X-Git-Url: https://git.distorted.org.uk/~mdw/doc/ips/blobdiff_plain/9907b634a7bc6c9db6381a578e192178a28e9799..57ea5481e3ef900bd2a23df2f117a3b5f382b885:/auth-mac.tex?ds=sidebyside

diff --git a/auth-mac.tex b/auth-mac.tex
index 60e0034..e9c9684 100644
--- a/auth-mac.tex
+++ b/auth-mac.tex
@@ -522,11 +522,10 @@
   Now we turn our attention to $T^1$.  It's clear that we can't simulate
   $T^1$ very easily using an oracle for $F$, since we don't know $K$
   (and indeed there might not be a key $K$).  The intuitive reason why
-  $T^1$ is insecure is that $F$ might have leak useful information if
-  its input matches its key.  This doesn't affect the strength of $F$ as
-  a PRF because you have to know the key before you can exploit this
-  leakage; but $T^1$ already knows the key, and this can be exploited to
-  break the MAC.
+  $T^1$ is insecure is that $F$ might leak useful information if  its input
+  matches its key.  This doesn't affect the strength of $F$ as a PRF
+  because you have to know the key before you can exploit this leakage; but
+  $T^1$ already knows the key, and this can be exploited to break the MAC.
 
   To show that this is insecure formally, let $F'$ be defined as
   follows:
@@ -853,7 +852,7 @@
   \next
     Algorithm $\Xid{V}{XUH}^{H, F}_{K, K'}(m, \tau)$: \+ \\
       $(s, \sigma) \gets \tau$; \\
-      \IF $\sigma = H_K(m) \xor F_{K'}(i)$ \THEN \RETURN $1$; \\
+      \IF $\sigma = H_K(m) \xor F_{K'}(s)$ \THEN \RETURN $1$; \\
       \ELSE \RETURN $0$;
   \end{program}
   Note that verification is stateless.
@@ -865,7 +864,7 @@
   For the stateful scheme presented earlier, provided $q_T \le 2^l$, we have
   \begin{eqnarray*}[Ll]
      \InSec{suf-cma}(\Xid{\mathcal{M}}{XUH}^{H, F}; t, q_T, q_V) \\
-     & \le (q_V + 1)(\InSec{prf}(F; t, q_T + 1) + \InSec{xuh}(H) + 2^{-L}).
+     & \le (q_V + 1)(\InSec{prf}(F; t, q_T + 1) + \InSec{xuh}(H)).
   \end{eqnarray*}
 \end{slide}
 
@@ -883,13 +882,13 @@
   \next
     Algorithm $\Xid{V}{XUH$\$$}^{H, F}_{K, K'}(m, \tau)$: \+ \\
       $(s, \sigma) \gets \tau$; \\
-      \IF $\sigma = H_K(m) \xor F_{K'}(i)$ \THEN \RETURN $1$; \\
+      \IF $\sigma = H_K(m) \xor F_{K'}(s)$ \THEN \RETURN $1$; \\
       \ELSE \RETURN $0$;
   \end{program}
   \begin{eqnarray*}[Ll]
      \InSec{suf-cma}(\Xid{\mathcal{M}}{XUH$\$$}^{H, F}; t, q_T, q_V) \\
      & \le (q_V + 1)
-         \Bigl(\InSec{prf}(F; t, q_T + 1) + \InSec{xuh}(H) + 2^{-L} +
+         \Bigl(\InSec{prf}(F; t, q_T + 1) + \InSec{xuh}(H) +
                \frac{q_T(q_T - 1)}{2^{l+1}}\Bigr).
   \end{eqnarray*}
 \end{slide}
@@ -946,7 +945,8 @@
   $F$ is random, and let $N$ be the event that the nonce $s$ returned by $A$
   is not equal to any nonce $s_i$ returned by the tagging oracle.  Suppose
   $N$ occurs: then the random function $F$ has never been queried before at
-  $F$, and $\Pr[F(s) = \sigma \xor H_K(m)]$ is precisely $2^{-L}$.
+  $F$, and $\Pr[S \mid N] = \Pr[F(s) = \sigma \xor H_K(m)]$ is precisely
+  $2^{-L}$.
   
   So suppose instead that $N$ doesn't occur.  Then, since the $s_i$ are
   distinct, there is a unique $i$ such that $s = s_i$.  For $A$ to win, we
@@ -960,26 +960,22 @@
   \[ H_K(m_i) \xor H_K(m) = \sigma \xor \sigma_i. \]
   Since the $s_i$ are distinct and $F$ is a random function, the $\sigma_i$
   are independent uniformly-distributed random strings from $\{0, 1\}^L$.
-  Hence the collision-finder $C$ succeeds with probability $\Pr[S \land
-  \lnot N] \le \InSec{xuh}(H)$.
+  Hence the collision-finder $C$ succeeds with probability $\Pr[S \mid
+  \bar{N}] \le \InSec{xuh}(H)$.
 
   Wrapping up, we have
   \begin{eqnarray*}[rl]
     \Adv{prf}{F}(D)
     & \ge \Succ{suf-cma}{\Xid{\mathcal{M}}{XUH}^{H, F}}(A) -
-      (\Pr[S \mid N] \Pr[N] + \Pr[S \mid \lnot N] \Pr[\lnot N]) \\
+      (\Pr[S \mid N] \Pr[N] + \Pr[S \mid \bar{N}] \Pr[\bar{N}]) \\
     & \ge \Succ{suf-cma}{\Xid{\mathcal{M}}{XUH}^{H, F}}(A) -
-      (2^{-L} + \InSec{xuh}(H)).
+      (2^{-L} \Pr[N] + \InSec{xuh}(H) \Pr[\bar{N}]) \\
+    & \ge \Succ{suf-cma}{\Xid{\mathcal{M}}{XUH}^{H, F}}(A) -
+      \InSec{xuh}(H).
   \end{eqnarray*}
   Maximizing and rearranging yields the required result.
 \end{proof}
 
-\begin{remark*}
-  Note that our bound has a $2^{-L}$ term in it that's missing from
-  \cite{Goldwasser:1999:LNC}.  We believe that their proof is wrong in its
-  handling of the XOR-collision probability.
-\end{remark*}
-
 \endinput
 
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