remove trailing space

chap-proofs.tex

@@ -42,7 +42,7 @@ Let us now define more formally the notions of reduction and computability using
   A TM $M$ is said to \emph{compute} a function $f: \Sigma^\star \to \Gamma^\star$, if for any finite input $x \in \Sigma^\star$ on tape $T_1$, blank tapes $T_2, \ldots, T_k$ with a beginning symbol $\triangleright$ and initial state $q_{start}$, $M$ halts in a finite number of steps with $f(x)$ written on its output tape $T_k$.
 
   A TM $M$ is said to \emph{recognize} a language $L \subseteq \Sigma^\star$ if on a finite input $x \in \Sigma^\star$ written on its input tape $T_1$, blank tapes $T_2, \ldots, T_k$ with a beginning symbol $\triangleright$ and initial state $q_{start}$, the machine $M$ eventually ends on the state $q_{halt}$ with $1$ written on its output tape if and only if $x \in L$.
-  
+
   A TM $M$ is said to run in $T(n)$-time if, on any input $x$, it eventually stops within $T(|x|)$ steps.
 
   A TM $M$ is said to run in $S(n)$-space if, on any input $x$, it eventually stops and had write at most $S(|x|)$ memory cells in its working tapes.
@@ -222,7 +222,7 @@ Two examples of security game are given in Figure~\ref{fig:sec-game-examples}: t
       \procedure{$\Exp{\mathrm{EU-CMA}}{\adv}(\lambda)$}{
         (vk,sk) \gets \Sigma.\mathsf{keygen}(1^\lambda)\\
         \mathsf{st} \gets \emptyset; \ensemble{sign} = \emptyset\\
-        \pcwhile \adv(\texttt{query}, vk, \mathsf{st}, \oracle{sign}{sk,\cdot} ) \pcdo 
+        \pcwhile \adv(\texttt{query}, vk, \mathsf{st}, \oracle{sign}{sk,\cdot} ) \pcdo
         ;\\
         (m^\star, \sigma^\star) \gets \adv(\texttt{forge}, vk, \mathsf{st}) \\
         \pcreturn (vk, \ensemble{sign},  m^\star, \sigma^\star)
@@ -255,7 +255,7 @@ Those signature queries are provided by an oracle \oracle{sign}{sk,\cdot}, which
 
 For EU-CMA, the advantage of an adversary $\adv$ is defined as
 \[
-  \advantage{\textrm{EU-CMA}}{\adv}(\lambda) 
+  \advantage{\textrm{EU-CMA}}{\adv}(\lambda)
   \triangleq
   \Pr\left[ \Sigma.\mathsf{verif}(vk, m^\star, \sigma^\star) = \top~\land~ \sigma^\star \notin \ensemble{sign} \right].
 \]