector $v \in \Bin^\ell$. We maintain a \emph{shift register} $s_i$, whose |
| 1615 | initial value is $v$. To encrypt a block $x_i$, we XOR it with the result of |
| 1616 | passing the shift register through the PRF, forming $y_i$, and then update |
| 1617 | the shift register by shifting in the ciphertext block $y_i$. That is, |
| 1618 | \begin{equation} |
| 1619 | s_0 = v \qquad |
| 1620 | y_i = x_i \xor F_K(s_i) \qquad |
| 1621 | s_{i+1} = s_i \shift{t} y_i \ \text{(for $0 \le i < n$)}. |
| 1622 | \end{equation} |
| 1623 | Decryption follows from noting that $x_i = y_i \xor F_K(s_i)$. See |
| 1624 | figure~\ref{fig:cfb} for a diagrammatic representation. |
| 1625 | |
| 1626 | Also, we observe that the final plaintext block needn't be $t$ bits long: we |
| 1627 | can pad it out to $t$ bits and truncate the result without affecting our |
| 1628 | ability to decrypt. |
| 1629 | |
| 1630 | \begin{figure} |
| 1631 | \begin{cgraph}{cfb-mode} |
| 1632 | [] !{<0.425cm, 0cm>: <0cm, 0.5cm>::} |
| 1633 | *+=(2, 0)+[F]{\mathstrut v} ="v" :|<>(0.35)@{/}_<>(0.35){\ell}[rrrrr] |
| 1634 | *+[o][F]{\shift{t}} ="shift" |
| 1635 | [ll] :|-@{/}^-{\ell}[dd] *+[F]{E} ="e" [ll] {K} :"e" |
| 1636 | :|-@{/}^-{t}[dd] *{\xor} ="xor" |
| 1637 | [lll] *+=(2, 0)+[F]{\mathstrut x_0} :|-@{/}_-{t} "xor" |
| 1638 | :|-@{/}^-{t}[ddd] *+=(2, 0)+[F]{\mathstrut y_0} |
| 1639 | "xor" [d] :`r "shift" "shift"|-@{/}_-{t} |
| 1640 | :|-@{/}_-{\ell}[rrrrrrr] *+[o][F]{\shift{t}} ="shift" |
| 1641 | [ll] :|-@{/}^-{\ell}[dd] *+[F]{E} ="e" [ll] {K} :"e" |
| 1642 | :|-@{/}^-{t}[dd] *{\xor} ="xor" |
| 1643 | [lll] *+=(2, 0)+[F]{\mathstrut x_1} :|-@{/}_-{t} "xor" |
| 1644 | :|-@{/}^-{t}[ddd] *+=(2, 0)+[F]{\mathstrut y_1} |
| 1645 | "xor" [d] :`r "shift" "shift"|-@{/}_-{t} |
| 1646 | :@{-->}|-@{/}_-{\ell}[rrrrrrr] *+[o][F]{\shift{t}} ="shift" |
| 1647 | [ll] :@{-->}|-@{/}^-{\ell}[dd] *+[F]{E} ="e" [ll] {K} :"e" |
| 1648 | :@{-->}|-@{/}^-{t}[dd] *{\xor} ="xor" |
| 1649 | [lll] *+=(2, 0)+[F--]{\mathstrut x_i} :@{-->}|-@{/}_-{t} "xor" |
| 1650 | :@{-->}|-@{/}^-{t}[ddd] *+=(2, 0)+[F--]{\mathstrut y_i} |
| 1651 | "xor" [d] :@{-->} `r "shift" "shift"|-@{/}_-{t} |
| 1652 | [rrrrrdd] *+[F]{E} ="e" |
| 1653 | "shift" :@{-->}`r "e" |-@{/}_-{\ell} "e" |
| 1654 | [ll] {K} :"e" |
| 1655 | :|-@{/}^-{t}[dd] *{\xor} ="xor" |
| 1656 | [lll] *+=(2, 0)+[F]{\mathstrut x_{n-1}} :|-@{/}_-{t} "xor" |
| 1657 | :|-@{/}^-{t}[ddd] *+=(2, 0)+[F]{\mathstrut y_{n-1}} |
| 1658 | \end{cgraph} |
| 1659 | |
| 1660 | \caption{Encryption using CFB mode} |
| 1661 | \label{fig:cfb} |
| 1662 | \end{figure} |
| 1663 | |
| 1664 | \begin{definition}[CFB algorithms] |
| 1665 | For any function $F\colon \Bin^\ell \to \Bin^t$, any initialization vector |
| 1666 | $v \in \Bin^\ell$, any plaintext $x \in \Bin^*$ and any ciphertext $y \in |
| 1667 | \Bin^*$, we define PRF encryption mode $\id{CFB} = (\id{cfb-encrypt}, |
| 1668 | \id{cfb-decrypt})$ as follows: |
| 1669 | \begin{program} |
| 1670 | Algorithm $\id{cfb-encrypt}(F, v, x)$: \+ \\ |
| 1671 | $s \gets v$; \\ |
| 1672 | $L \gets |x|$; \\ |
| 1673 | $x \gets x \cat 0^{t\lceil L/t \rceil - L}$; \\ |
| 1674 | $y \gets \emptystring$; \\ |
| 1675 | \FOR $i = 0$ \TO $(|x| - t')/t$ \DO \\ \ind |
| 1676 | $x_i \gets x[ti \bitsto t(i + 1)]$; \\ |
| 1677 | $y_i \gets x_i \xor F(s)$; \\ |
| 1678 | $s \gets s \shift{t} y_i$; \\ |
| 1679 | $y \gets y \cat y_i$; \- \\ |
| 1680 | \RETURN $(s, y[0 \bitsto L])$; |
| 1681 | \next |
| 1682 | Algorithm $\id{cfb-decrypt}(F, v, y)$: \+ \\ |
| 1683 | $s \gets v$; \\ |
| 1684 | $L \gets |y|$; \\ |
| 1685 | $y \gets y \cat 0^{t\lceil L/t \rceil - L}$; \\ |
| 1686 | $x \gets \emptystring$; \\ |
| 1687 | \FOR $i = 0$ \TO $(|x| - t')/t$ \DO \\ \ind |
| 1688 | $y_i \gets y[ti \bitsto t(i + 1)]$; \\ |
| 1689 | $x_i \gets x_i \xor F(s)$; \\ |
| 1690 | $s \gets s \shift{t} y_i$; \\ |
| 1691 | $x \gets x \cat x_i$; \- \\ |
| 1692 | \RETURN $x[0 \bitsto L]$; |
| 1693 | \end{program} |
| 1694 | We now define the schemes $\Xid{\E}{CFB$\$$}^F$, |
| 1695 | $\Xid{\E}{CFBC}^{F, c}$, $\Xid{\E}{CFBE}^{F, c}$, and |
| 1696 | $\Xid{\E}{CFBL}^{F, V_0}$ according to |
| 1697 | definition~\ref{def:enc-scheme}; and we define the hybrid scheme |
| 1698 | $\Xid{\E}{CFBH}^{F, V_0, c}_{n_L, n_C, n_E}$ according to |
| 1699 | definition~\ref{def:enc-hybrid}. |
| 1700 | \end{definition} |
| 1701 | |
| 1702 | \subsection{Sliding strings} |
| 1703 | |
