|
@@ -1,6 +1,21 @@
|
|
|
-\chapter{Dixon}
|
|
|
+\chapter{Dixon \label{chap:dixon}}
|
|
|
|
|
|
-dixon!
|
|
|
+~\cite{dixon} describes a class of ``probabilistic algorithms'' for finding a
|
|
|
+factor of any composite number, at a computational cost asymptotically best
|
|
|
+than all other ones previously described:
|
|
|
+\bigO{\beta(\log N \log \log N)^{\rfrac{1}{2}}}
|
|
|
+for some constant $\beta > 0$.
|
|
|
+
|
|
|
+\paragraph{Kraitchick} was the first one popularizing the idea the instead of
|
|
|
+looking for integers $\angular{x, y}$ such that $x^2 -y^2 = N$ -recall Fermat's
|
|
|
+problem, formulated in equation ~\ref{eq:fermat_problem}, it is sufficient to
|
|
|
+look for \emph{multiples} of $N$:
|
|
|
+\begin{align}
|
|
|
+ x^2 - y^2 \equiv 0 \pmod{N}
|
|
|
+\end{align}
|
|
|
+and, once found, claim that $\gcd(N, x \pm y)$ are non-trial divisors of $N$
|
|
|
+just as we did in \ref{sec:fermat:implementation}.
|
|
|
+On the top of this,
|
|
|
|
|
|
%%% Local Variables:
|
|
|
%%% mode: latex
|