From: mdw <mdw>
Date: Wed, 30 Jan 2002 09:57:06 +0000 (+0000)
Subject: Miscellaneous rewordings and fixings.
X-Git-Tag: 1.1.0~9
X-Git-Url: https://git.distorted.org.uk/~mdw/doc/ips/commitdiff_plain/9907b634a7bc6c9db6381a578e192178a28e9799

Miscellaneous rewordings and fixings.
---

diff --git a/auth-mac.tex b/auth-mac.tex
index cd3defa..60e0034 100644
--- a/auth-mac.tex
+++ b/auth-mac.tex
@@ -384,8 +384,8 @@
   for any adversary $A$ constrained to run in time $t$ and permitted $q$
   oracle queries,
   \[ \Pr[K \getsr \{0, 1\}^k;
-         (x, y) \gets A^{F_K(\cdot)}] \le \epsilon :
-         x \ne y \land F_K(x) = F_K(y) \]%
+         (x, y) \gets A^{F_K(\cdot)} :
+         x \ne y \land F_K(x) = F_K(y)] \le \epsilon \]%
   
   If $H_K$ is a $(t, q_T, q_V, \epsilon)$-secure MAC on $k$-bit messages, and
   moreover $(t, q_T + q_V, \epsilon')$-weakly collision resistant, then
@@ -397,8 +397,8 @@
 
   Let $A$ be an adversary which forges a $\Xid{T}{NMAC}^H$ tag in time $t$,
   using $q_T$ tagging queries and $q_V$ verification queries with probability
-  $\epsilon$.  We construct an adversary $A'$ which forces a $H$ tag for a
-  $k$-bit in essentially the same time.
+  $\epsilon$.  We construct an adversary $A'$ which forges an $H$-tag for a
+  $k$-bit message in essentially the same time.
   \begin{program}
     Adversary $A'^{T(\cdot), V(\cdot, \cdot)}$ \+ \\
       $K \getsr \{0, 1\}^k$; \\
@@ -485,38 +485,39 @@
   function $F$.  For the sake of simplicity, we allow the adversary $A$
   to query on \emph{padded} messages, rather than the raw unpadded
   messages.  We count the number $q'$ of individual message blocks.
-
-  As the game with $A$ progresses, we can construct a directed
-  \emph{graph} of the query results so far.  We start with a node
-  labelled $I$.  When processing an $H$-query, each time we compute $t'
-  = F(t \cat x_i)$, we add a node $t'$, and an edge $x_i$ from $t$ to
-  $t'$.  The `bad' event occurs whenever we add an edge to a previously
-  existing node.  We must show, firstly, that the
-  adversary cannot distinguish $H$ from a random function unless the bad
-  event occurs; and, secondly, that the bad event doesn't occur very
-  often.
+  
+  As the game with $A$ progresses, we can construct a directed \emph{graph}
+  of the query results so far.  We start with a node labelled $I$.  When
+  processing an $H$-query, each time we compute $t' = F(t \cat x_i)$, we add
+  a node $t'$, and an edge $x_i$ from $t$ to $t'$.  The `bad' event occurs
+  whenever we add an edge to a previously existing node.  We must show,
+  firstly, that the adversary cannot distinguish $H$ from a random function
+  unless the bad event occurs; and, secondly, that the bad event doesn't
+  occur very often.
 
   The latter is easier: our standard collision bound shows that the bad
-  event occurs during the game with probability at most $q'(q' - 1)/2$.
-
+  event occurs during the game with probability at most $q'(q' - 1)/2^{t+1}$.
+  
   The former is trickier.  This needs a lot more work to make it really
   rigorous, but we show the idea.  Assume that the bad event has not
-  occurred. Consider a query $x_0, x_1, \ldots, x_{n-1}$.  If it's the
-  same as an earlier query, then $A$ learns nothing (because it could
-  have remembered the answer from last time).  If it's \emph{not} a
-  prefix of some previous query, then we must add a new edge to our
-  graph; then either the bad event occurs or we create a new node for
-  the result, and because $F$ is a random function, the answer is
-  uniformly random.  Finally, we consider the case where the query is a
-  prefix of some earlier query, or queries.  But these were computed at
-  random at the time.
+  occurred. Consider a query $x_0, x_1, \ldots, x_{n-1}$.  If it's the same
+  as an earlier query, then $A$ learns nothing (because it could have
+  remembered the answer from last time).  If it's a \emph{prefix} of some
+  earlier query, then the answer is the value of some internal node which
+  hasn't been revealed before; however, the value of that internal node was
+  chosen uniformly at random (we claim).  Finally, if the query is not a
+  prefix of any previous query, then we add a new edge to our graph.  If the
+  bad event doesn't occur, we must add a new node for the result, and the
+  value at that node will be uniformly random, because $F$ is a random
+  function being evaluated at a new point -- this is the only time we add new
+  nodes to the graph, justifying the claim made earlier.
 
   At the end of all of this, we see that
   \[ \InSec{prf}(T^0; t, q) \le
-       \InSec{prf}(F; t, q') + \frac{q'(q' - 1)}{2} \]%
+       \InSec{prf}(F; t, q') + \frac{q'(q' - 1)}{2^{t+1}} \]%
   and hence
   \[ \InSec{suf-cma}(\mathcal{M}^0; t, q) \le 
-       \InSec{prf}(F; t, q') + \frac{q'(q' - 1)}{2} + \frac{1}{2^t}. \]%
+       \InSec{prf}(F; t, q') + \frac{2 q'(q' - 1) + 1}{2^{t+1}}. \]%
 
   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 $K$
diff --git a/basics.tex b/basics.tex
index 9c7401f..3874d3c 100644
--- a/basics.tex
+++ b/basics.tex
@@ -780,8 +780,9 @@ But if your cryptography is no good, you may never know.
   
   Let $A$ be an adversary which distinguishes~$F$ from a pseudorandom
   function in time~$t$, after making $q$ oracle queries.  We consider a
-  sequence of games $\G{i}$ played with the adversary.  In each, let $S_i$ be
-  the event that the adversary returns~$1$ at the end of the game.
+  sequence of games $\G{i}$ played with the adversary.  In each game $\G{i}$,
+  let $S_i$ be the event that the adversary returns~$1$ at the end of the
+  game.
 
   Game~$\G0$ is the `random function' game.  We given $A$ an oracle
   containing a random function $R \inr \Func{L}{L}$.