| 1704 | Consider for a moment the mode CFBL, i.e., with carry-over of IV from one |
| 1705 | plaintext to the next, with $t < \ell$. Then we find that some IVs are |
| 1706 | weak. |
| 1707 | |
| 1708 | Pretend for a moment that we're an adversary playing the LOR-CPA game using |
| 1709 | an ideal random function $F \inr \Func{\ell}{t}$, and that the initial IV |
| 1710 | $V_0 = 0^\ell$. We choose two distinct 8-bit plaintexts $l$ and $r$ as our |
| 1711 | first left-or-right query. With probability $2^{-t}$, the result of |
| 1712 | encrypting that first query is $0^t$. However, in this case, the IV for the |
| 1713 | \emph{next} query is $V_0 \shift{t} 0^t = 0^\ell = V_0$. If this happens, |
| 1714 | we have only to submit the pair $(l, l)$ as our second query. If the |
| 1715 | ciphertext to this second query also comes back zero, we guess that we're |
| 1716 | dealing with a left oracle; otherwise we guess right. If we don't get lucky |
| 1717 | with our first query, we just guess randomly. |
| 1718 | |
| 1719 | \begin{figure} |
| 1720 | \begin{program} |
| 1721 | Adversary $S^{E(\cdot, \cdot)}$: \+ \\ |
| 1722 | $l \gets 0^t$; $r \gets 0^{t - 1} 1$; \\ |
| 1723 | $y \gets E(l, r)$; \\ |
| 1724 | \IF $y[\ell \bitsto \ell + t] = 0^t$ \THEN \\ \ind |
| 1725 | \IF $E(l, l) = y$ \THEN $b \gets 0$ \ELSE $b \gets 1$; \- \\ |
| 1726 | \ELSE \\ \ind |
| 1727 | $b \getsr \{0, 1\}$; \- \\ |
| 1728 | \RETURN $b$; |
| 1729 | \end{program} |
| 1730 | \caption{Adversary $S$ attacking $\Xid{\E}{CFBL}^{\Func{\ell}{t}, 0^\ell}$} |
| 1731 | \label{fig:adv-sliding} |
| 1732 | \end{figure} |
| 1733 | |
| 1734 | This attack is shown more formally as adversary~$S$ in |
| 1735 | figure~\ref{fig:adv-sliding}. Its resource usage is almost trivial -- |
| 1736 | negligible computation and at most two encryption queries. However, its |
| 1737 | advantage is quite good: |
| 1738 | \[ \Adv{LOR-CPA}{\Xid{\E}{CFBL}^{\Func{\ell}{t}, 0^\ell}}(S) = |
| 1739 | \frac{1}{2^t} \biggl( 1 - \frac{1}{2^t} \biggr). |
| 1740 | \] |
| 1741 | |
| 1742 | This attack works because $V_0[t \bitsto \ell] = V_0[0 \bitsto \ell - t]$. |
| 1743 | There are similar attacks for other such relationships. The following |
| 1744 | definition characterizes these kinds of `bad' IVs. |
| 1745 | |
| 1746 | \begin{definition}[Sliding strings] |
| 1747 | \label{def:slide} |
| 1748 | We say that an $\ell$-bit string $x$ \emph{$t$-slides} if there exist |
| 1749 | integers $i$ and $j$ such that $0 \le j < i < \ell/t$ and $x[i t \bitsto |
| 1750 | \ell] = x[j t \bitsto \ell - (i - j) t]$. |
| 1751 | \end{definition} |
| 1752 | \begin{remark} |
| 1753 | For all $\ell > 0$ and $t < \ell$, the string $0^{\ell-1} 1$ does not |
| 1754 | $t$-slide. |
| 1755 | \end{remark} |
| 1756 | |
| 1757 | \subsection{Security of CFB mode} |
| 1758 | |
| 1759 | %% I suspect David will want to put some negative results here, and complain |
| 1760 | %% about Alkassar et al.'s alleged proof. I'll press on with the positive |
| 1761 | %% stuff. |
| 1762 | %% |
| 1763 | %% The problems come when $t < \ell$. Then C-mode isn't necessarily secure |
| 1764 | %% (well, we get a similar bound with $t$ instead of $\ell$, which isn't very |
| 1765 | %% impressive). The L-mode needs careful selection of the initial IV. |
| 1766 | |
| 1767 | \begin{theorem}[Security of CFB mode] |
| 1768 | \label{thm:cfb} |
| 1769 | Let $F$ be a pseudorandom function from $\Bin^\ell$ to $\Bin^t$; let $V_0 |
| 1770 | \in \Bin^\ell$ be a non-$t$-sliding string; let $c$ be a generalized |
| 1771 | counter in $\Bin^\ell$; and let $n_L$, $n_C$, $n_E$ and $q_E$ be |
| 1772 | nonnegative integers; and furthermore suppose that |
| 1773 | \begin{itemize} |
| 1774 | \item $n_L + n_C + n_E \le q_E$, |
| 1775 | \item $n_L = 0$, or $n_C = n_E = 0$, or $\ell \le t$ and $V_0 \ne c(i)$ |
| 1776 | for any $n_L \le i < n_L + n_C + n_E$, and |
| 1777 | \item either $n_C = 0$ or $\ell \le t$. |
| 1778 | \end{itemize} |
| 1779 | Then, for any $t_E$ and $\mu_E$, and whenever |
| 1780 | we have |
| 1781 | \[ \InSec{lor-cpa}(\Xid{\E}{CFBH}^{F, V_0, c}_{n_L, n_C, n_E}; |
| 1782 | t_E, q_E, \mu_E) \le |
| 1783 | 2 \cdot \InSec{prf}(F; t_E + q t_F, q) + \frac{q (q - 1)}{2^\ell} |
| 1784 | \] |
| 1785 | where $q = \bigl\lfloor \bigl(\mu_E + q_E (t - 1)\bigr)/t \bigr\rfloor + |
| 1786 | n_E$, and $t_F$ is some small constant. |
| 1787 | \end{theorem} |
| 1788 | |
| 1789 | The proof is a bit involved; we postpone it until |
