Fairness proof

1 files changed, 32 insertions, 5 deletions

diff --git a/games/games.tex b/games/games.tex
index e705649..2250f3f 100644
--- a/games/games.tex
+++ b/games/games.tex
@@ -435,11 +435,38 @@ could see in a game has the same probability, and he must win with probability $
 
 
 \subsection{Fairness}
-\paragraph{``Proof sketch''.}  We enumerate all things that can lead to a failure in the refresh step.
-\begin{itemize}
-  \item double-spending with different contract hash $\Rightarrow$ merchant can forge/modify deposit permission
-  \item double-spending with refresh $\Rightarrow$ linking protocol must be broken
-\end{itemize}
+
+\begin{proof}[Proof-sketch]
+We required that any refresh operations were run to conclusion, 
+which makes sense with our adversary $\prt{A}$ being a merchant
+unable to control either the customer or exchagne.  It follows 
+that $\prt{A}$ never called the refresh oracle on $C_n$.
+
+As the refresh $R$ must fail, $\prt{A}$ must have the custoemr
+spend $C_n$, and then either deposit $C_n$, refresh $C_n$, or
+spend $C_n$ with another merchant.
+In any case, $\prt{A}$ must not return the correct receipt to
+the customer, doing so concludes the transaction honestly.  
+
+If $\prt{A}$ deposits $C_n$, then the customer would also obtain
+the correct receipt from exchange by doing a refresh, so the
+adversary must distract them with signed message from the exchange
+indicating double spending or previous refresh. In either subcase,
+$\prt{A}$ provides signatures by both $C_n$ and $\V{pkE}$.
+
+If $\prt{A}$ either refreshes or spends $C_n$, then they provide
+a signature by $C_n$ too.
+
+TODO: Anything more about signature reduction?
+\end{proof}
+
+We have stated the game and theorem with $\prt{A}$ controlling only
+the merchant, but even if they control the exchagne as well they
+cannot forge the signature by $C_n$.  An exchange can however
+take actions like droping the refresh connection.  Attacks like
+these can only be thwarted with the aid of authorities who can
+witness the attack, like our auditor.
+
 
 \subsection{Income Transparency}
 To win the game, the adversary must produce enough coins that are not in the wallet of any non-corrupted user, but withdraw little