| 1 | \xcalways\section{Message authentication codes}\x |
| 2 | |
| 3 | \xcalways\subsection{Definitions and security notions}\x |
| 4 | |
| 5 | \begin{slide} |
| 6 | \head{Definition of a MAC} |
| 7 | |
| 8 | A MAC is a pair of algorithms $\mathcal{M} = (T, V)$: |
| 9 | \begin{itemize} |
| 10 | \item The \emph{tagging} algorithm $T\colon \{0, 1\}^k \times \{0, 1\}^* |
| 11 | \to \{0, 1\}^L$ is a probabilistic algorithm which, given a key and a |
| 12 | string, returns a \emph{tag}. We write $\tau \in T_K(m)$. |
| 13 | \item The \emph{verification} algorithm $V\colon \{0, 1\}^k \times \{0, |
| 14 | 1\}^* \times \{0, 1\}^L \to \{0, 1\}$ is a deterministic algorithm which, |
| 15 | given a key, a message and a tag, returns $1$ if the tag is valid, or $0$ |
| 16 | otherwise; i.e., we require that $V_K(m, \tau) = 1 \iff \tau \in T_K(m)$. |
| 17 | \end{itemize} |
| 18 | The basic idea is that it's hard for an adversary to \emph{forge} a tag for |
| 19 | a message it's not seen before. |
| 20 | \end{slide} |
| 21 | |
| 22 | \begin{slide} |
| 23 | \topic{informal security notion} |
| 24 | \resetseq |
| 25 | \head{Strong MACs, \seq: informal security notion} |
| 26 | |
| 27 | Our standard notion of security for MACs is \emph{strong unforgeability |
| 28 | against chosen message attack}, or SUF-CMA, for short |
| 29 | \cite{Abdalla:2001:DHIES, Bellare:2000:AER}. Let $A$ be an adversary which |
| 30 | is attempting to attack the MAC $\mathcal{M} = (T, V)$. |
| 31 | |
| 32 | We play a game with the adversary. We invent a key $K \inr \{0, 1\}^k$. |
| 33 | The adversary \emph{wins} if, after requesting tags for some messages of |
| 34 | its choice, and checking some guesses, it can return a pair $(m, \tau)$ |
| 35 | such that: |
| 36 | \begin{itemize} |
| 37 | \item the tag is correct, i.e., $V_K(m, \tau) = 1$; and |
| 38 | \item the tag is not one returned by the adversary's tagging oracle for |
| 39 | that message. |
| 40 | \end{itemize} |
| 41 | \end{slide} |
| 42 | |
| 43 | \begin{slide} |
| 44 | \topic{strong MACs} |
| 45 | \head{Strong MACs, \seq: the experiment} |
| 46 | |
| 47 | We perform the following experiment with the adversary. |
| 48 | \begin{program} |
| 49 | Experiment $\Expt{suf-cma}{\mathcal{M}}(A)$: \+ \\ |
| 50 | $K \getsr \{0, 1\}^k$; \\ |
| 51 | $\Xid{T}{list} \gets \emptyset$; \\ |
| 52 | $(m, \tau) \gets A^{\id{tag}(\cdot), V_K(\cdot, \cdot)}$; \\ |
| 53 | \IF $V_K(m, \tau) \land (m, \tau) \notin \Xid{T}{list}$ |
| 54 | \THEN \RETURN $1$; \\ |
| 55 | \ELSE \RETURN $0$; \- \\[\smallskipamount] |
| 56 | Oracle $\id{tag}(m)$: \+ \\ |
| 57 | $\tau \gets T_K(m)$; \\ |
| 58 | $\Xid{T}{list} \gets \Xid{T}{list} \cup \{(m, \tau)\}$; \\ |
| 59 | \RETURN $\tau$; |
| 60 | \end{program} |
| 61 | \end{slide} |
| 62 | |
| 63 | \begin{slide} |
| 64 | \head{Strong MACs, \seq: wrapping up the notation} |
| 65 | |
| 66 | The \emph{success probability} of an adversary $A$ against the MAC |
| 67 | $\mathcal{M}$ in the sense of SUF-CMA is |
| 68 | \[ \Succ{suf-cma}{\mathcal{M}}(A) = |
| 69 | \Pr[\Expt{suf-cma}{\mathcal{M}}(A) = 1]. \]% |
| 70 | The \emph{insecurity} of a MAC $\mathcal{M}$ in the SUF-CMA sense is then |
| 71 | \[ \InSec{suf-cma}(\mathcal{M}; t, q_T, q_V) = |
| 72 | \max_A \Succ{suf-cma}{\mathcal{M}}(A) \]% |
| 73 | where the maximum is taken over all adversaries $A$ running in time $t$ and |
| 74 | making $q_T$ tagging and $q_V$ verification queries. |
| 75 | |
| 76 | If $\InSec{suf-cma}(\mathcal{M}; t, q_T, q_V) \le \epsilon$ then we say |
| 77 | that $\mathcal{M}$ is a \emph{$(t, q_T, q_V, \epsilon)$-secure MAC in the |
| 78 | SUF-CMA sense}. |
| 79 | \end{slide} |
| 80 | |
| 81 | \begin{slide} |
| 82 | \topic{other notions} |
| 83 | \head{Other security notions for MACs} |
| 84 | |
| 85 | There are a number of weaker security notions in use: |
| 86 | \begin{itemize} |
| 87 | \item The definition of a \emph{weak MAC} restricts the adversary from |
| 88 | returning any message with which it queried its tagging oracle. The |
| 89 | strong MAC definition considers this OK, as long as the tag is different |
| 90 | from any returned by the oracle for that message. |
| 91 | \item Some definitions of MACs don't equip the adversary with a |
| 92 | verification oracle. Our definition considers these to be $(t, q_T, 0, |
| 93 | \epsilon)$-secure. |
| 94 | \item You can have a MAC with a bounded domain $\{0, 1\}^L$ rather than |
| 95 | $\{0, 1\}^*$ as shown previously. |
| 96 | \item Further quantification is possible, e.g., counting the total number |
| 97 | of bytes queried, or the maximum size of a tagging query. |
| 98 | \end{itemize} |
| 99 | \end{slide} |
| 100 | |
| 101 | \xcalways\subsection{Basic results}\x |
| 102 | |
| 103 | \begin{slide} |
| 104 | \topic{PRFs are MACs} |
| 105 | \resetseq |
| 106 | \head{PRFs are MACs, \seq} |
| 107 | |
| 108 | If $F_K\colon \{0, 1\}^* \to \{0, 1\}^L$ is a $(t, q, \epsilon)$-secure |
| 109 | PRF, then it's also a $(t', q_T, q_V, \epsilon')$-secure MAC, with $q = q_T |
| 110 | + q_V + 1$, $t = t' + O(q)$, and $\epsilon' \le \epsilon + (q_V + 1) |
| 111 | 2^{-L}$. The constant hidden by the $O(\cdot)$ is small and depends on the |
| 112 | model of computation. |
| 113 | |
| 114 | Suppose $A$ can break $F$ used as a MAC in time $t$ and with $q_T$ and |
| 115 | $q_V$ queries to its tagging and verification oracles respectively. |
| 116 | |
| 117 | If we can construct an adversary which distinguishes $F_K$ from a random |
| 118 | function using $A$ as an essential component, then we can prove the |
| 119 | result. |
| 120 | \end{slide} |
| 121 | |
| 122 | \begin{slide} |
| 123 | \head{PRFs are MACs, \seq: the distinguisher} |
| 124 | |
| 125 | \begin{program} |
| 126 | Distinguisher $D^{F(\cdot)}$: \+ \\ |
| 127 | $\Xid{T}{list} \gets \emptyset$; \\ |
| 128 | $(m, \tau) \gets A^{T_F(\cdot), V_F(\cdot, \cdot)}$; \\ |
| 129 | \IF $m \notin \Xid{T}{list} \land \tau = F(m)$ |
| 130 | \THEN \RETURN $1$; \\ |
| 131 | \ELSE \RETURN $0$; \- \\[\smallskipamount] |
| 132 | Oracle $T_F(m)$: \+ \\ |
| 133 | $\Xid{T}{list} \gets \Xid{T}{list} \cup \{m\}$; \\ |
| 134 | \RETURN $F(m)$; \- \\[\smallskipamount] |
