From b0a72dc66c8388105958cf4f943e17f3c0387dcd Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Sun, 1 Sep 2013 12:38:44 +0200 Subject: [PATCH] fixed bug and added implementation --- .../Calculate-Legendre/Calculate-Legendre.png | Bin 48548 -> 54026 bytes .../Calculate-Legendre/Calculate-Legendre.tex | 6 +- .../Calculate-Legendre/calculateLegendre.py | 63 ++++++++++++++++++ 3 files changed, 68 insertions(+), 1 deletion(-) create mode 100644 source-code/Pseudocode/Calculate-Legendre/calculateLegendre.py diff --git a/source-code/Pseudocode/Calculate-Legendre/Calculate-Legendre.png b/source-code/Pseudocode/Calculate-Legendre/Calculate-Legendre.png index 4ba811da9d4fcb710177bdbefaa7f678ccab0472..bb1306f13d2014e078432298b07d77a2b7bb87de 100644 GIT binary patch literal 54026 zcmaf)cQl*f|M%@ttJC-jPCh6(tCPe{ipt9WDM8;J;}+kb8nIwIRhS82blYS z1E5ZRE@V(BRL0HQ!{6D-*G0z1&ou|H$wfwXiwt=G&XeHWtvqNJ+k>f?MSf1|JC8fx zu(Aq?_9Un|4L7gmj+g%_t*CC^Xs*6F#Uoc$-5kdWsHXQHruV-uf4yh-9AXU68ft^$ 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z_1|p>EzBL}_%CWA##`ZkTe158csRHI2V3z5t!jkpm{G}jv0DCTj}#a|*~2o&LNkCRv`Z$k&o$r@ZkTS7l;?+zhnZ#qtuPbfDfp; Lo?7`so7evj*l%QX diff --git a/source-code/Pseudocode/Calculate-Legendre/Calculate-Legendre.tex b/source-code/Pseudocode/Calculate-Legendre/Calculate-Legendre.tex index 53a2e75..3326188 100644 --- a/source-code/Pseudocode/Calculate-Legendre/Calculate-Legendre.tex +++ b/source-code/Pseudocode/Calculate-Legendre/Calculate-Legendre.tex @@ -35,7 +35,11 @@ \State $p_1, p_2, \dots, p_n \gets \Call{Factorize}{a}$ \State \Return $\prod_{i=1}^n \Call{CalculateLegendre}{p_i, p}$ \Else \Comment{now: $a \in \mathbb{P}, \sqrt{p-2} \geq a \geq 3$} - \State \Return $(-1) \cdot \Call{CalculateLegendre}{p, a}$ + \If{$\frac{p-1}{2} \equiv 0 \mod 2$} + \State \Return $(-1) \cdot \Call{CalculateLegendre}{p, a}$ + \Else + \State \Return $(-1) \cdot \Call{CalculateLegendre}{p, a}$ + \EndIf \EndIf \EndProcedure \end{algorithmic} diff --git a/source-code/Pseudocode/Calculate-Legendre/calculateLegendre.py b/source-code/Pseudocode/Calculate-Legendre/calculateLegendre.py new file mode 100644 index 0000000..beb8b32 --- /dev/null +++ b/source-code/Pseudocode/Calculate-Legendre/calculateLegendre.py @@ -0,0 +1,63 @@ +#!/usr/bin/env python +# -*- coding: utf-8 -*- + +def isPrime(a): + return all(a % i for i in xrange(2, a)) + +# http://stackoverflow.com/a/14793082/562769 +def factorize(n): + factors = [] + + p = 2 + while True: + while(n % p == 0 and n > 0): #while we can divide by smaller number, do so + factors.append(p) + n = n / p + p += 1 #p is not necessary prime, but n%p == 0 only for prime numbers + if p > n / p: + break + if n > 1: + factors.append(n) + return factors + +def calculateLegendre(a, p): + """ + Calculate the legendre symbol (a, p) with p is prime. + The result is either -1, 0 or 1 + + >>> calculateLegendre(3, 29) + -1 + >>> calculateLegendre(111, 41) + -1 + >>> calculateLegendre(113, 41) + 1 + """ + if a >= p or a < 0: + return calculateLegendre(a % p, p) + elif a == 0 or a == 1: + return a + elif a == 2: + if a%8 == 1 or a%8 == 7: + return 1 + else: + return -1 + elif a == p-1: + if p%4 == 1: + return 1 + else: + return -1 + elif not isPrime(a): + factors = factorize(a) + product = 1 + for pi in factors: + product *= calculateLegendre(pi, p) + return product + else: + if ((p-1)/2)%2==0: + return calculateLegendre(p, a) + else: + return (-1)*calculateLegendre(p, a) + +if __name__ == "__main__": + import doctest + doctest.testmod()