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November 12, 2011 16:45
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#!/usr/bin/env python3 | |
import random | |
import collections | |
def primesbelow(N): | |
# http://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n-in-python/3035188#3035188 | |
#""" Input N>=6, Returns a list of primes, 2 <= p < N """ | |
correction = N % 6 > 1 | |
N = {0:N, 1:N-1, 2:N+4, 3:N+3, 4:N+2, 5:N+1}[N%6] | |
sieve = [True] * (N // 3) | |
sieve[0] = False | |
for i in range(int(N ** .5) // 3 + 1): | |
if sieve[i]: | |
k = (3 * i + 1) | 1 | |
sieve[k*k // 3::2*k] = [False] * ((N//6 - (k*k)//6 - 1)//k + 1) | |
sieve[(k*k + 4*k - 2*k*(i%2)) // 3::2*k] = [False] * ((N // 6 - (k*k + 4*k - 2*k*(i%2))//6 - 1) // k + 1) | |
return [2, 3] + [(3 * i + 1) | 1 for i in range(1, N//3 - correction) if sieve[i]] | |
smallprimeset = set(primesbelow(100000)) | |
_smallprimeset = 100000 | |
def isprime(n, precision=7): | |
# http://en.wikipedia.org/wiki/Miller-Rabin_primality_test#Algorithm_and_running_time | |
if n == 1 or n % 2 == 0: | |
return False | |
elif n < 1: | |
raise ValueError("Out of bounds, first argument must be > 0") | |
elif n < _smallprimeset: | |
return n in smallprimeset | |
d = n - 1 | |
s = 0 | |
while d % 2 == 0: | |
d //= 2 | |
s += 1 | |
for repeat in range(precision): | |
a = random.randrange(2, n - 2) | |
x = pow(a, d, n) | |
if x == 1 or x == n - 1: continue | |
for r in range(s - 1): | |
x = pow(x, 2, n) | |
if x == 1: return False | |
if x == n - 1: break | |
else: return False | |
return True | |
# https://comeoncodeon.wordpress.com/2010/09/18/pollard-rho-brent-integer-factorization/ | |
def pollard_brent(n): | |
if n % 2 == 0: return 2 | |
if n % 3 == 0: return 3 | |
y, c, m = random.randint(1, n-1), random.randint(1, n-1), random.randint(1, n-1) | |
g, r, q = 1, 1, 1 | |
while g == 1: | |
x = y | |
for i in range(r): | |
y = (pow(y, 2, n) + c) % n | |
k = 0 | |
while k < r and g==1: | |
ys = y | |
for i in range(min(m, r-k)): | |
y = (pow(y, 2, n) + c) % n | |
q = q * abs(x-y) % n | |
g = gcd(q, n) | |
k += m | |
r *= 2 | |
if g == n: | |
while True: | |
ys = (pow(ys, 2, n) + c) % n | |
g = gcd(abs(x - ys), n) | |
if g > 1: | |
break | |
return g | |
smallprimes = primesbelow(1000) # might seem low, but 1000*1000 = 1000000, so this will fully factor every composite < 1000000 | |
def primefactors(n, sort=False): | |
factors = [] | |
limit = int(n ** .5) + 1 | |
for checker in smallprimes: | |
if checker > limit: break | |
while n % checker == 0: | |
factors.append(checker) | |
n //= checker | |
limit = int(n ** .5) + 1 | |
if checker > limit: break | |
if n < 2: return factors | |
while n > 1: | |
if isprime(n): | |
factors.append(n) | |
break | |
factor = pollard_brent(n) # trial division did not fully factor, switch to pollard-brent | |
factors.extend(primefactors(factor)) # recurse to factor the not necessarily prime factor returned by pollard-brent | |
n //= factor | |
if sort: factors.sort() | |
return factors | |
def factorization(n): | |
factors = collections.defaultdict(int) | |
for p1 in primefactors(n): factors[p1] += 1 | |
return factors | |
totients = {} | |
def totient(n): | |
if n == 0: return 1 | |
try: return totients[n] | |
except KeyError: pass | |
tot = 1 | |
for p, exp in factorization(n).items(): | |
tot *= (p - 1) * p ** (exp - 1) | |
totients[n] = tot | |
return tot | |
def gcd(a, b): | |
if a == b: return a | |
while b > 0: a, b = b, a % b | |
return a | |
def lcm(a, b): | |
return abs(a * b) // gcd(a, b) | |
def isperm(a, b): | |
if a == b: return False | |
if type(a) == int: | |
c = [0] * 10 | |
while a: | |
c[a % 10] += 1 | |
a //= 10 | |
while b: | |
c[b % 10] -= 1 | |
b //= 10 | |
for count in c: | |
if count != 0: return False | |
return True | |
for char in set(a): | |
if b.count(char) != a.count(char): | |
return False | |
return True | |
# decorators | |
def profile(func): | |
import time | |
import os | |
def wrapper(*arg): | |
if os.name == "nt": | |
timer = time.clock | |
else: | |
timer = time.time | |
start = timer() | |
res = func(*arg) | |
stop = timer() | |
print("%s took %0.3f s" % (func.__name__, stop - start)) | |
return res | |
return wrapper |
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