| 135 | Oracle $V_F(m, \tau)$: \+ \\ |
| 136 | \IF $\tau = F(m)$ \THEN \RETURN $1$; \\ |
| 137 | \ELSE \RETURN $0$; |
| 138 | \end{program} |
| 139 | \end{slide} |
| 140 | |
| 141 | \begin{slide} |
| 142 | \head{PRFs are MACs, \seq: wrapping up} |
| 143 | |
| 144 | The distinguisher simulates the tagging and verification oracles for the |
| 145 | MAC forger, using its supplied oracle. If the forger succeeds, then the |
| 146 | distinguisher returns 1; otherwise it returns zero. |
| 147 | |
| 148 | The probability that the distinguisher returns 1 given an instance $F_K$ of |
| 149 | the PRF is precisely $\Succ{suf-cma}{F}(A)$. |
| 150 | |
| 151 | The probability that it returns 0 given a random function depends on what |
| 152 | $A$ does when it's given a random function. But we know that the |
| 153 | probability of it successfully guessing the MAC for a message for which it |
| 154 | didn't query $T$ can be at most $(q_V + 1) 2^{-L}$. So |
| 155 | \[ \Adv{prf}{F}(D) \ge \Succ{suf-cma}{F}(A) - (q_V + 1) 2^{-L}. \] |
| 156 | Let $q = q_T + q_V + 1$; then counting, rearranging, maximizing yields |
| 157 | \[ \InSec{suf-cma}(F; t, q_T, q_V) \le |
| 158 | \InSec{prf}(F; t + O(q), q) + (q_V + 1)2^{-L}. \]% |
| 159 | \end{slide} |
| 160 | |
| 161 | \begin{slide} |
| 162 | \head{PRFs are MACs, \seq: MACs aren't PRFs} |
| 163 | |
| 164 | The converse of our result is not true. Suppose $\mathcal{M} = (T, V)$ is |
| 165 | a deterministic MAC. Choose some integer $n$. Then define $\mathcal{M}' = |
| 166 | (T', V')$, as follows: |
| 167 | \[ T'_K(x) = 0^n \cat T_K(x); \qquad |
| 168 | V'_K(x, \tau) = \begin{cases} |
| 169 | 1 & if $T'_K(x) = \tau$ \\ |
| 170 | 0 & otherwise |
| 171 | \end{cases}. |
| 172 | \] |
| 173 | $T'$ is obviously not a PRF: an adversary checking for the string of $n$ |
| 174 | zero bits on the output will succeed with advantage $1 - 2^{-qn}$. |
| 175 | |
| 176 | However, $\mathcal{M}'$ is a secure MAC. Suppose $A'$ attacks |
| 177 | $\mathcal{M}'$. |
| 178 | \begin{program} |
| 179 | Adversary $A^{T(\cdot), V(\cdot)}$: \+ \\ |
| 180 | $(m, \tau') \gets A'^{\id{tag}(\cdot), \id{verify}(\cdot)}$; \\ |
| 181 | \PARSE $\tau'$ \AS $n\colon z, \tau$; \\ |
| 182 | \RETURN $(m, \tau)$; |
| 183 | \next |
| 184 | Oracle $\id{tag}(m)$: \+ \\ |
| 185 | \RETURN $0^n \cat T(m)$; \- \\[\smallskipamount] |
| 186 | Oracle $\id{verify}(m, \tau')$: \+ \\ |
| 187 | \PARSE $\tau'$ \AS $n\colon z, \tau$; \\ |
| 188 | \IF $z \ne 0^n$ \THEN \RETURN $0$; \\ |
| 189 | \ELSE \RETURN $V(m, \tau)$; |
| 190 | \end{program} |
| 191 | \end{slide} |
| 192 | |
| 193 | \begin{exercise} |
| 194 | \begin{parenum} |
| 195 | \item Suppose that $F\colon \{0, 1\}^k \times \{0, 1\}^* \to \{0, 1\}^L$ is |
| 196 | a $(t, q, \epsilon)$-secure PRF. Let $T^{(\ell)}_K(x)$ be the leftmost |
| 197 | $\ell$~bits of $F_K(x)$ for $\ell \le L$. Demonstrate the security of |
| 198 | $T^{(\ell)}(\cdot)$ as a MAC. |
| 199 | \item Discuss the merits of truncating MAC tags in practical situations. |
| 200 | \end{parenum} |
| 201 | \answer% |
| 202 | \begin{parenum} |
| 203 | \item The follows exactly the same pattern as the `PRFs are MACs' proof in |
| 204 | the slides: $T^{(\ell)}$ is a $(t, q_T, q_V, \epsilon + (q_V + |
| 205 | 1)2^{-\ell})$-secure MAC, where $q_T + q_V = q$. |
| 206 | \item Obviously, truncating tags saves bandwidth. There is a trade-off |
| 207 | between tag size and security, as the $2^{-\ell}$ term shows. Note that |
| 208 | this term represents the probability of the adversary guessing the |
| 209 | correct tag when it's actually attacking a random function, and |
| 210 | therefore, when this occurs, the adversary has one `by accident'. |
| 211 | Including sequence numbers in packets ensures that replay of accidental |
| 212 | forgery (or honest messages) will be detected. Hence, for some |
| 213 | applications, setting $\ell = 32$ or even lower is of no particular harm. |
| 214 | Perhaps more significantly, if the PRF isn't actually as good as it ought |
| 215 | to be, and (say) leaks key material very slowly, then truncating its |
| 216 | output can actually improve security. |
| 217 | \end{parenum} |
| 218 | \end{exercise} |
| 219 | |
| 220 | \begin{exercise} |
| 221 | A traditional MAC construction is the \emph{CBC-MAC}: it works like this. |
| 222 | Suppose $F\colon \{0, 1\}^k \times \{0, 1\}^l \to \{0, 1\}^l$ is a PRF. |
| 223 | Split a message~$x$ into $l$-bit blocks $x_0, x_1, \ldots, x_{n-1}$ |
| 224 | (applying some sort of padding if you need to). Then we define the CBC-MAC |
| 225 | as $F^{(n)}_K(x)$, where |
| 226 | \[ F^{(1)}_K(x) = F_K(x); |
| 227 | \qquad F^{(i+i)}(x) = F_K(x_i \xor F^{(i)}_K(x)). \]% |
| 228 | In \cite{Bellare:1994:SCB}, Mihir Bellare, Joe Kilian and Phil Rogaway |
| 229 | introduced the world to the concrete-security approach and, almost |
| 230 | incidentally, proved that the CBC-MAC is a PRF (and therefore a MAC) for |
| 231 | any \emph{fixed sized} input. |
| 232 | |
| 233 | Show that the CBC-MAC is easily broken if you're allowed to choose messages |
| 234 | of different lengths. |
| 235 | \answer% |
| 236 | Request tags $\tau$ for the message $x = x_0, x_1, \ldots, x_{n-1}$ and |
| 237 | $\tau'$ for $x' = x'_0 \xor \tau, x'_1, \ldots, x'_{n'-1}$. Let $y = x_0, |
| 238 | x_1, \ldots, x_{n-1}, x'_0 , x'_1, \ldots, x'_{n'-1}$. Note that |
| 239 | $F^{(n)}_K(y) = \tau$, and $F^{(n+1)}_K(y) = F_K(x'_0 \xor F^{(n)}_K(x)) = |
| 240 | F^{(1)}_K(x')$. Hence, $F^{(n+n')}_K(y) = \tau'$, and we have a correct |
| 241 | forgery. |
| 242 | \end{exercise} |
| 243 | |
| 244 | \begin{slide} |
| 245 | \topic{verification oracles} |
| 246 | \head{Verification oracles} |
| 247 | |
| 248 | We can leave verification oracles out of our analysis. This simplifies |
| 249 | analysis, but produces slightly less satisfactory quantitative results. |
| 250 | |
| 251 | Suppose that $\mathcal{M} = (T, V)$ is a $(t, q_T, 0, \epsilon)$-secure |
| 252 | MAC. Then, for any $q_V$, |
| 253 | \[ \InSec{suf-cma}(\mathcal{M}; t, q_T, q_V) \le |
| 254 | (q_V + 1)\InSec{suf-cma}(\mathcal{M}; t, q_T, 0). \]% |
| 255 | This bound is \emph{tight}: it's not possible for a general result like |
