我的 PollardP1_rho 代码有问题,但我不知道如何修复
Something wrong with my PollardP1_rho code but I don't know how to fix it
我尝试使用 MillerRabin + PollardP1_rho 方法将整数分解为 Python3 中的素数,以尽可能降低时间复杂度 could.But 它未能通过一些测试,我知道在哪里问题was.But我是算法初学者,我不知道如何解决it.So我会把所有相关代码放在这里。
import random
def gcd(a, b):
"""
a, b: integers
returns: a positive integer, the greatest common divisor of a & b.
"""
if a == 0:
return b
if a < 0:
return gcd(-a, b)
while b > 0:
c = a % b
a, b = b, c
return a
def mod_mul(a, b, n):
# Calculate a * b % n iterately.
result = 0
while b > 0:
if (b & 1) > 0:
result = (result + a) % n
a = (a + a) % n
b = (b >> 1)
return result
def mod_exp(a, b, n):
# Calculate (a ** b) % n iterately.
result = 1
while b > 0:
if (b & 1) > 0:
result = mod_mul(result, a, n)
a = mod_mul(a, a, n)
b = (b >> 1)
return result
def MillerRabinPrimeCheck(n):
if n in {2, 3, 5, 7, 11}:
return True
elif (n == 1 or n % 2 == 0 or n % 3 == 0 or n % 5 == 0 or n % 7 == 0 or n % 11 == 0):
return False
k = 0
u = n - 1
while not (u & 1) > 0:
k += 1
u = (u >> 1)
random.seed(0)
s = 5 #If the result isn't right, then add the var s.
for i in range(s):
x = random.randint(2, n - 1)
if x % n == 0:
continue
x = mod_exp(x, u, n)
pre = x
for j in range(k):
x = mod_mul(x, x, n)
if (x == 1 and pre != 1 and pre != n - 1):
return False
pre = x
if x != 1:
return False
return True
def PollardP1_rho(n, c):
'''
Consider c as a constant integer.
'''
i = 1
k = 2
x = random.randrange(1, n - 1) + 1
y = x
while 1:
i += 1
x = (mod_mul(x, x, n) + c) % n
d = gcd(y - x, n)
if 1 < d < n:
return d
elif x == y:
return n
elif i == k:
y = x
k = (k << 1)
result = []
def PrimeFactorsListGenerator(n):
if n <= 1:
pass
elif MillerRabinPrimeCheck(n) == True:
result.append(n)
else:
a = n
while a == n:
a = PollardP1_rho(n, random.randrange(1,n - 1) + 1)
PrimeFactorsListGenerator(a)
PrimeFactorsListGenerator(n // a)
当我尝试测试时:
PrimeFactorsListGenerator(4)
它没有停止并循环播放:
PollardP1_rho(4, random.randrange(1,4 - 1) + 1)
我已经测试了PollardP1_rho之前的函数,它们都正常工作,所以我知道函数PollardP1_rho不能正确处理数字4,还有数字5.How我可以修复吗那个?
我自己解决了。
代码中有 1 个错误。
我不应该在函数外使用 var 'result' 作为全局变量,我应该在函数中定义并使用 result.extend() 来确保整个递归的可用性 process.So 我重写了PollardP1_rho(n, c) 和 PrimeFactorsListGenerator(n):
def Pollard_rho(x, c):
'''
Consider c as a constant integer.
'''
i, k = 1, 2
x0 = random.randint(0, x)
y = x0
while 1:
i += 1
x0 = (mod_mul(x0, x0, x) + c) % x
d = gcd(y - x0, x)
if d != 1 and d != x:
return d
if y == x0:
return x
if i == k:
y = x0
k += k
def PrimeFactorsListGenerator(n):
result = []
if n <= 1:
return None
if MillerRabinPrimeCheck(n):
return [n]
p = n
while p >= n:
p = Pollard_rho(p, random.randint(1, n - 1))
result.extend(PrimeFactorsListGenerator(p))
result.extend(PrimeFactorsListGenerator(n // p))
return result
#PrimeFactorsListGenerator(400)
#PrimeFactorsListGenerator(40000)
还有一个额外的提示:你根本不需要写一个函数mod_mul(a, b, n),使用Python 内置的 pow(a, b, n) 可以解决问题,并且已经过全面优化。
我尝试使用 MillerRabin + PollardP1_rho 方法将整数分解为 Python3 中的素数,以尽可能降低时间复杂度 could.But 它未能通过一些测试,我知道在哪里问题was.But我是算法初学者,我不知道如何解决it.So我会把所有相关代码放在这里。
import random
def gcd(a, b):
"""
a, b: integers
returns: a positive integer, the greatest common divisor of a & b.
