libzahl

big integer library
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commit 60dd5110e21d1aedc047f2033af74330df552e40
parent 4d2e79e7eec793a557c26d1253bcfc13f6b555d6
Author: Mattias Andrée <maandree@kth.se>
Date:   Mon, 25 Jul 2016 16:15:08 +0200

Manual: How to calculate the Jacobi symbol

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

Diffstat:
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}