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75 lines
1.7 KiB
Python
75 lines
1.7 KiB
Python
#!/usr/bin/env python
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# -*- coding: utf-8 -*-
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def isPrime(a):
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return all(a % i for i in xrange(2, a))
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# http://stackoverflow.com/a/14793082/562769
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def factorize(n):
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factors = []
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p = 2
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while True:
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while(n % p == 0 and n > 0): #while we can divide by smaller number, do so
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factors.append(p)
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n = n / p
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p += 1 #p is not necessary prime, but n%p == 0 only for prime numbers
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if p > n / p:
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break
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if n > 1:
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factors.append(n)
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return factors
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def calculateLegendre(a, p):
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"""
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Calculate the legendre symbol (a, p) with p is prime.
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The result is either -1, 0 or 1
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>>> calculateLegendre(3, 29)
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-1
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>>> calculateLegendre(111, 41) # Beispiel aus dem Skript, S. 114
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-1
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>>> calculateLegendre(113, 41) # Beispiel aus dem Skript, S. 114
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1
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>>> calculateLegendre(2, 31)
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1
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>>> calculateLegendre(5, 31)
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1
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>>> calculateLegendre(150, 1009) # http://math.stackexchange.com/q/221223/6876
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1
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>>> calculateLegendre(25, 1009) # http://math.stackexchange.com/q/221223/6876
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1
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>>> calculateLegendre(2, 1009) # http://math.stackexchange.com/q/221223/6876
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1
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>>> calculateLegendre(3, 1009) # http://math.stackexchange.com/q/221223/6876
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1
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"""
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if a >= p or a < 0:
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return calculateLegendre(a % p, p)
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elif a == 0 or a == 1:
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return a
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elif a == 2:
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if p%8 == 1 or p%8 == 7:
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return 1
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else:
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return -1
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elif a == p-1:
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if p%4 == 1:
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return 1
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else:
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return -1
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elif not isPrime(a):
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factors = factorize(a)
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product = 1
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for pi in factors:
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product *= calculateLegendre(pi, p)
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return product
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else:
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if ((p-1)/2)%2==0 or ((a-1)/2)%2==0:
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return calculateLegendre(p, a)
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else:
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return (-1)*calculateLegendre(p, a)
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if __name__ == "__main__":
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import doctest
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doctest.testmod()
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