"""
if a == 0:
return b
if a < 0:
return gcd(-a, b)
while b > 0:
c = a % b
a, b = b, c
return a
def mod_mul(a, b, n):
# Calculate a * b % n iterately.
result = 0
while b > 0:
if (b & 1) > 0:
result = (result + a) % n
a = (a + a) % n
b = (b >> 1)
return result
def mod_exp(a, b, n):
# Calculate (a ** b) % n iterately.
result = 1
while b > 0:
if (b & 1) > 0:
result = mod_mul(result, a, n)
a = mod_mul(a, a, n)
b = (b >> 1)
return result
def MillerRabinPrimeCheck(n):
if n in {2, 3, 5, 7, 11}:
return True
elif (n == 1 or n % 2 == 0 or n % 3 == 0 or n % 5 == 0 or n % 7 == 0 or n % 11 == 0):
return False
k = 0
u = n - 1
while not (u & 1) > 0:
k += 1
u = (u >> 1)
random.seed(0)
s = 5 #If the result isn't right, then add the var s.
for i in range(s):
x = random.randint(2, n - 1)
if x % n == 0:
continue
x = mod_exp(x, u, n)
pre = x
for j in range(k):
x = mod_mul(x, x, n)
if (x == 1 and pre != 1 and pre != n - 1):
return False
pre = x
if x != 1:
return False
return True
def PollardP1_rho(n, c):
'''
Consider c as a constant integer.
'''
i = 1
k = 2
x = random.randrange(1, n - 1) + 1
y = x
while 1:
i += 1
x = (mod_mul(x, x, n) + c) % n
d = gcd(y - x, n)
if 1 < d < n:
return d
elif x == y:
return n
elif i == k:
y = x
k = (k << 1)
result = []
def PrimeFactorsListGenerator(n):
if n <= 1:
pass
elif MillerRabinPrimeCheck(n) == True:
result.append(n)
else:
a = n
while a == n:
a = PollardP1_rho(n, random.randrange(1,n - 1) + 1)
PrimeFactorsListGenerator(a)
PrimeFactorsListGenerator(n // a)
当我尝试测试时:
PrimeFactorsListGenerator(4)
它没有停止并循环播放:
PollardP1_rho(4, random.randrange(1,4 - 1) + 1)
我已经测试了PollardP1_rho之前的函数,它们都正常工作,所以我知道函数PollardP1_rho不能正确处理数字4,还有数字5.How我可以修复吗那个?
我自己解决了。 代码中有 1 个错误。 我不应该在函数外使用 var 'result' 作为全局变量,我应该在函数中定义并使用 result.extend() 来确保整个递归的可用性 process.So 我重写了PollardP1_rho(n, c) 和 PrimeFactorsListGenerator(n):
def Pollard_rho(x, c):
'''
Consider c as a constant integer.
'''
i, k = 1, 2
x0 = random.randint(0, x)
y = x0
while 1:
i += 1
x0 = (mod_mul(x0, x0, x) + c) % x
d = gcd(y - x0, x)
if d != 1 and d != x:
return d
if y == x0:
return x
if i == k:
y = x0
k += k
def PrimeFactorsListGenerator(n):
result = []
if n <= 1:
return None
if MillerRabinPrimeCheck(n):
return [n]
p = n
while p >= n:
p = Pollard_rho(p, random.randint(1, n - 1))
result.extend(PrimeFactorsListGenerator(p))
result.extend(PrimeFactorsListGenerator(n // p))
return result
#PrimeFactorsListGenerator(400)
#PrimeFactorsListGenerator(40000)
还有一个额外的提示:你根本不需要写一个函数mod_mul(a, b, n),使用Python 内置的 pow(a, b, n) 可以解决问题,并且已经过全面优化。