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More on exponentiation by squaring - libzahl - big integer library |
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--- |
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commit f1b60d7365ca1437efd3f6f8c05d2cdc46bd24b6 |
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parent fba103dca7a472eb58159d45ff8700736f89a136 |
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Author: Mattias Andrée <[email protected]> |
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Date: Thu, 12 May 2016 01:14:52 +0200 |
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More on exponentiation by squaring |
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Signed-off-by: Mattias Andrée <[email protected]> |
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Diffstat: |
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M doc/arithmetic.tex | 55 +++++++++++++++++++++++++++++… |
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1 file changed, 54 insertions(+), 1 deletion(-) |
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--- |
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diff --git a/doc/arithmetic.tex b/doc/arithmetic.tex |
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@@ -187,9 +187,62 @@ can be expressed as a simple formula |
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\vspace{-1em} |
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\[ \hspace*{-0.4cm} |
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a^b = \prod_{i = 0}^{\lceil \log_2 b \rceil} |
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- a^{2^i} \left \lfloor {b \over 2^i} \hspace*{-1ex} \mod 2 \right \rfloor |
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+ \left ( a^{2^i} \right )^{\left \lfloor {\displaystyle{b \over 2^i}} \hspa… |
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\] |
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+\noindent |
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+This is a natural extension to the observations |
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+ |
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+\vspace{-1em} |
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+\[ \hspace*{-0.4cm} |
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+ \forall n \in \textbf{Z}_{+} \exists B \subset \textbf{Z}_{+} : b = \sum_{… |
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+ ~~~~ \textrm{and} ~~~~ |
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+ a^{\sum x} = \prod a^x. |
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+\] |
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+ |
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+\noindent |
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+The algorithm can be expressed in psuedocode as |
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+ |
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+\vspace{1em} |
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+\hspace{-2.8ex} |
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+\begin{minipage}{\linewidth} |
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+\begin{algorithmic} |
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+ \STATE $r \gets 1$ |
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+ \STATE $f \gets a$ |
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+ \WHILE{$b \neq 0$} |
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+ \IF{$b \equiv 1 ~(\textrm{Mod}~ 2)$} |
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+ \STATE $r \gets r \cdot f$ |
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+ \ENDIF |
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+ \STATE $f \gets f^2$ \qquad \textcolor{c}{\{$f \gets f \cdot f$\}} |
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+ \STATE $b \gets \lfloor b / 2 \rfloor$ |
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+ \ENDWHILE |
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+ \RETURN $r$ |
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+\end{algorithmic} |
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+\end{minipage} |
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+\vspace{1em} |
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+ |
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+\noindent |
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+Modular exponentiation ($a^b \mod m$) by squaring can be |
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+expressed as |
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+ |
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+\vspace{1em} |
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+\hspace{-2.8ex} |
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+\begin{minipage}{\linewidth} |
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+\begin{algorithmic} |
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+ \STATE $r \gets 1$ |
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+ \STATE $f \gets a$ |
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+ \WHILE{$b \neq 0$} |
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+ \IF{$b \equiv 1 ~(\textrm{Mod}~ 2)$} |
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+ \STATE $r \gets r \cdot f \hspace*{-1ex}~ \mod m$ |
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+ \ENDIF |
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+ \STATE $f \gets f^2 \hspace*{-1ex}~ \mod m$ |
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+ \STATE $b \gets \lfloor b / 2 \rfloor$ |
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+ \ENDWHILE |
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+ \RETURN $r$ |
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+\end{algorithmic} |
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+\end{minipage} |
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+\vspace{1em} |
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+ |
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{\tt zmodpow} does \emph{not} calculate the |
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modular inverse if the exponent is negative, |
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rather, you should expect the result to be |
