Manual: how to calculate the legendre symbol - libzahl - big integer library
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---
DIR commit 243a542dce0f8da6fc3ac43d5e5fcb144559b507
DIR parent 802b2b18704f1b04ab3c3195d49333a546dc0ff4
HTML 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:
M doc/not-implemented.tex | 18 +++++++++++++++++-
1 file changed, 17 insertions(+), 1 deletion(-)
---
DIR 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}