Small random corrections.

book/dixon.tex

@@ -46,7 +46,7 @@ that $\mod{N}$ is equivalent to:
 \end{align}
 and voil\`a our congruence of squares (\cite{discretelogs} \S 4). For what
 concerns the generation of $x_i$ with the property \ref{eq:dixon:x_sequence},
-they can simply taken at random and tested using trial division.
+they can simply be taken at random and tested using trial division.
 
 \paragraph{Brillhart and Morrison} later proposed (\cite{morrison-brillhart}
 p.187) a better approach than trial division to find such $x$. Their idea aims
@@ -222,7 +222,7 @@ $e^{\sqrt{\ln N \ln \ln N}}$.
     \Comment finding linearity requires redundance
     \While{$i < r$}
     \Comment search for suitable pairs
-    \State $x_i \getsRandom \{0, \ldots N\}$
+    \State $x_i \getsRandom \naturalN_{< N}$
     \State $y_i \gets x_i^2 - N$
     \State $v_i \gets \texttt{smooth}(y_i)$
     \If{$v_i$} $i \gets i+1$ \EndIf

book/pollardrho.tex

@@ -58,10 +58,10 @@ factor of $N$.
 \paragraph{Choosing the function} Ideally, $F$ should be easily computable, but
 at the same time random enough to reduce as much as possible the epacts
 ~\cite{Crandall} \S 5.2.1. Any quadratic function $F(x) = x^2 + b$ should be
-enough \footnote{
+enough\footnote{
   Note that this has been only empirically verified, and so far not been proved
-  (~\cite{riesel}, p. 177)}
-, provided that $b \in \naturalN \setminus \{0, 2\}$.
+  (~\cite{riesel}, p. 177)},
+provided that $b \in \naturalN \setminus \{0, 2\}$.
 For example, ~\cite{pollardMC} uses $x^2 -1$, meanwhile we are going to choose
 $F(x) = x^2 + 1$.
 
@@ -150,7 +150,7 @@ this algorithm, based on the birthday paradox.
 
 We can obviously substitute the $365$ with any set of cardinality $\zeta$
 to express the probability that a random function from $\integerZ_{\epsilon}$
-to $\integerZ_{\zeta}$ is injective. Back to our particular case,
+to $\integerZ_{\zeta}$ is injective. However, back to our particular case,
 we want to answer the question:
 
 \emph{