Hello all , I have implemented Elliptic curve prime factorisation
using wikipedia [
http://en.wikipedia.org/wiki/Lenstra_elliptic_curve_factorization].
I think that this code is not optimised and posting for further
improvement. Feel free to comment and if you have any link regarding
Elliptic curve prime factorisation , kindly post it.
Thank you
import math
import random
#y^2=x^3+ax+b mod n
def extended_gcd(a,b): # taken from wikipedia
x,y,lastx,lasty=0,1,1,0
while b!=0:
q=a/b
a,b=b,a%b
x,lastx=(lastx-q*x,x)
y,lasty=(lasty-q*y,y)
if a<0:
return (-a,-lastx,-lasty)
else:
return (a,lastx,lasty)
def gcd(a,b):
if a < 0: a = -a
if b < 0: b = -b
if a == 0: return b
if b == 0: return a
while b != 0:
(a, b) = (b, a%b)
return a
def randomCurve(N):
A,u,v=random.randrange(N),random.randrange(N),random.randrange(N)
B=(v*v-u*u*u-A*u)%N
return [(A,B,N),(u,v)]
def addPoint(E,p_1,p_2):
if p_1=="Identity": return [p_2,1]
if p_2=="Identity": return [p_1,1]
a,b,n=E
(x_1,y_1)=p_1
(x_2,y_2)=p_2
x_1%=n
y_1%=n
x_2%=n
y_2%=n
if x_1 != x_2 :
d,u,v=extended_gcd(x_1-x_2,n)
s=((y_1-y_2)*u)%n
x_3=(s*s-x_1-x_2)%n
y_3=(-y_1-s*(x_3-x_1))%n
else:
if (y_1+y_2)%n==0:return ["Identity",1]
else:
d,u,v=extended_gcd(2*y_1,n)
s=((3*x_1*x_1+a)*u)%n
x_3=(s*s-2*x_1)%n
y_3=(-y_1-s*(x_3-x_1))%n
return [(x_3,y_3),d]
def mulPoint(E,P,m):
Ret="Identity"
d=1
while m!=0:
if m%2!=0: Ret,d=addPoint(E,Ret,P)
if d!=1 : return [Ret,d] # as soon as i got anything otherthan
1
return
P,d=addPoint(E,P,P)
if d!=1 : return [Ret,d]
m>>=1
return [Ret,d]
def ellipticFactor(N,m,times=5):
for i in xrange(times):
E,P=randomCurve(N);
Q,d=mulPoint(E,P,m)
if d!=1 : return d
return N
if __name__=="__main__":
n=input()
m=int(math.factorial(1000))
while n!=1:
k=ellipticFactor(n,m)
n/=k
print k
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