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	Manual: How to calculate the Jacobi symbol - libzahl - big integer library
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	---
	commit 60dd5110e21d1aedc047f2033af74330df552e40
	parent 4d2e79e7eec793a557c26d1253bcfc13f6b555d6
	Author: Mattias Andrée <[email protected]>
	Date: Mon, 25 Jul 2016 16:15:08 +0200

	Manual: How to calculate the Jacobi symbol

	Signed-off-by: Mattias Andrée <[email protected]>

	Diffstat:
	M doc/not-implemented.tex \| 46 +++++++++++++++++++++++++++++…

	1 file changed, 43 insertions(+), 3 deletions(-)
	---
	diff --git a/doc/not-implemented.tex b/doc/not-implemented.tex
	@@ -141,10 +141,11 @@ TODO
	\left ( \frac{a}{p} \right ) \in \{-1,~0,~1\},~
	p \in \textbf{P},~ p > 2
	}\)
	+\vspace{1em}

	\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
	+That is, unless $\displaystyle{a^{\frac{p - 1}{2}} \mod p \le 1}$,
	+$\displaystyle{a^{\frac{p - 1}{2}} \mod p = p - 1}$, so
	$\displaystyle{\left ( \frac{a}{p} \right ) = -1}$.

	It should be noted that
	@@ -158,7 +159,46 @@ so a compressed lookup table can be used for small $p$.
	\subsection{Jacobi symbol}
	\label{sec:Jacobi symbol}

	-TODO
	+\( \displaystyle{
	+ \left ( \frac{a}{n} \right ) =
	+ \prod_k \left ( \frac{a}{p_k} \right )^{n_k},
	+}\)
	+where $n$ = $\displaystyle{\prod_k p_k^{n_k}}$, and $p_k \in \textbf{P}$.
	+\vspace{1em}
	+
	+Like the Legendre symbol, the Jacobi symbol is $n$-period over $a$.
	+If $n$, is prime, the Jacobi symbol is the Legendre symbol, but
	+the Jacobi symbol is defined for all odd numbers $n$, where the
	+Legendre symbol is defined only for odd primes $n$.
	+
	+Use the following algorithm to calculate the Jacobi symbol:
	+
	+\vspace{1em}
	+\hspace{-2.8ex}
	+\begin{minipage}{\linewidth}
	+\begin{algorithmic}
	+ \STATE $a \gets a \mod n$
	+ \STATE $r \gets \mbox{lsb}~ a$
	+ \STATE $a \gets a \cdot 2^{-r}$
	+ \STATE \(\displaystyle{
	+ r \gets \left \lbrace \begin{array}{rl}
	+ 1 &
	+ \text{if}~ n \equiv 1, 7 ~(\text{Mod}~ 8)
	+ ~\text{or}~ r \equiv 0 ~(\text{Mod}~ 2) \\
	+ -1 & \text{otherwise} \\
	+ \end{array} \right .
	+ }\)
	+ \IF{$a = 1$}
	+ \RETURN $r$
	+ \ELSIF{$\gcd(a, n) \neq 1$}
	+ \RETURN $0$
	+ \ENDIF
	+ \STATE $(a, n) = (n, a)$
	+ \STATE \textbf{start over}
	+\end{algorithmic}
	+\end{minipage}
	+\vspace{1em}
	+


	\subsection{Kronecker symbol}