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@@ -1,6 +1,63 @@
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-\chapter{Pollard's $\rho$ factorization method}
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+\chapter{Pollard's $\rho$ factorization method \label{chap:pollardrho}}
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-$\rho$!
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+Pollard's $\rho$ factorization method is based on the statistical idea behind
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+the birthday paradox. It consists into indentifying a periodically recurrent
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+sequence of integers in the ring of remainders with respect to the public
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+modulus $N$, and claim that the period $\psi$ is one of the two primes
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+factorizing $N$.
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+
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+\paragraph{Origins of the name} The $\rho$ name is devoted to the graphical
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+representation of the algorithm: as we can see in figure ~\ref{fig:pollardrho},
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+if we graphically represent the lookup over a graphic
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+
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+\begin{center}
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+ \begin{tikzpicture}[scale=0.7, thick]
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+ \tikzstyle{every node}=[draw,circle,fill=white,minimum size=4pt,
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+ inner sep=0pt]
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+ \node (1) at (1.4, 0.2) [label=left:$x_1$] {};
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+ \node (2) at (2.5, 3) [label=left:$x_{i-2}$] {};
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+ \node (3) at (3.25, 5) [label=left:$x_{i-1}$] {};
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+ \node (4) at (4, 7) [label=left:$ x_i \equiv x_j $] {};
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+ \node (5) at (6, 9) [label=above:$x_{i+1}$] {};
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+ \node (6) at (8, 7) [label=right:$x_{i+2}$] {};
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+ \node (7) at (6, 5) [label=below:$x_{j-1}$] {};
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+
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+ \path (1) edge [dashed] (2);
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+ \path (2) edge (3);
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+ \path (3) edge (4);
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+ \path (4) edge [bend left] (5);
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+ \path (5) edge [bend left] (6);
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+ \path (6) edge [bend left, dashed] (7);
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+ \path (7) edge [bend left] (4);
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+
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+ %%\draw [decorate,decoration={brace, raise=1.5cm}] (1) -- (3)
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+ %%node[draw=no] at (-1.5, 4) {tail};
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+ \draw [decorate,decoration={brace, raise=3cm}] (5) -- (7)
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+ node[draw=none] at (13, 7) {\footnotesize {periodic sequence}};
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+
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+\end{tikzpicture}
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+\end{center}
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+
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+
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+\paragraph{A more rigourous description}
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+\begin{proof}
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+\end{proof}
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+
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+\section{A Computer program for Pollard's $\rho$ method}
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+
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+Using the same trick we saw in section ~\ref{sec:pollard-1:implementing}, we
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+chose to apply occasionally Euclid's algorithm by computing the accumulated
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+product; algorithm ~\ref{alg:pollardrho} outlines what we have so far discussed,
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+considering also the pascal transcript present in ~\cite{riesel} \S 5.
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+
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+\begin{algorithm}
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+ \caption{Pollard's $\rho$ factorization \label{alg:pollardrho}}
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+ \begin{algorithmic}[1]
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+ \State $a \getsRandom \naturalN \setminus \{0, 2\}$
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+ \State $x \getsRandom \naturalN$
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+ \State $y \gets x$
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+ \end{algorithmic}
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+\end{algorithm}
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "question_authority"
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Michele Orrù