| 1790 | section~\ref{sec:cfb-proof}. |
| 1791 | |
| 1792 | \begin{corollary} |
| 1793 | \label{cor:cfb-prf} |
| 1794 | Let $F$, $c$ and $V_0$ be as in theorem~\ref{thm:cfb}. Then for any $t_E$, |
| 1795 | $q_E$ and $\mu_E$, |
| 1796 | \begin{eqnarray*}[rl] |
| 1797 | \InSec{lor-cpa}(\Xid{\E}{CFB$\$$}^F; t_E, q_E, \mu_E) |
| 1798 | & \le 2 \cdot \InSec{prf}(F; t_E + q t_F, q) + \frac{q (q - 1)}{2^\ell} |
| 1799 | \\ |
| 1800 | \InSec{lor-cpa}(\Xid{\E}{CFBE}^{F, c}; t_E, q_E, \mu_E) |
| 1801 | & \le 2 \cdot \InSec{prf}(F; t_E + q' t_F, q') + |
| 1802 | \frac{q' (q' - 1)}{2^\ell} |
| 1803 | \\ |
| 1804 | \InSec{lor-cpa}(\Xid{\E}{CFBL}^{F, V_0}; t_E, q_E, \mu_E) |
| 1805 | & \le 2 \cdot \InSec{prf}(F; t_E + q t_F, q) + \frac{q (q - 1)}{2^\ell} |
| 1806 | \\ |
| 1807 | \tabpause{and, if $\ell \le t$,} |
| 1808 | \InSec{lor-cpa}(\Xid{\E}{CFBC}^{F, c}; t_E, q_E, \mu_E) |
| 1809 | & \le 2 \cdot \InSec{prf}(F; t_E + q t_F, q) + \frac{q (q - 1)}{2^\ell} |
| 1810 | \end{eqnarray*} |
| 1811 | where $q = \bigl\lfloor \bigl(\mu_E + q_E (t - 1)\bigr)/t \bigr\rfloor + |
| 1812 | n_E$, $q' = q + q_E$, and $t_F$ is some small constant. |
| 1813 | \end{corollary} |
| 1814 | \begin{proof} |
| 1815 | Follows from theorem~\ref{thm:cfb} and proposition~\ref{prop:enc-hybrid}. |
| 1816 | \end{proof} |
| 1817 | |
| 1818 | \begin{corollary} |
| 1819 | \label{cor:cfb-prp} |
| 1820 | Let $P$ be a pseudorandom permutation on $\Bin^\ell$, and let $c$ and $V_0$ |
| 1821 | be as in theorem~\ref{thm:cfb}. Then for any $t_E$, $q_E$ and $\mu_E$, |
| 1822 | \begin{eqnarray*}[rl] |
| 1823 | \InSec{lor-cpa}(\Xid{\E}{CFB$\$$}^P; t_E, q_E, \mu_E) |
| 1824 | & \le 2 \cdot \InSec{prp}(P; t_E + q t_F, q) + |
| 1825 | \frac{q (q - 1)}{2^{\ell-1}} |
| 1826 | \\ |
| 1827 | \InSec{lor-cpa}(\Xid{\E}{CFBE}^{P, c}; t_E, q_E, \mu_E) |
| 1828 | & \le 2 \cdot \InSec{prp}(P; t_E + q' t_F, q') + |
| 1829 | \frac{q' (q' - 1)}{2^{\ell-1}} |
| 1830 | \\ |
| 1831 | \InSec{lor-cpa}(\Xid{\E}{CFBL}^{P, V_0}; t_E, q_E, \mu_E) |
| 1832 | & \le 2 \cdot \InSec{prp}(P; t_E + q t_F, q) + |
| 1833 | \frac{q (q - 1)}{2^{\ell-1}} |
| 1834 | \\ |
| 1835 | \tabpause{and, if $\ell \le t$,} |
| 1836 | \InSec{lor-cpa}(\Xid{\E}{CFBC}^{P, c}; t_E, q_E, \mu_E) |
| 1837 | & \le 2 \cdot \InSec{prp}(P; t_E + q t_F, q) + |
| 1838 | \frac{q (q - 1)}{2^{\ell-1}} |
| 1839 | \end{eqnarray*} |
| 1840 | where $q = \bigl\lfloor \bigl(\mu_E + q_E (t - 1)\bigr)/t \bigr\rfloor + |
| 1841 | n_E$, $q' = q + q_E$, and $t_F$ is some small constant. |
| 1842 | \end{corollary} |
| 1843 | \begin{proof} |
| 1844 | Follows from corollary~\ref{cor:cfb-prf} and |
| 1845 | proposition~\ref{prop:prps-are-prfs}. |
| 1846 | \end{proof} |
| 1847 | |
| 1848 | \subsection{Proof of theorem~\ref{thm:cfb}} |
| 1849 | \label{sec:cfb-proof} |
| 1850 | |
| 1851 | Our proof follows the same lines as for CBC mode: we show the ROG-CPA |
| 1852 | security of hybrid-CFB mode using an ideal random function, and then apply |
| 1853 | our earlier results to complete the proof. However, the ROG-CPA result will |
| 1854 | be useful later when we consider the security of OFB mode, so we shall be a |
| 1855 | little more formal about defining it. |
| 1856 | |
| 1857 | The garbage emitter is in some sense the `perfect' one: it emits a `correct' |
| 1858 | IV followed by a uniform random string of the correct length. |
| 1859 | |
| 1860 | \begin{definition}[The $W_\$$ garbage emitter] |
| 1861 | Let natural numbers $n_L$, $n_C$, and $V_0 \in \Bin^\ell$ be given; then we |
| 1862 | define the garbage emitter $W_\$$ as follows. |
| 1863 | \begin{program} |
| 1864 | Initialization: \+ \\ |
| 1865 | $i \gets 0$; \\ |
| 1866 | $v \gets V_0$; |
| 1867 | \- \\[\medskipamount] |
| 1868 | Garbage emitter $W_\$(m)$: \+ \\ |
| 1869 | \IF $i < n_L$ \THEN $v' \gets v$; \\ |
| 1870 | \ELSE \IF $n_L \le i < n_L + n_C$ \THEN $v' \gets c(i)$; \\ |
| 1871 | \ELSE \IF $n_L + n_C \le i$ \THEN $v' \getsr \Bin^\ell$; \\ |
| 1872 | $i \gets i + 1$; \\ |
| 1873 | $m' \gets t \lfloor (m + t - 1)/t\rfloor$; \\ |
| 1874 | $y \getsr \Bin^{m'}$; \\ |
| 1875 | $v \gets v' \shift{m'} y$; \\ |
| 1876 | \RETURN $(v', y[0 \bitsto m])$ |
| 1877 | \end{program} |
| 1878 | \end{definition} |
| 1879 | |
| 1880 | We now show that CFB mode with a random function is hard to distinguish from |
| 1881 | $W_\$$. |
| 1882 | \begin{lemma}[Pseudorandomness of CFB mode] |
| 1883 | \label{lem:cfb-rog} |