| 256 | this to do better. |
| 257 | \end{slide} |
| 258 | |
| 259 | \begin{proof} |
| 260 | Consider an adversary $A$ which uses $q_V$ verification queries. We assume |
| 261 | the following properties of $A$'s behaviour: |
| 262 | \begin{itemize} |
| 263 | \item No verification query contains a message and a tag for that message |
| 264 | received from the tagging oracle. |
| 265 | \item If a verification query succeeds, the message is not given in a query |
| 266 | to the tagging oracle. |
| 267 | \item Once a verification query succeeds, all subsequent verification |
| 268 | queries also succeed and the adversary returns a correct forgery (e.g., |
| 269 | by simply repeating the successful query). |
| 270 | \end{itemize} |
| 271 | It's clear that any adversary can be transformed into one which has these |
| 272 | properties and succeeds with probability at least as high. |
| 273 | |
| 274 | Let $V$ be the event that at least one verification query succeeds, and let |
| 275 | $S$ be the event that $A$ succeeds. Then |
| 276 | \begin{eqnarray*}[rl] |
| 277 | \Succ{suf-cma}{\mathcal{M}}(A) |
| 278 | &= \Pr[S \mid V] \Pr[V] + \Pr[S \mid \lnot V] \Pr[\lnot V] \\ |
| 279 | &= \Pr[V] + \Pr[S \mid \lnot V] \Pr[\lnot V]. |
| 280 | \end{eqnarray*} |
| 281 | Now consider these two adversaries: |
| 282 | \begin{program} |
| 283 | Adversary $A'^{T(\cdot), V(\cdot, \cdot)}$: \+ \\ |
| 284 | $i \gets 0$; \\ |
| 285 | $(m, \tau) \gets A^{T(\cdot), \Xid{V}{sim}(\cdot, \cdot)}$; \\ |
| 286 | \RETURN $(m^*, \tau^*)$; \- \\[\smallskipamount] |
| 287 | Oracle $\Xid{V}{sim}(m, \tau)$: \+ \\ |
| 288 | $i \gets i + 1$; \\ |
| 289 | \IF $i < q_V$ \THEN \RETURN $V(m, \tau)$; \\ |
| 290 | $(m^*, \tau^*) \gets (m, \tau)$; \\ |
| 291 | \RETURN $1$; |
| 292 | \next |
| 293 | Adversary $Z^{T(\cdot), V(\cdot, \cdot)}$: \+ \\ |
| 294 | \\ |
| 295 | $(m, \tau) \gets A^{T(\cdot), \Xid{V}{zero}(\cdot, \cdot)}$; \\ |
| 296 | \RETURN $(m, \tau)$; \- \\[\smallskipamount] |
| 297 | Oracle $\Xid{V}{zero}(m, \tau)$: \+ \\ |
| 298 | \RETURN $0$; |
| 299 | \end{program} |
| 300 | The adversary $A'$ uses $q_V - 1$ verification queries. It ignores the |
| 301 | output of $A$, returning instead $A$'s $q_V$-th verification query. Thus, |
| 302 | by our assumptions on the behaviour of $A$, we have that $A'$ succeeds |
| 303 | precisely whenever one of $A$'s verification queries succeeds. Thus: |
| 304 | \[ \Pr[V] = \Succ{suf-cma}{\mathcal{M}}(A') |
| 305 | \le \InSec{suf-cma}(\mathcal{M}; t, q_T, q_V - 1). \]% |
| 306 | Similarly, the adversary $Z$ succeeds with precisely the same probability |
| 307 | as $A$, given that all of its verification queries failed; i.e., |
| 308 | \[ \Pr[S \mid \lnot V] \Pr[\lnot V] = \Succ{suf-cma}{\mathcal{M}}(Z) |
| 309 | \le \InSec{suf-cma}(\mathcal{M}; t, q_T, 0). \]% |
| 310 | Because $A$ was chosen arbitrarily, we can maximize: |
| 311 | \begin{eqnarray*}[rl] |
| 312 | \InSec{suf-cma}(\mathcal{M}; t, q_T, q_V) |
| 313 | & \le \InSec{suf-cma}(\mathcal{M}; t, q_T, q_V - 1) + |
| 314 | \InSec{suf-cma}(\mathcal{M}; t, q_T, 0) \\ |
| 315 | & \le (q_V + 1)\InSec{suf-cma}(\mathcal{M}; t, q_T, 0) |
| 316 | \end{eqnarray*} |
| 317 | as required. |
| 318 | |
| 319 | To show that the bound is tight, consider a random function $F$ used as a |
| 320 | MAC, with $L$-bit output. Then we claim that $\InSec{suf-cma}(F; t, q_T, |
| 321 | q_V) = (q_V + 1)/2^L$. To save typing, let $e_q = \InSec{suf-cma}(F; t, |
| 322 | q_T, q)$. We prove the claim by induction on $q$. Firstly, note that if |
| 323 | $q = 0$ then necessarily $e_q = 1/2^L$. Now suppose that $e_{q-1} = |
| 324 | q/2^L$. We're now allowed an extra query, so rather than returning the |
| 325 | result, we feed it to the verification oracle. If it answers `yes' then we |
| 326 | return it; otherwise we guess randomly from the remaining $2^L - q$ |
| 327 | possibilities. Now |
| 328 | \begin{eqnarray*}[rl] |
| 329 | e_q &= e_{q-1} + \frac{1 - e_{q-1}}{2^L - q} \\ |
| 330 | &= \frac{q}{2^L} + \frac{2^L - q}{2^L} \cdot \frac{1}{2^L - q} \\ |
| 331 | &= \frac{q + 1}{2^L} |
| 332 | \end{eqnarray*} |
| 333 | as claimed. |
| 334 | \end{proof} |
| 335 | |
| 336 | \xcalways\subsection{The HMAC construction}\x |
| 337 | |
| 338 | \begin{slide} |
| 339 | \resetseq |
| 340 | \head{The HMAC construction \cite{Bellare:1996:KHF}, \seq: motivation} |
| 341 | |
| 342 | It ought to be possible to construct a decent MAC using a hash function. |
| 343 | Many attempts have failed, however. For example, these constructions are |
| 344 | weak if used with standard one-pass Merkle-Damg\aa{}rd iterated hashes. |
| 345 | \begin{itemize} |
| 346 | \item Secret prefix: $T_K(m) = H(K \cat m)$. Given $H(K \cat m)$, it's |
| 347 | easy to compute $H(K \cat m \cat p \cat m')$ for a padding string $p$ and |
| 348 | arbitrary suffix $m'$. |
| 349 | \item Secret suffix: $T_K(m) = H(m \cat K)$. Finding a collision $H(m) = |
| 350 | H(m')$ yields $H(m \cat K) = H(m' \cat K)$. We saw earlier that |
| 351 | adversaries which know collisions \emph{exist} even if we don't know how |
| 352 | to describe them. |
| 353 | \end{itemize} |
| 354 | |
| 355 | It would be nice to have a construction whose security was provably related |
| 356 | to some plausible property of the underlying hash function. |
| 357 | \end{slide} |
| 358 | |
| 359 | \begin{slide} |
| 360 | \head{The HMAC construction, \seq: definition of NMAC} |
| 361 | |
| 362 | Let $H\colon \{0, 1\}^* \to \{0, 1\}^k$ be an iterated hash, constructed |
| 363 | from the compression function $F\colon \{0, 1\}^k \times \{0, 1\}^L \to |
| 364 | \{0, 1\}^k$. We define a keyed version of $H$. Let $K \in \{0, 1\}^k$; |
| 365 | then we compute $H_K(x)$ as follows: |
| 366 | \begin{enumerate} |
| 367 | \item Pad and split $x$ into the $L$-bit blocks $x_0$, $x_1$, \ldots, |
| 368 | $x_{n-1}$ as before. |
| 369 | \item Set $I_0 = K$. Let $I_{i+1} = F(I_i \cat x_i)$ for $0 \le i < n$. |
| 370 | \item The result $H_K(x) = I_n$. |
| 371 | \end{enumerate} |
| 372 | The NMAC (nested-MAC) construction requires two independent $k$-bit keys |
| 373 | $K_0$ and $K_1$. The construction itself is simply: |
