X-Git-Url: https://git.distorted.org.uk/~mdw/doc/ips/blobdiff_plain/76f457cbe78101034f0254a9ea940ca65cee1535..53aa10b5d9431ced4e98673cc872c6627cac1d5f:/enc-pub.tex

diff --git a/enc-pub.tex b/enc-pub.tex
index 985690c..2da1c82 100644
--- a/enc-pub.tex
+++ b/enc-pub.tex
@@ -947,7 +947,7 @@
 \begin{slide}
   \head{\PKCS1 encryption padding for RSA (cont.)}
 
-  Diagramatically, \PKCS1 encryption padding looks like this:
+  Diagrammatically, \PKCS1 encryption padding looks like this:
   \begin{tabular}[C]{r|c|c|c|c|} \hlx{c{2-5}v}
     \hex{00} & \hex{01} &
     \ldots\ random nonzero bytes \ldots &
@@ -1029,7 +1029,7 @@
 \end{slide}
 
 \begin{slide}
-  \head{Security of OAEP, 1: chosen plaintext attacks}
+  \head{Security of OAEP, \seq: chosen plaintext attacks}
   
   We write the encryption scheme formed by using the trapdoor one-way
   function $\mathcal{T}$ with OAEP embedding as
@@ -1151,7 +1151,7 @@ for OAEP+ which is presented in full later.
   definition is used to feed the challenge ciphertext to the plaintext
   extractor.
 
-  Quantatively, if $\mathcal{E}$ is $\epsilon$-plaintext aware, and the
+  Quantitatively, if $\mathcal{E}$ is $\epsilon$-plaintext aware, and the
   plaintext extractor runs in time $t_X(q_H)$, then
   \[ \InSec{ind-cca2}(\mathcal{E}; t, q_D, q_H) \le
      \InSec{ind-cpa}(\mathcal{E}; t + q_D t_X(q_H), q_H) + q_D \epsilon. \]%
@@ -1215,7 +1215,7 @@ for OAEP+ which is presented in full later.
       \RETURN $x$;
   \end{program}
   Firstly, we show that it still meets IND-CCA2.  Let $A'$ be an adversary
-  attacking $\mathcal{E}'$ in the IND-CCA2 sense.  Then $A$, below, acheives
+  attacking $\mathcal{E}'$ in the IND-CCA2 sense.  Then $A$, below, achieves
   the same advantage against $\mathcal{E}$.
   \begin{program}
     Adversary $A^{D(\cdot), H(\cdot)}(\cookie{find}, P)$: \+ \\
@@ -1275,9 +1275,10 @@ for OAEP+ which is presented in full later.
 
 \begin{slide}
   \topic{OAEP+}
-  \head{Plaintext-aware encryption: OAEP+ \cite{Shoup:2001:OAEPR}, 1}
+  \resetseq
+  \head{Plaintext-aware encryption: OAEP+ \cite{Shoup:2001:OAEPR}, \seq}
   
-  OAEP+ is a simple embedding scheme, very similar to OAEP, which acheives
+  OAEP+ is a simple embedding scheme, very similar to OAEP, which achieves
   plaintext-awareness.
   
   We assume that our one-way function $f_P$ operates on $n$-bit strings.  Fix
@@ -1307,7 +1308,7 @@ for OAEP+ which is presented in full later.
 \end{slide}
 
 \begin{slide}
-  \head{Plaintext-aware encryption: OAEP+, 2}
+  \head{Plaintext-aware encryption: OAEP+, \seq}
 
   %%        x <- {0, 1}^{n-2k}                r <-R {0, 1}^k
   %%                |                              |
@@ -1350,7 +1351,7 @@ for OAEP+ which is presented in full later.
 \end{slide}
 
 \begin{slide}
-  \head{Plaintext-aware encryption: OAEP+, 3}
+  \head{Plaintext-aware encryption: OAEP+, \seq}
   
   Let $\mathcal{T}$ be a trapdoor one-way function.  Then the insecurity of
   the public-key encryption scheme $\Xid{\mathcal{E}}{OAEP+}^{\mathcal{T}}$
@@ -1375,27 +1376,6 @@ for OAEP+ which is presented in full later.
   \end{eqnarray*}
 \end{slide}
 
-Before we move on to the OAEP+ security results, we state and prove the
-following lemma, due (I believe) to Victor Shoup.
-
-\begin{lemma}[Shoup]
-  \label{lem:shoup}
-  If $X$, $Y$ and $F$ are events, and
-  \[ \Pr[X \land \lnot F] = \Pr[Y \land \lnot F] \]
-  then
-  \[ |{\Pr[X]} - \Pr[Y]| \le \Pr[F]. \]
-\end{lemma}
-\begin{proof}
-  We have:
-  \begin{eqnarray*}[rll]
-    \Pr[X] &= \Pr[X \land F] &+ \Pr[X \land \lnot F] \\
-    \Pr[Y] &= \Pr[Y \land F] &+ \Pr[Y \land \lnot F]
-  \end{eqnarray*}
-  Subtracting gives
-  \[ |{\Pr[X]} - \Pr[Y]| = |{\Pr[X \land F]} - \Pr[Y \land F]| \le \Pr[F] \]
-  as required.
-\end{proof}
-
 \begin{proof}[Proof of the main OAEP+ result]
   
   We assume throughout that, before querying its $H'$ oracle at $x \cat r$,
@@ -1412,12 +1392,12 @@ following lemma, due (I believe) to Victor Shoup.
   
   \paragraph{Game $\G0$}
   This is effectively the original attack game: the adversary chooses two
-  plaintexts: one is chosen unformly at random, the adversary is given the
+  plaintexts: one is chosen uniformly at random, the adversary is given the
   ciphertext and asked to identify which plaintext was chosen.  Label the
   selected plaintext $x^*$, and the target ciphertext $y^*$.  Let $c^*$,
   $s^*$, $t^*$ and $w^*$ be the intermediate results of the OAEP+ padding, as
   described above.  Let $S_0$ be the event that the adversary wins the game.
-  Our objective is to bound the probabilty $\Pr[S_0] =
+  Our objective is to bound the probability $\Pr[S_0] =
   (\Adv{ind-cpa}{\Xid{\mathcal{E}}{OAEP+}^{\mathcal{T}}}(A) + 1)/2$.
 
   \paragraph{Game $\G1$}
@@ -1499,7 +1479,7 @@ following lemma, due (I believe) to Victor Shoup.
   Now observe that, if $F_1$ doesn't occur, Games $\G0$ and~$\G1$ are
   indistinguishable.  So
   \[ \Pr[S_0 \land \lnot F_1] = \Pr[S_1 \land \lnot F_1], \]
-  so by Shoup's lemma,
+  so by Lemma~\ref{lem:shoup} (slide~\pageref{lem:shoup}),
   \[ |{\Pr[S_0]} - \Pr[S_1]| \le \Pr[F_1]. \]
   Rearranging yields:
   \[ \Adv{ind-cpa}{\Xid{\mathcal{E}}{OAEP+}^{\mathcal{T}}}(A)