| 1884 | Let $\ell$, $t$, $n_L$, $n_C$, $n_E$, $q_E$, $c$, $V_0$, and $q$ be as in |
| 1885 | theorem~\ref{thm:cfb}. Then, for any $t_E$ and $\mu_E$, |
| 1886 | \[ \InSec{rog-cpa-$W_\$$} |
| 1887 | (\Xid{\E}{CFBH}^{\Func{\ell}{t}, V_0, c}_{n_L, n_C, n_E}; |
| 1888 | t, q_E, \mu_E) \le |
| 1889 | \frac{q (q - 1)}{2^{\ell+1}}. |
| 1890 | \] |
| 1891 | \end{lemma} |
| 1892 | Theorem~\ref{thm:cfb} follows from this result by application of |
| 1893 | propositions \ref{prop:rog-and-lor} and~\ref{prop:enc-info-to-real}. It |
| 1894 | remains therefore for us to prove lemma~\ref{lem:cfb-rog}. |
| 1895 | |
| 1896 | To reduce the weight of notation, let us agree to suppress the adornments on |
| 1897 | $\Adv{}{}$ and $\InSec{}$ symbols. Also, let $m_L = n_L$; let $m_C$ = $n_L + |
| 1898 | n_C$; and let $m_E = n_L + n_C + n_E$. (Remember: the $m$s are |
| 1899 | cu\textit{m}ulative.) |
| 1900 | |
| 1901 | The truncation of ciphertext blocks makes matters complicated. Let us say |
| 1902 | that an adversary is \emph{block-respecting} if all of its plaintext queries |
| 1903 | are a multiple of $t$ bits in length; obviously all of the oracle responses |
| 1904 | for a block-respecting adversary are also a multiple of $t$ bits in length. |
| 1905 | \begin{claim*} |
| 1906 | Let $A'$ be a block-respecting adversary querying a total of $\mu_E$ bits of |
| 1907 | plaintext queries; then |
| 1908 | \[ \Adv{}{}(A') \le \frac{q (q - 1)}{2^{\ell+1}} \] |
| 1909 | where $q = \mu_E/t$. |
| 1910 | \end{claim*} |
| 1911 | Lemma~\ref{lem:cfb-rog} follows from this claim: if $A$ is any adversary, |
| 1912 | then we construct a block-respecting adversary $A'$ by padding $A$'s |
| 1913 | plaintext queries and truncating the oracle responses; and if $A$ makes $q_E$ |
| 1914 | queries totalling $\mu_E$ bits, then the total bits queried by $A'$ is no |
| 1915 | more than $\bigl\lfloor\bigl( \mu_E + q_E (t - 1) \bigr)\bigr\rfloor$ bits. |
| 1916 | We now proceed to the proof of the above claim. |
| 1917 | |
| 1918 | Suppose, then, that we are given a block-respecting adversary $A$ which makes |
| 1919 | $q$ queries to its encryption oracle. Let $F(\cdot)$ denote the application |
| 1920 | of the random function. We want to show that, provided all of the $F$-inputs |
| 1921 | are distinct, the $F$-outputs are uniformly random, and hence the CFB |
| 1922 | ciphertexts are uniformly random. As for the CBC case, life isn't that good |
| 1923 | to us: we have to deal with the case where the adversary can see that two |
| 1924 | $F$-inputs would have collided, and therefore that a garbage string couldn't |
| 1925 | have been generated by CFB encryption of his plaintext. |
| 1926 | |
| 1927 | Our notation will be similar to, yet slightly different from, that of |
| 1928 | section~\ref{sec:cbc-proof}. |
| 1929 | |
| 1930 | Let $q' = q - n_E$ be the number of $t$-bit plaintext blocks the adversary |
| 1931 | submits, and for $0 \le i < q'$, let $x_i$ be the $i$th plaintext block |
| 1932 | queried, and let $y_i$ be the $i$th ciphertext block returned. |
| 1933 | |
| 1934 | For $m_L \le i < m_E$, let $c_i = c(i)$ be the $i$th counter value. For $0 |
| 1935 | \le i < q_E$ let $v_i$ be the $i$th initialization vector, i.e., |
| 1936 | \[ v_i = \begin{cases} |
| 1937 | V_0 & if $i = 0$ and $n_L > 0$; \\ |
| 1938 | v_{i-1} \shift{t} Y_{i-1} |
| 1939 | & if $1 \le i < m_L$ and $Y_{i-1}$ was the ciphertext |
| 1940 | from query $i - 1$; \\ |
| 1941 | c_i & if $m_L \le i < m_C$; \\ |
| 1942 | F(c_i) & if the oracle is `result', and $m_C \le i < m_E$; |
| 1943 | or \\ |
| 1944 | R_i & for some $R_i \inr \Bin^\ell$, otherwise. |
| 1945 | \end{cases} |
| 1946 | \] |
| 1947 | Note that the only difference in the $v_i$ between the `result' and `garbage' |
| 1948 | games occurs in the encrypted-counters phase. Furthermore, if no other |
| 1949 | $F$-input is equal to any $c_i$ for $m_C \le i < m_E$ then the IVs are |
| 1950 | identically distributed. |
| 1951 | |
| 1952 | Now, for $0 \le i < q'$, define |
| 1953 | \[ z_i = \begin{cases} |
| 1954 | v_j & if block $i$ is the first block of the |
| 1955 | $j$th query, or \\ |
| 1956 | z_{i-1} \shift{t} y_{i-1} & otherwise |
| 1957 | \end{cases} |
| 1958 | \] |
| 1959 | and let $w_i = x_i \xor y_i$. In the `result' game, we have $w_i = F(z_i)$, |
| 1960 | of course. All of this notation is summarized diagrammatically in |
| 1961 | figure~\ref{fig:cfb-proof-notation}. The $F$-inputs are precisely the $z_i$ |