| 374 | \[ \Xid{T}{NMAC}^H_{K_0, K_1}(x) = H_{K_0}(H_{K_1}(x)). \] |
| 375 | NMAC is deterministic, so verification consists of computing the tag and |
| 376 | comparing. |
| 377 | \end{slide} |
| 378 | |
| 379 | \begin{slide} |
| 380 | \head{The HMAC construction, \seq: security of NMAC} |
| 381 | |
| 382 | Consider a function $F\colon \{0, 1\}^k \times \{0, 1\}^* \to \{0, 1\}^k$. |
| 383 | We say that $F$ is \emph{$(t, q, \epsilon)$-weakly collision resistant} if, |
| 384 | for any adversary $A$ constrained to run in time $t$ and permitted $q$ |
| 385 | oracle queries, |
| 386 | \[ \Pr[K \getsr \{0, 1\}^k; |
| 387 | (x, y) \gets A^{F_K(\cdot)} : |
| 388 | x \ne y \land F_K(x) = F_K(y)] \le \epsilon \]% |
| 389 | |
| 390 | If $H_K$ is a $(t, q_T, q_V, \epsilon)$-secure MAC on $k$-bit messages, and |
| 391 | moreover $(t, q_T + q_V, \epsilon')$-weakly collision resistant, then |
| 392 | $\Xid{T}{NMAC}^H$ is a $(t, q_T, q_V, \epsilon + \epsilon')$-secure MAC. |
| 393 | \end{slide} |
| 394 | |
| 395 | \begin{slide} |
| 396 | \head{The HMAC construction, \seq: NMAC security proof} |
| 397 | |
| 398 | Let $A$ be an adversary which forges a $\Xid{T}{NMAC}^H$ tag in time $t$, |
| 399 | using $q_T$ tagging queries and $q_V$ verification queries with probability |
| 400 | $\epsilon$. We construct an adversary $A'$ which forges an $H$-tag for a |
| 401 | $k$-bit message in essentially the same time. |
| 402 | \begin{program} |
| 403 | Adversary $A'^{T(\cdot), V(\cdot, \cdot)}$ \+ \\ |
| 404 | $K \getsr \{0, 1\}^k$; \\ |
| 405 | $(m, \tau) \gets A^{T(H_K(\cdot)), V(H_K(\cdot), \cdot)}$; \\ |
| 406 | \RETURN $(H_K(m), \tau)$; |
| 407 | \end{program} |
| 408 | $A'$ might fail even though $A$ succeeded only if the message it returns, |
| 409 | $H_K(m)$, collides with one of its tagging queries. But since $H_K$ is |
| 410 | $(t, q_T + q_V, \epsilon')$-weakly collision resistant, this happens with |
| 411 | at most probability $\epsilon'$. Hence, $A'$ succeeds with probability at |
| 412 | least $\epsilon - \epsilon'$. Rearrangement yields the required result. |
| 413 | \end{slide} |
| 414 | |
| 415 | \begin{slide} |
| 416 | \head{The HMAC construction, \seq: from NMAC to HMAC} |
| 417 | |
| 418 | Implementing NMAC involves using strange initialization vectors and |
| 419 | generally messing about with your hash function. HMAC is an attempt to |
| 420 | capture the provable security properties using a plain ol' hash function. |
| 421 | |
| 422 | Suppose $H$ is an iterated hash function with a $k$-bit output and $L$-bit |
| 423 | blocks (with $L \ge k$). We set $\id{ipad}$ to be the byte $\hex{36}$ |
| 424 | repeated $L/8$ times, and $\id{opad}$ to be the byte $\hex{5C}$ repeated |
| 425 | $L/8$ times. Select a key $K$ of $L$ bits: if your key is shorter, pad it |
| 426 | by appending zero bits; if it's longer, hash it with $H$ and then pad. |
| 427 | |
| 428 | The HMAC tagging function is then defined as |
| 429 | \[ \Xid{T}{HMAC}^H_K(m) = |
| 430 | H(K \xor \id{opad} \cat H(K \xor \id{ipad} \cat m)). \]% |
| 431 | \end{slide} |
| 432 | |
| 433 | \begin{slide} |
| 434 | \head{The HMAC construction, \seq: comparison with NMAC} |
| 435 | |
| 436 | Comparing the two constructions, we see that |
| 437 | \[ \Xid{T}{HMAC}^H_K = |
| 438 | \Xid{T}{NMAC}^{H'}_{F(I \cat K \xor \id{opad}), |
| 439 | F(I \cat K \xor \id{ipad})}. \]% |
| 440 | Here, $I$ is $H$'s initialization vector, $F$ is the compression function; |
| 441 | $H'$ denotes a keyed hash function that is `like' $H$ but performs padding |
| 442 | as if there were an extra initial block of message data for each message. |
| 443 | |
| 444 | The proof of NMAC assumes that the two keys are random and independent. |
| 445 | This isn't the case in HMAC, but a little handwaving about pseudorandomness |
| 446 | of the compression function makes the problem go away. |
| 447 | \end{slide} |
| 448 | |
| 449 | \begin{exercise} |
| 450 | Suppose that $F\colon \{0, 1\}^k \times \{0, 1\}^{t+\ell} \to \{0, |
| 451 | 1\}^t$ is a PRF. Let $x \in \{0, 1\}^*$ be a message. We define the |
| 452 | function $H_K(x)$ as follows: |
| 453 | \begin{itemize} |
| 454 | \item Pad $x$ to a multiple of $\ell$ bits using some injective |
| 455 | mapping. Break the image of $x$ under this mapping into $\ell$-bit |
| 456 | blocks $x_0, x_1, \ldots, x_{n-1}$. |
| 457 | \item For $0 \le i \le n$, define $H^{(i)}_K(x)$ by |
| 458 | \[ H^{(0)}_K(x) = I; \qquad |
| 459 | H^{(i+1)}_K(x) = F_K(H^{(i)}(x) \cat x_i) \]% |
| 460 | where $I$ is some fixed $t$-bit string (e.g., $I = 0^t$). |
| 461 | \item Then set $H_K(x) = H^{(n)}_K(x)$. |
| 462 | \end{itemize} |
| 463 | We define two (deterministic) MACs $\mathcal{M}^i = (T^i, V^i)$ (for |
| 464 | $i \in \{0, 1\}$) using the $H_K$ construction. Verification in each |
| 465 | case consists of computing the tag and comparing to the one offered. |
| 466 | \begin{eqlines*} |
| 467 | T^0_K(x) = H_K(x); \qquad T^1_K(x) = H_K(x \cat K); \\ |
| 468 | V^i_K(x, \tau) = \begin{cases} |
| 469 | 1 & if $\tau = T^i_K(x)$ \\ |
| 470 | 0 & otherwise |
| 471 | \end{cases}. |
| 472 | \end{eqlines*} |
| 473 | Decide whether each of these constructions is secure. A full proof is |
| 474 | rather hard: an informal justification would be good. |
| 475 | \answer% |
| 476 | $\mathcal{M}^0$ is secure; $\mathcal{M}^1$ isn't, under the sole |
| 477 | assumption that $F$ is a PRF. |
| 478 | |
| 479 | To see that $\mathcal{M}^0$ is secure, it suffices to show that $T^0$ |
| 480 | is a PRF. This is actually quite involved. Given an adversary $A$ |
| 481 | attacking $T^1$ as a PRF, we construct an adversary $B$ attacking $F$, |
| 482 | which simply computes $H$ as required, using the oracle supplied. To |
| 483 | complete the proof, we need to show a bound on the |
| 484 | information-theoretic security of $H$ when instantiated using a random |
| 485 | function $F$. For the sake of simplicity, we allow the adversary $A$ |
| 486 | to query on \emph{padded} messages, rather than the raw unpadded |
| 487 | messages. We count the number $q'$ of individual message blocks. |
| 488 | |
| 489 | As the game with $A$ progresses, we can construct a directed \emph{graph} |
