fixes

1 files changed, 10 insertions, 6 deletions

diff --git a/games/games.tex b/games/games.tex
index 22dd209..b57722b 100644
--- a/games/games.tex
+++ b/games/games.tex
@@ -242,7 +242,7 @@ given in literature.
 \section{Proofs}
 
 \subsection{Anonymity}
-\paragraph{``Proof sketch''.}  Proof by looking at withdraw/deposit/refresh transcripts.  Do game hopping, replace
+\paragraph{``Proof sketch''.}  Proof by looking at withdraw/deposit/refresh transcripts seen by the adversary.  Do game hopping, replace
 everything that's not random by something random (with computationally indistinguishable distribution) until everything that the adversary
 could see in a game has the same probability, and he must win with probability $1/2$.
 
@@ -255,7 +255,8 @@ could see in a game has the same probability, and he must win with probability $
 \end{itemize}
 
 \subsection{Income Transparency}
-To win the game, the adversary must produce coins that are not in the wallet of any non-corrupted user.
+To win the game, the adversary must produce enough coins that are not in the wallet of any non-corrupted user, but withdraw little
+coins via corrupted users.
 
 \begin{lemma}
   The signatures on $C_1, \dots, C_\ell$ must have been obtained by using either \ora{Withdraw}, \ora{RefreshReveal} or \ora{Link}.
@@ -267,15 +268,18 @@ To win the game, the adversary must produce coins that are not in the wallet of
 
 \begin{lemma}
   To win the Income Transparency game, the adversary must call $\ora{RefreshReveal}$ at least $(\kappa V_r - V_w) > 0$ times
-  in a way that causes the linking step to fail.
+  in a way that causes the linking step to fail.  (This implies unforgeability.)
+\end{lemma}
+
+\begin{lemma}
+  The probability to use \ora{RefreshCommit} and \ora{RefreshReveal} in a way that causes \algo{Link} to return $\bot$ at most $1/\kappa$.
 \end{lemma}
 
 \begin{theorem}
-  The adversary wins the Income Transparency game with with probability ...
+  The adversary wins the Income Transparency game with with probability at most $1 / \kappa$.
 \end{theorem}
 
-Conclusion:  Even the best adversary needs to waste $\kappa$ coins to obtain one untaxable coin.
-
+FIXME:  We should also include what the expected ``untaxed'' money for the adversary is for each won game, vs. the money wasted.
 
 \section{Standard Definitions}
 \begin{definition}[One-more forgery]