| 1962 | and $c_i$ for $m_C \le i < m_E$. |
| 1963 | |
| 1964 | We'll denote probabilities in the `result' game as $\Pr_R[\cdot]$ and in the |
| 1965 | `garbage' game as $\Pr_G[\cdot]$. |
| 1966 | |
| 1967 | \begin{figure} |
| 1968 | \begin{vgraphs} |
| 1969 | \begin{vgraph}{cfb-notation-a} |
| 1970 | [] !{<1.333cm, 0cm>: <0cm, 1cm>::} |
| 1971 | {z_i} ="z" :|-@{/}_-{\ell}[r] *+[F]{F} ="F" |
| 1972 | :|-@{/}_-{t}[r] {w_i} ="w" :|-@{/}_-{t}[r] *{\xor} ="xor" |
| 1973 | "xor" [u] {x_i} ="x" :|-@{/}^-{t}"xor" :|-@{/}^-{t}[d] {y_i} ="y" |
| 1974 | "z" [lu] {v_j} ="v" :"z" |
| 1975 | "z" [ru] {z_{i-1} \shift{t} y_{i-1}} ="y" :"z" |
| 1976 | "v" :@{.}|-*+\hbox{or} "y" |
| 1977 | \end{vgraph} |
| 1978 | \end{vgraphs} |
| 1979 | |
| 1980 | \caption{Notation for the proof of lemma~\ref{lem:cfb-rog}.} |
| 1981 | \label{fig:cfb-proof-notation} |
| 1982 | \end{figure} |
| 1983 | |
| 1984 | Let $C_r$ be the event, in either game, that $z_i = z_j$ for some $0 \le i < |
| 1985 | j < r$, or that $z_i = c_j$ for some $0 \le i < r$ and some $m_C \le j < |
| 1986 | m_E$. |
| 1987 | |
| 1988 | Let's assume that $C_r$ didn't happen; we want the probability that $C_{r+1}$ |
| 1989 | did, which is just the probability that $z_r$ collides with some $z_i$ where |
| 1990 | $0 \le i < r$, or some $c_i$ for $m_C \le i < m_E$. Observe that, under this |
| 1991 | assumption, all the $w_i$, and hence the $y_i$, are uniformly distributed, |
| 1992 | and that therefore the two games are indistinguishable. |
| 1993 | |
| 1994 | One of the following cases holds. |
| 1995 | \begin{enumerate} |
| 1996 | \item If $r = 0$ and $m_L > 0$ then $z_r = V_0$. There is no other $z_i$ yet |
| 1997 | for $z_r$ to collide with, though it might collide with some encrypted |
| 1998 | counter $F(c_i)$, with probability $n_E/2^\ell$. |
| 1999 | \item If $z_r = c_i$ is the IV for some message $i$ where $m_L \le i < m_C$, |
| 2000 | life is a bit complicated. It can't collide with $V_0$ or other $c_i$ by |
| 2001 | assumption; the encrypted counters and random IVs haven't been chosen yet; |
| 2002 | and either $n_C = 0$ (in which case there's nothing to do here anyway) or |
| 2003 | $\ell \le t$, so there are no $z_i$ containing partial copies of $V_0$ to |
| 2004 | worry about. This leaves non-IV $z_i$: again, $\ell \le t$, so $z_i = |
| 2005 | y_i[t - \ell \bitsto t]$, which is random by our assumption of $\bar{C}_r$; |
| 2006 | hence a collision with one of these $z_i$ occurs with probability at most |
| 2007 | $r/2^\ell$. |
| 2008 | \item If $z_r$ is the IV for some message $i$ where $m_C \le i < m_E$, then |
| 2009 | it can collide with previous $z_i$ or either previous or future $c_i$. We |
| 2010 | know, however, that no $F$-input has collided with $c_i$, so in the |
| 2011 | `result' game, $z_r = F(c_r)$ is uniformly distributed; in the `garbage' |
| 2012 | game, $W_\$$ generates $z_r$ at random anyway. It collides, therefore, |
| 2013 | with probability at most $(r + n_E)/2^\ell$. |
| 2014 | \item If $z_r$ is the IV for some message $i$ where $m_E \le i < q'$ then |
| 2015 | $z_r$ was chosen uniformly at random. Hence it collides with probability |
| 2016 | at most $(r + n_E)/2^\ell$. |
| 2017 | \item Finally, either $z_r$ is not the IV for a message, or it is, but the |
| 2018 | message number $i < n_L$, so in either case, $z_r = z_{r-1} \shift{t} |
| 2019 | y_{r-1}$. We have two subcases to consider. |
| 2020 | \begin{enumerate} |
| 2021 | \item If $1 \le r < \ell/t$ (we dealt with the case $r = 0$ above) then |
| 2022 | some of $V_0$ remains in the shift register. If $z_r$ collides with some |
| 2023 | $z_i$, for $0 \le i < r$, then we must have $z_r[0 \bitsto \ell - t r] = |
| 2024 | z_i[0 \bitsto \ell - t r]$; but $z_r[0 \bitsto \ell - t r] = V_0[t r |
| 2025 | \bitsto \ell]$, and $z_i[0 \bitsto \ell - t r] = V_0[t i \bitsto \ell - t |
| 2026 | (r - i)]$, i.e., we have found a $t$-sliding of $V_0$, which is |
| 2027 | impossible by hypothesis. Hence, $z_r$ cannot collide with any earlier |
| 2028 | $z_i$. Also by hypothesis, $n_C = n_E = 0$ if $\ell > t$, so $z_r$ |
| 2029 | cannot collide with any counters $c_i$. |
| 2030 | \item Suppose, then, that $r \ge \ell/t$. For $0 \le j < \ell/t$, let $H_j |
| 2031 | = \ell - t j$, $L_j = \max(0, H_j - t)$, and $N_j = H_j - L_j$. (Note |
| 2032 | that $\sum_{0\le j<\ell/t} N_j = \ell$.) Then $z_r[L_j \bitsto H_j] = |