| 490 | of the query results so far. We start with a node labelled $I$. When |
| 491 | processing an $H$-query, each time we compute $t' = F(t \cat x_i)$, we add |
| 492 | a node $t'$, and an edge $x_i$ from $t$ to $t'$. The `bad' event occurs |
| 493 | whenever we add an edge to a previously existing node. We must show, |
| 494 | firstly, that the adversary cannot distinguish $H$ from a random function |
| 495 | unless the bad event occurs; and, secondly, that the bad event doesn't |
| 496 | occur very often. |
| 497 | |
| 498 | The latter is easier: our standard collision bound shows that the bad |
| 499 | event occurs during the game with probability at most $q'(q' - 1)/2^{t+1}$. |
| 500 | |
| 501 | The former is trickier. This needs a lot more work to make it really |
| 502 | rigorous, but we show the idea. Assume that the bad event has not |
| 503 | occurred. Consider a query $x_0, x_1, \ldots, x_{n-1}$. If it's the same |
| 504 | as an earlier query, then $A$ learns nothing (because it could have |
| 505 | remembered the answer from last time). If it's a \emph{prefix} of some |
| 506 | earlier query, then the answer is the value of some internal node which |
| 507 | hasn't been revealed before; however, the value of that internal node was |
| 508 | chosen uniformly at random (we claim). Finally, if the query is not a |
| 509 | prefix of any previous query, then we add a new edge to our graph. If the |
| 510 | bad event doesn't occur, we must add a new node for the result, and the |
| 511 | value at that node will be uniformly random, because $F$ is a random |
| 512 | function being evaluated at a new point -- this is the only time we add new |
| 513 | nodes to the graph, justifying the claim made earlier. |
| 514 | |
| 515 | At the end of all of this, we see that |
| 516 | \[ \InSec{prf}(T^0; t, q) \le |
| 517 | \InSec{prf}(F; t, q') + \frac{q'(q' - 1)}{2^{t+1}} \]% |
| 518 | and hence |
| 519 | \[ \InSec{suf-cma}(\mathcal{M}^0; t, q) \le |
| 520 | \InSec{prf}(F; t, q') + \frac{2 q'(q' - 1) + 1}{2^{t+1}}. \]% |
| 521 | |
| 522 | Now we turn our attention to $T^1$. It's clear that we can't simulate |
| 523 | $T^1$ very easily using an oracle for $F$, since we don't know $K$ |
| 524 | (and indeed there might not be a key $K$). The intuitive reason why |
| 525 | $T^1$ is insecure is that $F$ might leak useful information if its input |
| 526 | matches its key. This doesn't affect the strength of $F$ as a PRF |
| 527 | because you have to know the key before you can exploit this leakage; but |
| 528 | $T^1$ already knows the key, and this can be exploited to break the MAC. |
| 529 | |
| 530 | To show that this is insecure formally, let $F'$ be defined as |
| 531 | follows: |
| 532 | \[ F'_K(x) = \begin{cases} |
| 533 | K & if $x = p \cat K \cat q$ where |
| 534 | $|p| = t$ and $|q| = \ell - k$ \\ |
| 535 | F_K(x) & otherwise |
| 536 | \end{cases}. \]% |
| 537 | We choose a simple injective padding scheme: if $x$ is a message then |
| 538 | we form $x' = x \cat 1 \cat 0^n$, where $0 \le n < \ell$ and $|x'|$ is |
| 539 | a multiple of $\ell$. If $T^1$ is instantiated with this PRF then it |
| 540 | is insecure as a MAC: submitting a tagging query for the empty string |
| 541 | $\emptystring$ reveals the key $K$, which can be used to construct a |
| 542 | forgery. |
| 543 | |
| 544 | To complete the proof, we must show that $F'$ is a PRF. Let $A$ be an |
| 545 | adversary attacking $F'$. We consider a series of games; for each |
| 546 | $\G{i}$, let $S_i$ be the event that $A$ returns $1$ in that game. |
| 547 | Game~$\G0$ is the standard attack game with $A$ given an oracle for a |
| 548 | random function; game~$\G1$ is the same, except that $A$ is given an |
| 549 | oracle for $F'_K$ for some $K \inr \{0, 1\}^k$. Then |
| 550 | $\Adv{prf}{F'}(A) = \Pr[S_1] - \Pr[S_0]$. Let game~$\G2$ be the same |
| 551 | as $\G1$, except that if $A$ makes any query of the form $p \cat K |
| 552 | \cat q$ with $|p| = t$ and $|q| = \ell - k$ then the game halts |
| 553 | immediately, and let $F_2$ be the event that this occurs. By |
| 554 | Lemma~\ref{lem:shoup} (slide~\pageref{lem:shoup}), then, $|{\Pr[S_2]} - |
| 555 | \Pr[S_1]| \le \Pr[F_2]$. Let game~$\G3$ be the same as $\G2$ except |
| 556 | that we give $A$ an oracle for $F_K$ rather than $F'_K$. Since $F$ |
| 557 | and $F'$ differ only on queries of the form $p \cat K \cat q$, we have |
| 558 | $\Pr[S_3] = \Pr[S_2]$. But $\Pr[S_3] - \Pr[S_0] = \Adv{prf}{F}(A) \le |
| 559 | \InSec{prf}(F; t, q)$. Hence, $\Adv{prf}{F'}(A) \le \InSec{prf}{F}(A) |
| 560 | - \Pr[F_2]$. |
| 561 | |
| 562 | Finally, we bound the probability of $F_2$. Fix an integer $n$. |
| 563 | Consider an adversary $B$ attacking $F$ which runs as follows. It |
| 564 | initially requests $F(0), F(1), \ldots, F(n - 1)$ from its oracle. It |
| 565 | then runs $A$, except that, for each oracle query $x$, it parses $x$ |
| 566 | as $p \cat K' \cat q$ with $|p| = t$, $|K'| = k$ and $|q| = \ell - k$; |
| 567 | then, if $F_{K'}(0) = F(0) \land F_{K'}(1) = F(1) \land \cdots \land |
| 568 | F_{K'}(n - 1) = F(n - 1)$, $B$ immediately returns $1$, claiming that |
| 569 | its oracle $F$ is the function $F_{K'}$; if this never occurs, $B$ |
| 570 | returns $0$. Clearly, if $B$ is given an instance $F_K$ of $F$ then |
| 571 | it succeeds with probability $\Pr[F_2]$; however, if $F$ is a random |
| 572 | function then $B$ returns $1$ with probability at most $q 2^{-nk}$. |
| 573 | Hence, $\Adv{prf}{F}(B) \le \Pr[F_2] - q 2^{-nk}$. $B$ issues $q + n$ |
| 574 | queries, and takes time $t + O(n q)$. Wrapping everything up, we get |
| 575 | \[ \InSec{prf}(F'; t, q) \le |
| 576 | 2\cdot\InSec{prf}(F; t + O(q n), q + n) + \frac{q}{2^{nk}}. \]% |
| 577 | This completes the proof of generic insecurity for $\mathcal{M}^1$. |
| 578 | \end{exercise} |
| 579 | |
| 580 | \xcalways\subsection{Universal hashing}\x |
| 581 | |
| 582 | \begin{slide} |
| 583 | \topic{almost-universal hash functions} |
| 584 | \resetseq |
| 585 | \head{Universal hashing, \seq: definition} |
| 586 | |
| 587 | Consider a family of hash functions $H\colon \keys H \times \dom H \to |
