#!/usr/local/bin/python
# -*- coding: utf-8 -*-
import math
import random
#y^2=x^3+ax+b mod n
# ax+by=gcd(a,b). This function returns [gcd(a,b),x,y]. Source Wikipedia
def extended_gcd(a,b):
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
# pick first a point P=(u,v) with random non-zero coordinates u,v (mod N), then pick a random non-zero A (mod N),
# then take B = u^2 - v^3 - Ax (mod N).
# http://en.wikipedia.org/wiki/Lenstra_elliptic_curve_factorization
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)]
# Given the curve y^2 = x^3 + ax + b over the field K (whose characteristic we assume to be neither 2 nor 3), and points
# P = (xP, yP) and Q = (xQ, yQ) on the curve, assume first that xP != xQ. Let the slope of the line s = (yP − yQ)/(xP − xQ); since K # is a field, s is well-defined. Then we can define R = P + Q = (xR, − yR) by
# s=(xP-xQ)/(yP-yQ) Mod N
# xR=s^2-xP-xQ Mod N
# yR=yP+s(xR-xP) Mod N
# If xP = xQ, then there are two options: if yP = −yQ, including the case where yP = yQ = 0, then the sum is defined as 0[Identity]. # thus, the inverse of each point on the curve is found by reflecting it across the x-axis. If yP = yQ != 0, then R = P + P = 2P = # (xR, −yR) is given by
# s=3xP^2+a/(2yP) Mod N
# xR=s^2-2xP Mod N
# yR=yP+s(xR-xP) Mod N
# http://en.wikipedia.org/wiki/Elliptic_curve#The_group_law
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]
# http://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication
# Q=0 [Identity element]
# while m:
# if (m is odd) Q+=P
# P+=P
# m/=2
# return Q
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
Diff to Previous Revision
--- revision 1 2011-01-14 18:46:04
+++ revision 2 2011-01-15 04:32:04
@@ -1,21 +1,21 @@
+#!/usr/local/bin/python
+# -*- coding: utf-8 -*-
import math
import random
-
#y^2=x^3+ax+b mod n
-def extended_gcd(a,b):
- if a == 0 and b == 0: return (0, 0, 1)
- if a == 0: return (abs(b), 0, b/abs(b))
- if b == 0: return (abs(a), a/abs(a), 0)
- x_sign = 1; y_sign = 1
- if a < 0: a = -a; x_sign = -1
- if b < 0: b = -b; y_sign = -1
- x = 1; y = 0; r = 0; s = 1
- while b != 0:
- (c, q) = (a%b, a/b)
- (a, b, r, s, x, y) = (b, c, x-q*r, y-q*s, r, s)
- return (a, x*x_sign, y*y_sign )
-
+# ax+by=gcd(a,b). This function returns [gcd(a,b),x,y]. Source Wikipedia
+def extended_gcd(a,b):
+ 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
@@ -25,11 +25,26 @@
(a, b) = (b, a%b)
return a
+# pick first a point P=(u,v) with random non-zero coordinates u,v (mod N), then pick a random non-zero A (mod N),
+# then take B = u^2 - v^3 - Ax (mod N).
+# http://en.wikipedia.org/wiki/Lenstra_elliptic_curve_factorization
+
def randomCurve(N):
A,u,v=random.randrange(N),random.randrange(N),random.randrange(N)
- C=(v*v-u*u*u-A*u)%N
- return [(A,C,N),(u,v)]
+ B=(v*v-u*u*u-A*u)%N
+ return [(A,B,N),(u,v)]
+ # Given the curve y^2 = x^3 + ax + b over the field K (whose characteristic we assume to be neither 2 nor 3), and points
+ # P = (xP, yP) and Q = (xQ, yQ) on the curve, assume first that xP != xQ. Let the slope of the line s = (yP − yQ)/(xP − xQ); since K # is a field, s is well-defined. Then we can define R = P + Q = (xR, − yR) by
+ # s=(xP-xQ)/(yP-yQ) Mod N
+ # xR=s^2-xP-xQ Mod N
+ # yR=yP+s(xR-xP) Mod N
+ # If xP = xQ, then there are two options: if yP = −yQ, including the case where yP = yQ = 0, then the sum is defined as 0[Identity]. # thus, the inverse of each point on the curve is found by reflecting it across the x-axis. If yP = yQ != 0, then R = P + P = 2P = # (xR, −yR) is given by
+ # s=3xP^2+a/(2yP) Mod N
+ # xR=s^2-2xP Mod N
+ # yR=yP+s(xR-xP) Mod N
+ # http://en.wikipedia.org/wiki/Elliptic_curve#The_group_law
+
def addPoint(E,p_1,p_2):
if p_1=="Identity": return [p_2,1]
if p_2=="Identity": return [p_1,1]
@@ -55,6 +70,14 @@
return [(x_3,y_3),d]
+ # http://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication
+ # Q=0 [Identity element]
+ # while m:
+ # if (m is odd) Q+=P
+ # P+=P
+ # m/=2
+ # return Q
+
def mulPoint(E,P,m):
Ret="Identity"
d=1
@@ -73,12 +96,13 @@
for i in xrange(times):
E,P=randomCurve(N);
Q,d=mulPoint(E,P,m)
- if d!=1: return d
+ if d!=1 : return d
return N
if __name__=="__main__":
n=input()
+ m=int(math.factorial(1000))
while n!=1:
- k=ellipticFactor(n,int(math.factorial(1000)))
+ k=ellipticFactor(n,m)
n/=k
print k