| 2033 | y_{r-j-1}[t - N_j \bitsto t]$; but the $y_i$ for $i < r$ are uniformly |
| 2034 | distributed. Thus, $z_r$ collides with some specific other value $z'$ |
| 2035 | only with probability $1/2^{\sum_j N_j} = 1/2^\ell$. The overall |
| 2036 | collision probablity for $z_r$ is then at most $(r + n_E)/2^\ell$. |
| 2037 | \end{enumerate} |
| 2038 | \end{enumerate} |
| 2039 | In all these cases, it's clear that the collision probability is no more than |
| 2040 | $(r + n_E)/2^\ell$. |
| 2041 | |
| 2042 | The probability that there is a collision during the course of the game is |
| 2043 | $\Pr[C_{q'}]$, which we can now bound thus: |
| 2044 | \begin{equation} |
| 2045 | \Pr[C_q'] \le \sum_{0<i\le q'} \Pr[C_i | \bar{C}_{i-1}] |
| 2046 | \le \sum_{0<i\le q'} \frac{i + n_E}{2^\ell}. |
| 2047 | \end{equation} |
| 2048 | If we set $i' = i + n_E$, then we get |
| 2049 | \begin{equation} |
| 2050 | \Pr[C_q'] \le \sum_{0\le i'\le q} \frac{i'}{2^\ell} |
| 2051 | = \frac{q (q - 1)}{2^{\ell+1}}. |
| 2052 | \end{equation} |
| 2053 | Finally, then, we can apply the same argument as we used at the end of |
| 2054 | section~\ref{sec:cbc-proof} to show that |
| 2055 | \begin{equation} |
| 2056 | \Adv{}{}(A') \le \frac{q (q - 1)}{2^{\ell+1}} |
| 2057 | \end{equation} |
| 2058 | as claimed. This completes the proof. |
| 2059 | |
| 2060 | %%%-------------------------------------------------------------------------- |
| 2061 | |
| 2062 | \section{OFB mode encryption} |
| 2063 | \label{sec:ofb} |
| 2064 | |
| 2065 | \subsection{Description} |
| 2066 | \label{sec:ofb-desc} |
| 2067 | |
| 2068 | Suppose $F$ is an $\ell$-bit-to-$L$-bit pseudorandom function, and let $t \le |
| 2069 | L$. OFB mode works as follows. Given a message $X$, we divide it into |
| 2070 | $t$-bit blocks $x_0$, $x_1$, $\ldots$, $x_{n-1}$. Choose an initialization |
| 2071 | vector $v \in \Bin^\ell$. We maintain a \emph{shift register} $s_i$, whose |
| 2072 | initial value is $v$. To encrypt a block $x_i$, we XOR it with the result |
| 2073 | $z_i$ of passing the shift register through the PRF, forming $y_i$, and then |
| 2074 | update the shift register by shifting in the PRF output $z_i$. That |
| 2075 | is, |
| 2076 | \begin{equation} |
| 2077 | s_0 = v \qquad |
| 2078 | z_i = F_K(s_i) \qquad |
| 2079 | y_i = x_i \xor z_i \qquad |
| 2080 | s_{i+1} = s_i \shift{t} z_i \ \text{(for $0 \le i < n$)}. |
| 2081 | \end{equation} |
| 2082 | Decryption is precisely the same operation. |
| 2083 | |
| 2084 | Also, we observe that the final plaintext block needn't be $t$ bits long: we |
| 2085 | can pad it out to $t$ bits and truncate the result without affecting our |
| 2086 | ability to decrypt. |
| 2087 | |
| 2088 | \begin{figure} |
| 2089 | \begin{cgraph}{ofb-mode} |
| 2090 | [] !{<0.425cm, 0cm>: <0cm, 0.5cm>::} |
| 2091 | *+=(2, 0)+[F]{\mathstrut v} ="v" :|<>(0.35)@{/}_<>(0.35){\ell}[rrrrr] |
| 2092 | *+[o][F]{\shift{t}} ="shift" |
| 2093 | [ll] :|-@{/}^-{\ell}[dd] *+[F]{E} ="e" [ll] {K} :"e" |
| 2094 | :|-@{/}^-{t}[ddd] *{\xor} ="xor" |
| 2095 | [lll] *+=(2, 0)+[F]{\mathstrut x_0} :|-@{/}_-{t} "xor" |
| 2096 | :|-@{/}^-{t}[dd] *+=(2, 0)+[F]{\mathstrut y_0} |
| 2097 | "xor" [u] :`r "shift" "shift"|-@{/}_-{t} |
| 2098 | :|-@{/}_-{\ell}[rrrrrrr] *+[o][F]{\shift{t}} ="shift" |
| 2099 | [ll] :|-@{/}^-{\ell}[dd] *+[F]{E} ="e" [ll] {K} :"e" |
| 2100 | :|-@{/}^-{t}[ddd] *{\xor} ="xor" |
| 2101 | [lll] *+=(2, 0)+[F]{\mathstrut x_1} :|-@{/}_-{t} "xor" |
| 2102 | :|-@{/}^-{t}[dd] *+=(2, 0)+[F]{\mathstrut y_1} |
| 2103 | "xor" [u] :`r "shift" "shift"|-@{/}_-{t} |
| 2104 | :@{-->}|-@{/}_-{\ell}[rrrrrrr] *+[o][F]{\shift{t}} ="shift" |
| 2105 | [ll] :@{-->}|-@{/}^-{\ell}[dd] *+[F]{E} ="e" [ll] {K} :"e" |
| 2106 | :@{-->}|-@{/}^-{t}[ddd] *{\xor} ="xor" |
| 2107 | [lll] *+=(2, 0)+[F--]{\mathstrut x_i} :@{-->}|-@{/}_-{t} "xor" |
| 2108 | :@{-->}|-@{/}^-{t}[dd] *+=(2, 0)+[F--]{\mathstrut y_i} |
| 2109 | "xor" [u] :@{-->} `r "shift" "shift"|-@{/}_-{t} |
| 2110 | [rrrrrdd] *+[F]{E} ="e" |
| 2111 | "shift" :@{-->}`r "e" |-@{/}_-{\ell} "e" |
| 2112 | [ll] {K} :"e" |
| 2113 | :|-@{/}^-{t}[ddd] *{\xor} ="xor" |
| 2114 | [lll] *+=(2, 0)+[F]{\mathstrut x_{n-1}} :|-@{/}_-{t} "xor" |
| 2115 | :|-@{/}^-{t}[dd] *+=(2, 0)+[F]{\mathstrut y_{n-1}} |
| 2116 | \end{cgraph} |
| 2117 | |
| 2118 | \caption{Encryption using OFB mode} |
| 2119 | \label{fig:ofb} |
| 2120 | \end{figure} |
| 2121 | |
| 2122 | \begin{definition}[OFB algorithms] |