| 588 | \ran H$. We define |
| 589 | \[ \InSec{uh}(H) = |
| 590 | \max_{x \ne y} \Pr[K \getsr \keys H : H_K(x) = H_K(y)]. \]% |
| 591 | If $\InSec{uh}(H) \le \epsilon$ then we say that $H$ is |
| 592 | \emph{$\epsilon$-almost universal}. Note that the concept of |
| 593 | almost-universality is not quantified by running times. |
| 594 | |
| 595 | If $H$ is $1/|{\ran H}|$-almost universal, then we say that $H$ is |
| 596 | \emph{universal}. Sometimes it's said that this is the best possible |
| 597 | insecurity: this isn't true. |
| 598 | \end{slide} |
| 599 | |
| 600 | \begin{proof}[Counterexample] |
| 601 | Here's a function $H\colon \{0, 1, 2\} \times \{0, 1, 2, 3\} \to \{0, 1\}$ |
| 602 | which is $\frac{1}{3}$-almost universal, though $|{\ran H}| = 2$: |
| 603 | \begin{quote} \item |
| 604 | \begin{tabular}[C]{c|cccc} |
| 605 | & 0 & 1 & 2 & 3 \\ \hlx{vhv} |
| 606 | 0 & 0 & 1 & 0 & 1 \\ |
| 607 | 1 & 0 & 0 & 1 & 1 \\ |
| 608 | 2 & 0 & 1 & 1 & 0 |
| 609 | \end{tabular} |
| 610 | \end{quote} |
| 611 | \end{proof} |
| 612 | |
| 613 | \begin{slide} |
| 614 | \topic{dynamic view} |
| 615 | \head{Universal hashing, \seq: a dynamic view} |
| 616 | |
| 617 | Suppose that $H$ is $\epsilon$-almost universal. Consider this experiment: |
| 618 | \begin{program} |
| 619 | Experiment $\Expt{uh}{H}(A)$: \+ \\ |
| 620 | $(x, y) \gets A$; \\ |
| 621 | $K \getsr \keys H$; \\ |
| 622 | \IF $x \ne y \land H_K(x) = H_K(y)$ \THEN \RETURN $1$; \\ |
| 623 | \ELSE \RETURN $0$; |
| 624 | \end{program} |
| 625 | The adversary may make random decisions before outputting its selection |
| 626 | $x$, $y$. We show that $\Pr[\Expt{uh}{H}(A) = 1] \le \InSec{uh}(H) = |
| 627 | \epsilon$. |
| 628 | |
| 629 | Let $\rho \in \{0, 1\}^*$ be $A$'s coin tosses: $A$ chooses $x$ and $y$ as |
| 630 | functions of $\rho$. For some \emph{fixed} $\rho$, |
| 631 | \[ \Pr[K \getsr \keys H : H_K(x(\rho)) = H_K(y(\rho))] \le |
| 632 | \InSec{uh}(H). \]% |
| 633 | \end{slide} |
| 634 | |
| 635 | \begin{slide} |
| 636 | \head{Universal hashing, \seq: the dynamic view (cont.)} |
| 637 | |
| 638 | Now we treat $\rho$ as a random variable, selected from some distribution |
| 639 | $P$ on the set $\{0, 1\}^*$. We see that |
| 640 | \begin{eqnarray*}[Ll] |
| 641 | \Pr[\rho \getsr P; K \getsr \keys H : H_K(x(\rho)) = H_K(y(\rho))] \\ |
| 642 | &= \sum_{\rho \in \{0, 1\}^*} |
| 643 | P(\rho) \cdot \Pr[K \getsr \keys H : H_K(x(\rho)) = H_K(y(\rho))] \\ |
| 644 | &\le \sum_{\rho \in \{0, 1\}^*} P(\rho) \cdot \InSec{uh}(H) |
| 645 | = \InSec{uh}(H). |
| 646 | \end{eqnarray*} |
| 647 | Thus, no adversary can succeed in producing a collision in an |
| 648 | $\epsilon$-almost universal hash with probability better than $\epsilon$. |
| 649 | But obviously the adversary can ignore its coin tosses and simply return |
| 650 | the best colliding pair. Hence the two notions are completely equivalent. |
| 651 | \end{slide} |
| 652 | |
| 653 | \begin{slide} |
| 654 | \topic{composition} |
| 655 | \head{Universal hashing, \seq: composition} |
| 656 | |
| 657 | Suppose that $G$ is $\epsilon$-almost universal, and $G'$ is |
| 658 | $\epsilon'$-almost universal, and $\dom G = \ran G'$. We define the |
| 659 | composition $G \compose G'$ to be the family $H\colon (\keys G \times |
| 660 | \keys G') \times \dom G' \to \ran G$ by $H_{k, k'}(m) = |
| 661 | G_k(G'_{k'}(m))$. |
| 662 | |
| 663 | Then $H$ is $(\epsilon + \epsilon')$-almost universal. To see this, fix $x |
| 664 | \ne y$, and choose $K = (k, k') \inr \keys G \times \keys G'$. Let $x' = |
| 665 | G'_{k'}(x)$ and $y' = G'_{k'}(y)$. Following our previous result, we see: |
| 666 | \begin{eqnarray*}[rl] |
| 667 | \Pr[H_K(x) = H_K(y)] |
| 668 | &= \Pr[G_k(G'_{k'}(x)) = G_k(G'_{k'}(y))] \\ |
| 669 | &= \Pr[G_k(x') = G_k(y')] \\ |
| 670 | &= \Pr[G_k(x') = G_k(y') \mid x' \ne y'] \Pr[G'_{k'}(x) \ne G'_{k'}(y)]\\ |
| 671 | &\le \epsilon + \epsilon'. |
| 672 | \end{eqnarray*} |
| 673 | \end{slide} |
| 674 | |
| 675 | \begin{slide} |
| 676 | \topic{the collision game} |
| 677 | \head{Universal hashing, \seq: the collision game} |
| 678 | |
| 679 | Suppose that, instead of merely a pair $(x, y)$, our adversary was allowed |
| 680 | to return a \emph{set} $Y$ of $q$ elements, and measure the probability |
| 681 | that $H_K(x) = H_K(y)$ for some $x \ne y$ with $x, y \in Y$, and for $K |
| 682 | \inr \keys H$. |
| 683 | |
| 684 | Let $\InSec{uh-set}(H; q)$ be maximum probability achievable for sets $Y$ |
| 685 | with $|Y| \le q$. Then |
| 686 | \[ \InSec{uh-set}(H; q) \le \frac{q(q - 1)}{2} \cdot \InSec{uh}(H) .\] |
| 687 | \end{slide} |
| 688 | |
| 689 | \begin{proof} |
| 690 | This is rather tedious. We use the dynamic view. Suppose $A$ returns $(x, |
| 691 | Y)$ with $|Y| = q$, and succeeds with probability $\epsilon$. Consider |
| 692 | \begin{program} |
| 693 | Adversary $A'$: \+ \\ |
| 694 | $(x, Y) \gets A$; \\ |
| 695 | $y \getsr Y$; \\ |
| 696 | \RETURN $(x, Y \setminus \{y\})$; |
| 697 | \end{program} |
| 698 | The worst that can happen is that $A'$ accidentally removes the one |
| 699 | colliding element from $Y$. This occurs with probability $2/q$. So |
| 700 | \[ \Succ{uh-set}{H}(A') \ge \frac{q - 2}{q} \Succ{uh-set}{H}(A). \] |
| 701 | Rearranging and maximizing gives |
| 702 | \[ \InSec{uh-set}(H; q) \le |
| 703 | \frac{q}{q - 2} \cdot \InSec{uh-set}(H; q - 1). \] |
| 704 | Note that $\InSec{uh-set}(H; 2) = \InSec{uh}(H)$ is our initial notion. A |
| 705 | simple inductive argument completes the proof. |
| 706 | \end{proof} |
| 707 | |
| 708 | \begin{slide} |
| 709 | \topic{a MAC} |
| 710 | \head{Universal hashing, \seq: a MAC} |
| 711 | |
| 712 | Suppose that $H\colon \{0, 1\}^k \times \{0, 1\}^* \to \{0, 1\}^l$ is an |
| 713 | almost universal hash function, and $F\colon \{0, 1\}^{k'} \times \{0, |
| 714 | 1\}^l \to \{0, 1\}^L$ is a PRF\@. Define a MAC $\Xid{\mathcal{M}}{UH}^{H, |
| 715 | F} = (\Xid{T}{UH}^{H, F}, \Xid{V}{UH}^{H, F})$ where: |
| 716 | \begin{eqnarray*}[rl] |
| 717 | \Xid{T}{UH}^{H, F}_{K, K'}(m) &= F_{K'}(H_K(m)) \\ |
| 718 | \Xid{V}{UH}^{H, F}_{K, K'}(m, \tau) &= \begin{cases} |
| 719 | 1 & if $\tau = F_{K'}(H_K(m))$ \\ |
| 720 | 0 & otherwise |
