libzahl

big integer library
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commit 243a542dce0f8da6fc3ac43d5e5fcb144559b507
parent 802b2b18704f1b04ab3c3195d49333a546dc0ff4
Author: Mattias Andrée <maandree@kth.se>
Date:   Mon, 25 Jul 2016 15:40:04 +0200

Manual: how to calculate the legendre symbol

Signed-off-by: Mattias Andrée <maandree@kth.se>

Diffstat:
doc/not-implemented.tex | 18+++++++++++++++++-
1 file changed, 17 insertions(+), 1 deletion(-)

diff --git a/doc/not-implemented.tex b/doc/not-implemented.tex @@ -136,7 +136,23 @@ TODO \subsection{Legendre symbol} \label{sec:Legendre symbol} -TODO +\( \displaystyle{ + \left ( \frac{a}{p} \right ) \equiv a^{\frac{p - 1}{2}} ~(\text{Mod}~p),~ + \left ( \frac{a}{p} \right ) \in \{-1,~0,~1\},~ + p \in \textbf{P},~ p > 2 +}\) + +\noindent +That is, unless $\displaystyle{a^{\frac{p - 1}{2}} ~\text{Mod}~ p \le 1}$, +$\displaystyle{a^{\frac{p - 1}{2}} ~\text{Mod}~ p = p - 1}$, so +$\displaystyle{\left ( \frac{a}{p} \right ) = -1}$. + +It should be noted that +\( \displaystyle{ + \left ( \frac{a}{p} \right ) = + \left ( \frac{a ~\text{Mod}~ p}{p} \right ), +}\) +so a compressed lookup table can be used for small $p$. \subsection{Jacobi symbol}