| 2123 | For any function $F\colon \Bin^\ell \to \Bin^t$, any initialization vector |
| 2124 | $v \in \Bin^\ell$, any plaintext $x \in \Bin^*$ and any ciphertext $y \in |
| 2125 | \Bin^*$, we define PRF encryption mode $\id{OFB} = (\id{ofb-encrypt}, |
| 2126 | \id{ofb-decrypt})$ as follows: |
| 2127 | \begin{program} |
| 2128 | Algorithm $\id{ofb-encrypt}(F, v, x)$: \+ \\ |
| 2129 | $s \gets v$; \\ |
| 2130 | $L \gets |x|$; \\ |
| 2131 | $x \gets x \cat 0^{t\lceil L/t \rceil - L}$; \\ |
| 2132 | $y \gets \emptystring$; \\ |
| 2133 | \FOR $i = 0$ \TO $(|x| - t')/t$ \DO \\ \ind |
| 2134 | $x_i \gets x[ti \bitsto t(i + 1)]$; \\ |
| 2135 | $z_i \gets F(s)$; \\ |
| 2136 | $y_i \gets x_i \xor z_i$; \\ |
| 2137 | $s \gets s \shift{t} z_i$; \\ |
| 2138 | $y \gets y \cat y_i$; \- \\ |
| 2139 | \RETURN $(s, y[0 \bitsto L])$; |
| 2140 | \next |
| 2141 | Algorithm $\id{ofb-decrypt}(F, v, y)$: \+ \\ |
| 2142 | \RETURN $\id{ofb-encrypt}(F, v, y)$; |
| 2143 | \end{program} |
| 2144 | We now define the schemes $\Xid{\E}{OFB$\$$}^F$, $\Xid{\E}{OFBC}^{F, c}$, |
| 2145 | $\Xid{\E}{OFBE}^{F, c}$, and $\Xid{\E}{OFBL}^{F, V_0}$ according to |
| 2146 | definition~\ref{def:enc-scheme}; and we define the hybrid scheme |
| 2147 | $\Xid{\E}{OFBH}^{F, V_0, c}_{n_L, n_C, n_E}$ according to |
| 2148 | definition~\ref{def:enc-hybrid}. |
| 2149 | \end{definition} |
| 2150 | |
| 2151 | \begin{remark}[Similarity to CFB mode] |
| 2152 | \label{rem:ofb-like-cfb} |
| 2153 | OFB mode is strongly related to CFB mode: we can OFB encrypt a message $x$ |
| 2154 | by \emph{CFB-encrypting} the all-zero string $0^{|x|}$ with the same key |
| 2155 | and IV. That is, we could have written $\id{ofb-encrypt}$ and |
| 2156 | $\id{ofb-decrypt}$ like this: |
| 2157 | \begin{program} |
| 2158 | Algorithm $\id{ofb-encrypt}(F, v, x)$: \+ \\ |
| 2159 | $(s, z) \gets \id{cfb-encrypt}(F, v, 0^{|x|})$; \\ |
| 2160 | \RETURN $(s, x \xor z)$; |
| 2161 | \next |
| 2162 | Algorithm $\id{ofb-decrypt}(F, v, y)$: \+ \\ |
| 2163 | \RETURN $\id{ofb-encrypt}(F, v, y)$; |
| 2164 | \end{program} |
| 2165 | We shall use this fact to prove the security of OFB mode in the next |
| 2166 | section. |
| 2167 | \end{remark} |
| 2168 | |
| 2169 | \subsection{Security of OFB mode} |
| 2170 | |
| 2171 | \begin{theorem}[Security of OFB mode] |
| 2172 | \label{thm:ofb} |
| 2173 | Let $F$ be a pseudorandom function from $\Bin^\ell$ to $\Bin^t$; let $V_0 |
| 2174 | \in \Bin^\ell$ be a non-$t$-sliding string; let $c$ be a generalized |
| 2175 | counter in $\Bin^\ell$; and let $n_L$, $n_C$, $n_E$ and $q_E$ be |
| 2176 | nonnegative integers; and furthermore suppose that |
| 2177 | \begin{itemize} |
| 2178 | \item $n_L + n_C + n_E \le q_E$, |
| 2179 | \item $n_L = 0$, or $n_C = n_E = 0$, or $\ell \le t$ and $V_0 \ne c(i)$ |
| 2180 | for any $n_L \le i < n_L + n_C + n_E$, and |
| 2181 | \item either $n_C = 0$ or $\ell \le t$. |
| 2182 | \end{itemize} |
| 2183 | Then, for any $t_E$ and $\mu_E$, and whenever |
| 2184 | we have |
| 2185 | \[ \InSec{lor-cpa}(\Xid{\E}{OFBH}^{F, V_0, c}_{n_L, n_C, n_E}; |
| 2186 | t_E, q_E, \mu_E) \le |
| 2187 | 2 \cdot \InSec{prf}(F; t_E + q t_F, q) + \frac{q (q - 1)}{2^\ell} |
| 2188 | \] |
| 2189 | where $q = \bigl\lfloor \bigl(\mu_E + q_E (t - 1)\bigr)/t \bigr\rfloor + |
| 2190 | n_E$, and $t_F$ is some small constant. |
| 2191 | \end{theorem} |
| 2192 | \begin{proof} |
| 2193 | We claim that |
| 2194 | \[ \InSec{rog-cpa-$W_\$$} |
| 2195 | (\Xid{\E}{OFBH}^{\Func{\ell}{t}, V_0, c}_{n_L, n_C, n_E}; |
| 2196 | t, q_E, \mu_E) \le |
| 2197 | \frac{q (q - 1)}{2^{\ell+1}}. |
| 2198 | \] |
| 2199 | This follows from lemma~\ref{lem:cfb-rog}, which makes the same statement |
| 2200 | about CFB mode, and the observation in remark~\ref{rem:ofb-like-cfb}. |
| 2201 | Suppose $A$ attempts to distinguish OFBH encryption from $W_\$$. We define |
| 2202 | the adversary $B$ which uses $A$ to attack CFBH encryption, as follows: |
| 2203 | \begin{program} |
| 2204 | Adversary $B^{E(\cdot)}$: \+ \\ |
| 2205 | \RETURN $A^{\id{ofb}(\cdot)}$; \- |
| 2206 | \next |
| 2207 | Function $\id{ofb}(x)$: \+ \\ |
| 2208 | $(v, z) \gets E(0^{|x|})$; \\ |
| 2209 | \RETURN $(v, x \xor z)$; |
| 2210 | \end{program} |
| 2211 | Now we apply proposition~\ref{prop:rog-and-lor}; the theorem follows. |
| 2212 | \end{proof} |
| 2213 | |
| 2214 | \begin{corollary} |
| 2215 | \label{cor:ofb-prf} |