| 721 | \end{cases}. |
| 722 | \end{eqnarray*} |
| 723 | We have |
| 724 | \begin{eqnarray*}[Ll] |
| 725 | \InSec{suf-cma}(\Xid{\mathcal{M}}{UH}^{H, F}; t, q_T, q_V) \\ |
| 726 | & \le |
| 727 | (q_V + 1) \biggl(\InSec{prf}(F; t, q_T + 1) + \frac{1}{2^L} + |
| 728 | \frac{q_T(q_T - 1)}{2} \cdot \InSec{uh}(H)\biggr). |
| 729 | \end{eqnarray*} |
| 730 | \end{slide} |
| 731 | |
| 732 | \begin{proof} |
| 733 | We shall prove the result for $q_V = 0$ and $q_T = q$, and appeal to the |
| 734 | earlier result on verification oracles. |
| 735 | |
| 736 | Suppose $A$ attacks the scheme $\Xid{\mathcal{M}}{UH}^{H, F}$ in time $t$, |
| 737 | issuing $q$ tagging queries. Consider a distinguisher $D$, constructed |
| 738 | from a forger $A$: |
| 739 | \begin{program} |
| 740 | Distinguisher $D^{F(\cdot)}$: \+ \\ |
| 741 | $K \getsr \{0, 1\}^k$; \\ |
| 742 | $\Xid{T}{list} \gets \emptyset$; \\ |
| 743 | $(m, \tau) \gets A^{\id{tag}(K, \cdot)}$; \\ |
| 744 | \IF $m \notin \Xid{T}{list} \land \tau = F(H_K(m))$ |
| 745 | \THEN \RETURN $1$; \\ |
| 746 | \ELSE \RETURN $0$; \- \\[\smallskipamount] |
| 747 | Oracle $\id{tag}(K, m)$: \+ \\ |
| 748 | $\Xid{T}{list} \gets \Xid{T}{list} \cup \{m\}$; \\ |
| 749 | \RETURN $F(H_K(m))$; \- \\[\smallskipamount] |
| 750 | \end{program} |
| 751 | Note that $A$ isn't provided with a verification oracle: that's because we |
| 752 | didn't allow it any verification queries. |
| 753 | |
| 754 | We can see immediately that |
| 755 | \[ \Pr[K \getsr \{0, 1\}^{k'} : D^{F_K(\cdot)} = 1] = |
| 756 | \Succ{suf-cma}{\Xid{\mathcal{M}}{UH}^{H, F}}(A). \]% |
| 757 | |
| 758 | We must now find an upper bound for $\Pr[F \getsr \Func{l}{L} : |
| 759 | D^{F(\cdot)}]$. Suppose that the adversary returns the pair $(m^*, |
| 760 | \tau^*)$, and that its tagging oracle queries and their answers are $(m_i, |
| 761 | \tau_i)$ for $0 \le i < q$. Consider the event $C$ that $H_K(m) = |
| 762 | H_K(m')$ for some $m \ne m'$, with $m, m' \in \{m^*\} \cup \{\,m_i \mid 0 |
| 763 | \le i < q\,\}$. |
| 764 | |
| 765 | If $C$ doesn't occur, then $F$ has not been queried before at $H_K(m)$, but |
| 766 | there's a $2^{-L}$ probability that the adversary guesses right anyway. If |
| 767 | $C$ does occur, then we just assume that the adversary wins, even though it |
| 768 | might not have guessed the right tag. |
| 769 | |
| 770 | By our result on the collision game, $\Pr[C] \le q \cdot \InSec{uh}(H)$. |
| 771 | Then |
| 772 | \[ \Succ{prf}{F}(D) \ge |
| 773 | \Succ{suf-cma}{\Xid{\mathcal{M}}{UH}^{H, F}}(A) - |
| 774 | \frac{1}{2^L} - \frac{q(q - 1)}{2} \cdot \InSec{uh}(H). \]% |
| 775 | The result follows. |
| 776 | \end{proof} |
| 777 | |
| 778 | \begin{slide} |
| 779 | \topic{almost XOR-universality} |
| 780 | \resetseq |
| 781 | \head{Almost XOR-universality, \seq: definition} |
| 782 | |
| 783 | Consider a family of hash functions $H\colon \keys H \times \dom H \to |
| 784 | \{0, 1\}^L$. Define |
| 785 | \[ \InSec{xuh}(H) = |
| 786 | \max_{x \ne y, \delta} |
| 787 | \Pr[K \getsr \keys H : H_K(x) \xor H_K(y) = \delta]. \]% |
| 788 | If $\InSec{xuh}(H) < \epsilon$ then we say that $H$ is |
| 789 | \emph{$\epsilon$-almost XOR-universal}, or \emph{AXU}. Setting $\delta = |
| 790 | 0$ shows that |
| 791 | \begin{eqnarray*}[rl] |
| 792 | \InSec{xuh}(H) |
| 793 | & \ge \max_{x \ne y} \Pr[K \getsr \keys H : H_K(x) \xor H_K(y) = 0] \\ |
| 794 | & = \InSec{uh}(H). |
| 795 | \end{eqnarray*} |
| 796 | |
| 797 | We can take a dynamic view of almost XOR-universality using the same |
| 798 | technique as for almost universal functions. |
| 799 | |
| 800 | If $H$ is $2^{-L}$-almost XOR universal then we say that $H$ is |
| 801 | \emph{XOR-universal}. This is the best achievable. |
| 802 | \end{slide} |
| 803 | |
| 804 | \begin{proof} |
| 805 | Fix some pair $x \ne y$. Choose $\delta \inr \{0, 1\}^L$. Then for any |
| 806 | \emph{fixed} $K \in \keys H$, $H_K(x)$ and $H_K(y)$ are fixed, so $H_K(x) |
| 807 | \xor H_K(y)$ is fixed, and $\Pr[H_K(x) \xor H_K(y) = \delta] = 2^{-L}$. |
| 808 | Using the same trick as for proving the dynamic view: |
| 809 | \begin{eqnarray*}[Ll] |
| 810 | \Pr[\delta \getsr \{0, 1\}^L; K \getsr \keys H : |
| 811 | H_K(x) \xor H_K(y) = \delta] \\ |
| 812 | & = \sum_{K \in \keys H} \frac{1}{|{\keys H}|} |
| 813 | \Pr[\delta \getsr \{0, 1\}^L; K \getsr \keys H : |
| 814 | H_K(x) \xor H_K(y) = \delta] \\ |
| 815 | & = \sum_{K \in \keys H} \frac{1}{|{\keys H}|} 2^{-L} = 2^{-L}. |
| 816 | \end{eqnarray*} |
| 817 | Since $H$ is arbitrary, this proves the lower bound on the almost |
| 818 | XOR-universality. |
| 819 | \end{proof} |
| 820 | |
| 821 | \begin{slide} |
| 822 | \topic{composition} |
| 823 | \head{Almost XOR-universality, \seq: composition} |
| 824 | |
| 825 | We extend our result about composition of almost-universal functions. |
| 826 | Suppose that $G$ is $\epsilon$-almost XOR universal, and $G'$ is |
| 827 | $\epsilon'$-almost universal (it doesn't have to be almost XOR-universal), |
| 828 | and $\dom G = \ran G'$. |
| 829 | |
| 830 | Then the composition $H = G \compose G'$ is $(\epsilon + \epsilon')$-almost |
| 831 | XOR-universal. The proof is simple, and very similar to the |
| 832 | almost-universal case. |
| 833 | \end{slide} |
| 834 | |
| 835 | \begin{slide} |
| 836 | \topic{a better MAC} |
| 837 | \head{Almost XOR-universality, \seq: a better MAC} |
| 838 | |
| 839 | The security result for the UH-based MAC contains a factor $q_T$, which |
| 840 | it'd be nice to remove. Our new scheme uses an AXU hash $H\colon \keys H |
| 841 | \times \{0, 1\}^* \to \{0, 1\}^l$ and a PRF $F\colon \keys F \times \{0, |
| 842 | 1\}^l \to \{0, 1\}^L$. |
| 843 | |
| 844 | We first present a stateful version $\Xid{\mathcal{M}}{XUH}^{H, F}$. |
| 845 | Choose $(K, K') \inr \keys H \times \keys F$, and initialize a counter $i |
| 846 | \gets 0$. The tagging and verification algorithms are then: |
| 847 | \begin{program} |
| 848 | Algorithm $\Xid{T}{XUH}^{H, F}_{K, K'}(m)$: \+ \\ |
| 849 | $\tau \gets (i, H_K(m) \xor F_{K'}(i))$; \\ |
| 850 | $i \gets i + 1$; \\ |
| 851 | \RETURN $\tau$; |
| 852 | \next |
| 853 | Algorithm $\Xid{V}{XUH}^{H, F}_{K, K'}(m, \tau)$: \+ \\ |
