Grammar mostly

1 files changed, 5 insertions, 4 deletions

diff --git a/games/games.tex b/games/games.tex
index e60a756..12dc31e 100644
--- a/games/games.tex
+++ b/games/games.tex
@@ -790,15 +790,16 @@ $E[{b \over l-w}] \ge 1-{1\over\kappa}$.
 In our actual refresh operation, our commitment phase sends only the
 hash of the planchets to reduce bandwidth.  We could however commit
 to the full planchets without damaging anything else, including
-unforgeability.  We may transform our our adversary $\cal A$ into
+unforgeability.  We may thus transform our our adversary $\cal A$ into
 any adversary for the protocol that commits to full planchets by
 rewinding $\cal A$ to try each $\gamma \in 1,\ldots,\kappa$ during
 each refresh operation to obtain all planchets.  We observe a hash 
 collision if this fails to provide the correct coins. 
 
 We consider the degree one directed graph on coins induced by the
-refresh protocol.  It follows from unforgeability that any coin
-must originate from some user's withdraw in this graph.  
+refresh protocol, thanks to assuming a single denomination.  
+It follows from unforgeability that any coin must originate from
+some user's withdraw in this graph.  
 Let $X$ denote the coins from $\{C_1,\ldots,C_\ell\}$ that originate
 from a non-corrupted user.  So $\ell \leq w + |X|$. 
 
@@ -808,7 +809,7 @@ via the linking protocol, but no non-corrupted user could obtain the
 coin provided by the refresh, as otherwise $C$ gets marked as spend. 
 
 In each $R_C$, we know $\cal A$ must submit a planchet for which the
-linking protocol fails produce $C$ correctly.  In this case, either
+linking protocol fails to produce $C$ correctly.  In this case, either
 \begin{enumerate}
 \item the planchet must be false outright, meaning either $C$ itself
  or the blinding factor does not arise form $t C'$, or else