import random from itertools import combinations import math def eucli

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import random
from itertools import combinations
import math
def euclid(a, b):
"""returns the Greatest Common Divisor of a and b"""
a = abs(a)
b = abs(b)
if a < b:
a, b = b, a
while b != 0:
a, b = b, a % b
return a
def coPrime(l):
"""returns 'True' if the values in the list L are all co-prime
otherwise, it returns 'False'. """
for i, j in combinations(l, 2):
if euclid(i, j) != 1:
return False
return True
def extractTwos(m):
"""m is a positive integer. A tuple (s, d) of integers is returned
such that m = (2 ** s) * d."""
# the problem can be break down to count how many '0's are there in
# the end of bin(m). This can be done this way: m & a stretch of '1's
# which can be represent as (2 ** n) - 1.
assert m >= 0
i = 0
while m & (2 ** i) == 0:
i += 1
return i, m >> i
def int2baseTwo(x):
"""x is a positive integer. Convert it to base two as a list of integers
in reverse order as a list."""
# repeating x >>= 1 and x & 1 will do the trick
assert x >= 0
bitInverse = []
while x != 0:
bitInverse.append(x & 1)
x >>= 1
return bitInverse
def modExp(a, d, n):
"""returns a ** d (mod n)"""
assert d >= 0
assert n >= 0
base2D = int2baseTwo(d)
base2DLength = len(base2D)
modArray = []
result = 1
for i in range(1, base2DLength + 1):
if i == 1:
modArray.append(a % n)
else:
modArray.append((modArray[i - 2] ** 2) % n)
for i in range(0, base2DLength):
if base2D[i] == 1:
result *= base2D[i] * modArray[i]
return result % n
def millerRabin(n, k):
"""
Miller Rabin pseudo-prime test
return True means likely a prime, (how sure about that, depending on k)
return False means definitely a composite.
Raise assertion error when n, k are not positive integers
and n is not 1
"""
assert n >= 1
# ensure n is bigger than 1
assert k > 0
# ensure k is a positive integer so everything down here makes sense
if n == 2:
return True
# make sure to return True if n == 2
if n % 2 == 0:
return False
# immediately return False for all the even numbers bigger than 2
extract2 = extractTwos(n - 1)
s = extract2[0]
d = extract2[1]
assert 2 ** s * d == n - 1
def tryComposite(a):
"""Inner function which will inspect whether a given witness
will reveal the true identity of n. Will only be called within
millerRabin"""
x = modExp(a, d, n)
if x == 1 or x == n - 1:
return None
else:
for j in range(1, s):
x = modExp(x, 2, n)
if x == 1:
return False
elif x == n - 1:
return None
return False
for i in range(0, k):
a = random.randint(2, n - 2)
if tryComposite(a) == False:
return False
return True # actually, we should return probably true.
def primeSieve(k):
"""return a list with length k + 1, showing if list[i] == 1, i is a prime
else if list[i] == 0, i is a composite, if list[i] == -1, not defined"""
def isPrime(n):
"""return True is given number n is absolutely prime,
return False is otherwise."""
for i in range(2, int(n ** 0.5) + 1):
if n % i == 0:
return False
return True
result = [-1] * (k + 1)
for i in range(2, int(k + 1)):
if isPrime(i):
result[i] = 1
else:
result[i] = 0
return result
def findAPrime(a, b, k):
"""Return a pseudo prime number roughly between a and b,
(could be larger than b). Raise ValueError if cannot find a
pseudo prime after 10 * ln(x) + 3 tries. """
x = random.randint(a, b)
for i in range(0, int(10 * math.log(x) + 3)):
if millerRabin(x, k):
return x
else:
x += 1
raise ValueError