| 854 | $(s, \sigma) \gets \tau$; \\ |
| 855 | \IF $\sigma = H_K(m) \xor F_{K'}(s)$ \THEN \RETURN $1$; \\ |
| 856 | \ELSE \RETURN $0$; |
| 857 | \end{program} |
| 858 | Note that verification is stateless. |
| 859 | \end{slide} |
| 860 | |
| 861 | \begin{slide} |
| 862 | \head{Almost XOR-universality, \seq: security of AXU-based MACs} |
| 863 | |
| 864 | For the stateful scheme presented earlier, provided $q_T \le 2^l$, we have |
| 865 | \begin{eqnarray*}[Ll] |
| 866 | \InSec{suf-cma}(\Xid{\mathcal{M}}{XUH}^{H, F}; t, q_T, q_V) \\ |
| 867 | & \le (q_V + 1)(\InSec{prf}(F; t, q_T + 1) + \InSec{xuh}(H) + 2^{-L}). |
| 868 | \end{eqnarray*} |
| 869 | \end{slide} |
| 870 | |
| 871 | \begin{slide} |
| 872 | \head{Almost XOR-universality, \seq: randomized AXU-based MAC} |
| 873 | |
| 874 | We can avoid statefulness by using randomization. This new scheme is |
| 875 | $\Xid{\mathcal{M}}{XUH$\$$}^{H, F} = (\Xid{T}{XUH$\$$}^{H, F}, |
| 876 | \Xid{V}{XUH$\$$}^{H, F})$: |
| 877 | \begin{program} |
| 878 | Algorithm $\Xid{T}{XUH$\$$}^{H, F}_{K, K'}(m)$: \+ \\ |
| 879 | $s \getsr \{0, 1\}^l$; \\ |
| 880 | $\tau \gets (s, H_K(m) \xor F_{K'}(s))$; \\ |
| 881 | \RETURN $\tau$; |
| 882 | \next |
| 883 | Algorithm $\Xid{V}{XUH$\$$}^{H, F}_{K, K'}(m, \tau)$: \+ \\ |
| 884 | $(s, \sigma) \gets \tau$; \\ |
| 885 | \IF $\sigma = H_K(m) \xor F_{K'}(s)$ \THEN \RETURN $1$; \\ |
| 886 | \ELSE \RETURN $0$; |
| 887 | \end{program} |
| 888 | \begin{eqnarray*}[Ll] |
| 889 | \InSec{suf-cma}(\Xid{\mathcal{M}}{XUH$\$$}^{H, F}; t, q_T, q_V) \\ |
| 890 | & \le (q_V + 1) |
| 891 | \Bigl(\InSec{prf}(F; t, q_T + 1) + \InSec{xuh}(H) + 2^{-L} + |
| 892 | \frac{q_T(q_T - 1)}{2^{l+1}}\Bigr). |
| 893 | \end{eqnarray*} |
| 894 | \end{slide} |
| 895 | |
| 896 | \begin{proof} |
| 897 | We prove the result with $q_V = 0$ and $q_T = q$, and appeal to the result |
| 898 | on verification oracles. Let $m_i$ be the message specified in the $i$-th |
| 899 | tagging query ($0 \le i < q$), and let $(s_i, \sigma_i) = (s_i, H_K(m) \xor |
| 900 | F_{K'}(s_i))$ be the tag returned. We call the $s_i$ the \emph{nonce}. |
| 901 | |
| 902 | We prove the result for the stateless scheme. The bound $q \le 2^l$ |
| 903 | ensures that the nonces are all distinct (we have $s_i = i$). The security |
| 904 | bound for the randomized version merely has as an extra term upper bound |
| 905 | for the probability of a nonce collision. |
| 906 | |
| 907 | Let $A$ be an adversary attacking the MAC in time $t$, and using $q$ |
| 908 | tagging queries. Then we present the following pair of adversaries: |
| 909 | \begin{program} |
| 910 | Distinguisher $D^{F(\cdot)}$: \+ \\ |
| 911 | $i \gets 0$; \\ |
| 912 | $\Xid{T}{list} \gets \emptyset$; \\ |
| 913 | $K \getsr \keys H$; \\ |
| 914 | $(m, \tau) \gets A^{\id{tag}}$; \\ |
| 915 | $(s, \sigma) \gets \tau$; \\ |
| 916 | \IF $(m, \tau) \notin \Xid{T}{list} \land |
| 917 | \sigma = H_K(m) \xor F(s)$ |
| 918 | \THEN \RETURN $1$; \\ |
| 919 | \ELSE \RETURN $0$; \- \\[\smallskipamount] |
| 920 | Oracle $\id{tag}(m)$: \+ \\ |
| 921 | $\tau \gets (i, H_K(m) \xor F(i))$; \\ |
| 922 | $\Xid{T}{list} \gets \Xid{T}{list} \cup \{(m, \tau)\}$; \\ |
| 923 | $i \gets i + 1$; |
| 924 | \RETURN $\tau$; |
| 925 | \next |
| 926 | Collision-finder $C$: \+ \\ |
| 927 | $i \gets 0$; \\ |
| 928 | $(m, \tau) \gets A^{\id{tag}}$; \\ |
| 929 | $(s, \sigma) \gets \tau$; \\ |
| 930 | \IF $s \ge i \lor m = m_s$ \THEN \ABORT; \\ |
| 931 | \RETURN $(m, m_s, \sigma \xor \sigma_s)$; \- \\[\smallskipamount] |
| 932 | Oracle $\id{tag}(m)$: \+ \\ |
| 933 | $m_i \gets m$; \\ |
| 934 | $\sigma_i \getsr \{0, 1\}^L$; \\ |
| 935 | $\tau \gets (i, \sigma_i)$; \\ |
| 936 | $i \gets i + 1$; \\ |
| 937 | \RETURN $\tau$; |
| 938 | \end{program} |
| 939 | |
| 940 | We need to find a lower bound on the advantage of $D$. If $F$ is chosen |
| 941 | from the PRF then $D$ returns $1$ precisely when $A$ finds a valid |
| 942 | forgery. We now examine the setting in which $F$ is a random function. |
| 943 | |
| 944 | Let $S$ be the event that $A$ succeeds in returning a valid forgery when |
| 945 | $F$ is random, and let $N$ be the event that the nonce $s$ returned by $A$ |
| 946 | is not equal to any nonce $s_i$ returned by the tagging oracle. Suppose |
| 947 | $N$ occurs: then the random function $F$ has never been queried before at |
| 948 | $F$, and $\Pr[F(s) = \sigma \xor H_K(m)]$ is precisely $2^{-L}$. |
| 949 | |
| 950 | So suppose instead that $N$ doesn't occur. Then, since the $s_i$ are |
| 951 | distinct, there is a unique $i$ such that $s = s_i$. For $A$ to win, we |
| 952 | must have $m \ne m_i$ (for if $m = m_i$ then the only valid tag is |
| 953 | $H_K(m_i) \xor F(s_i) = \sigma_i$, which was already returned by the |
| 954 | tagging oracle). If $A$'s forgery is successful then |
| 955 | \[ \sigma = H_K(m) \xor F(s) |
| 956 | \qquad \text{and} \qquad |
| 957 | \sigma_i = H_K(m_i) \xor F(s_i) \]% |
| 958 | but $s = s_i$, whence |
| 959 | \[ H_K(m_i) \xor H_K(m) = \sigma \xor \sigma_i. \] |
| 960 | Since the $s_i$ are distinct and $F$ is a random function, the $\sigma_i$ |
| 961 | are independent uniformly-distributed random strings from $\{0, 1\}^L$. |
| 962 | Hence the collision-finder $C$ succeeds with probability $\Pr[S \land |
| 963 | \lnot N] \le \InSec{xuh}(H)$. |
| 964 | |
| 965 | Wrapping up, we have |
| 966 | \begin{eqnarray*}[rl] |
| 967 | \Adv{prf}{F}(D) |
| 968 | & \ge \Succ{suf-cma}{\Xid{\mathcal{M}}{XUH}^{H, F}}(A) - |
| 969 | (\Pr[S \mid N] \Pr[N] + \Pr[S \mid \lnot N] \Pr[\lnot N]) \\ |
| 970 | & \ge \Succ{suf-cma}{\Xid{\mathcal{M}}{XUH}^{H, F}}(A) - |
| 971 | (2^{-L} + \InSec{xuh}(H)). |
| 972 | \end{eqnarray*} |
| 973 | Maximizing and rearranging yields the required result. |
| 974 | \end{proof} |
| 975 | |
| 976 | \begin{remark*} |
| 977 | Note that our bound has a $2^{-L}$ term in it that's missing from |
| 978 | \cite{Goldwasser:1999:LNC}. We believe that their proof is wrong in its |
| 979 | handling of the XOR-collision probability. |
| 980 | \end{remark*} |
| 981 | |
| 982 | \endinput |
| 983 | |
| 984 | %%% Local Variables: |
| 985 | %%% mode: latex |
| 986 | %%% TeX-master: "ips" |
| 987 | %%% End: |