| 2216 | Let $F$, $c$ and $V_0$ be as in theorem~\ref{thm:ofb}. Then for any $t_E$, |
| 2217 | $q_E$ and $\mu_E$, |
| 2218 | \begin{eqnarray*}[rl] |
| 2219 | \InSec{lor-cpa}(\Xid{\E}{OFB$\$$}^F; t_E, q_E, \mu_E) |
| 2220 | & \le 2 \cdot \InSec{prf}(F; t_E + q t_F, q) + \frac{q (q - 1)}{2^\ell} |
| 2221 | \\ |
| 2222 | \InSec{lor-cpa}(\Xid{\E}{OFBE}^{F, c}; t_E, q_E, \mu_E) |
| 2223 | & \le 2 \cdot \InSec{prf}(F; t_E + q' t_F, q') + |
| 2224 | \frac{q' (q' - 1)}{2^\ell} |
| 2225 | \\ |
| 2226 | \InSec{lor-cpa}(\Xid{\E}{OFBL}^{F, V_0}; t_E, q_E, \mu_E) |
| 2227 | & \le 2 \cdot \InSec{prf}(F; t_E + q t_F, q) + \frac{q (q - 1)}{2^\ell} |
| 2228 | \\ |
| 2229 | \tabpause{and, if $\ell \le t$,} |
| 2230 | \InSec{lor-cpa}(\Xid{\E}{OFBC}^{F, c}; t_E, q_E, \mu_E) |
| 2231 | & \le 2 \cdot \InSec{prf}(F; t_E + q t_F, q) + \frac{q (q - 1)}{2^\ell} |
| 2232 | \end{eqnarray*} |
| 2233 | where $q = \bigl\lfloor \bigl(\mu_E + q_E (t - 1)\bigr)/t \bigr\rfloor + |
| 2234 | n_E$, $q' = q + q_E$, and $t_F$ is some small constant. |
| 2235 | \end{corollary} |
| 2236 | \begin{proof} |
| 2237 | Follows from theorem~\ref{thm:ofb} and proposition~\ref{prop:enc-hybrid}. |
| 2238 | \end{proof} |
| 2239 | |
| 2240 | \begin{corollary} |
| 2241 | \label{cor:ofb-prp} |
| 2242 | Let $P$ be a pseudorandom permutation on $\Bin^\ell$, and let $c$ and $V_0$ |
| 2243 | be as in theorem~\ref{thm:ofb}. Then for any $t_E$, $q_E$ and $\mu_E$, |
| 2244 | \begin{eqnarray*}[rl] |
| 2245 | \InSec{lor-cpa}(\Xid{\E}{OFB$\$$}^P; t_E, q_E, \mu_E) |
| 2246 | & \le 2 \cdot \InSec{prp}(P; t_E + q t_F, q) + \frac{q (q - 1)}{2^\ell} |
| 2247 | \\ |
| 2248 | \InSec{lor-cpa}(\Xid{\E}{OFBE}^{P, c}; t_E, q_E, \mu_E) |
| 2249 | & \le 2 \cdot \InSec{prp}(P; t_E + q' t_F, q') + |
| 2250 | \frac{q' (q' - 1)}{2^\ell} |
| 2251 | \\ |
| 2252 | \InSec{lor-cpa}(\Xid{\E}{OFBL}^{P, V_0}; t_E, q_E, \mu_E) |
| 2253 | & \le 2 \cdot \InSec{prp}(P; t_E + q t_F, q) + \frac{q (q - 1)}{2^\ell} |
| 2254 | \\ |
| 2255 | \tabpause{and, if $\ell \le t$,} |
| 2256 | \InSec{lor-cpa}(\Xid{\E}{OFBC}^{P, c}; t_E, q_E, \mu_E) |
| 2257 | & \le 2 \cdot \InSec{prp}(P; t_E + q t_F, q) + \frac{q (q - 1)}{2^\ell} |
| 2258 | \end{eqnarray*} |
| 2259 | where $q = \bigl\lfloor \bigl(\mu_E + q_E (t - 1)\bigr)/t \bigr\rfloor + |
| 2260 | n_E$, $q' = q + q_E$, and $t_F$ is some small constant. |
| 2261 | \end{corollary} |
| 2262 | \begin{proof} |
| 2263 | Follows from corollary~\ref{cor:ofb-prf} and |
| 2264 | proposition~\ref{prop:prps-are-prfs}. |
| 2265 | \end{proof} |
| 2266 | |
| 2267 | %%%-------------------------------------------------------------------------- |
| 2268 | |
| 2269 | \section{CBCMAC mode message authentication} |
| 2270 | \label{sec:cbcmac} |
| 2271 | |
| 2272 | |
| 2273 | |
| 2274 | \begin{figure} |
| 2275 | \begin{cgraph}{cbc-mac} |
| 2276 | []!{<0.425cm, 0cm>: <0cm, 0.75cm>::} |
| 2277 | *+=(2, 0)+[F]{\mathstrut x_0} |
| 2278 | :`d [dr] [rrr] *+[F]{E} ="e" [d] {K} :"e" |
| 2279 | :[rrr] *{\xor} ="xor" |
| 2280 | [u] *+=(2, 0)+[F]{\mathstrut x_1} :"xor" |
| 2281 | :[rrr] *+[F]{E} ="e" [d] {K} :"e" |
| 2282 | :@{-->}[rrr] *{\xor} ="xor" |
| 2283 | [u] *+=(2, 0)+[F--]{\mathstrut x_i} :@{-->}"xor" |
| 2284 | :@{-->}[rrr] *+[F]{E} ="e" [d] {K} :@{-->}"e" |
| 2285 | :@{-->}[rrr] *{\xor} ="xor" |
| 2286 | [u] *+=(2, 0)+[F]{\mathstrut x_{n-1}} :"xor" |
| 2287 | :[rrr] *+[F]{E} ="e" [d] {K} :"e" |
| 2288 | :[rrr] *+=(2, 0)+[F]{\mathstrut \tau} |
| 2289 | \end{cgraph} |
| 2290 | |
| 2291 | \caption{Message authentication using CBCMAC mode} |
| 2292 | \label{fig:cbcmac} |
| 2293 | \end{figure} |
| 2294 | |
| 2295 | \leavevmode\fixme |
| 2296 | Alas, it's been so long since I've picked this up that I've (a) forgotten how |
| 2297 | the proof for this mode went, and (b) lost my notes. You'll either have to |
| 2298 | wait, or I'll have to drop this bit. |
| 2299 | |
| 2300 | %%%-------------------------------------------------------------------------- |
| 2301 | |
| 2302 | \section{Acknowledgements} |
| 2303 | |
| 2304 | Thanks to Clive Jones for his suggestions on notation, and his help in |
| 2305 | structuring the proofs. |
| 2306 | |
| 2307 | %%%----- That's all, folks -------------------------------------------------- |
| 2308 | |
| 2309 | \bibliography{mdw-crypto,cryptography2000,cryptography,rfc} |
| 2310 | |
| 2311 | \end{document} |
| 2312 | |
| 2313 | %%% Local Variables: |
| 2314 | %%% mode: latex |
| 2315 | %%% TeX-master: t |
| 2316 | %%% End: |