[catapult] - Copy Telemetry's gsutilz over to third_party.

third_party/gsutil/third_party/rsa/rsa/prime.py

Index: third_party/gsutil/third_party/rsa/rsa/prime.py
diff --git a/third_party/gsutil/third_party/rsa/rsa/prime.py b/third_party/gsutil/third_party/rsa/rsa/prime.py
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index 0000000000000000000000000000000000000000..7422eb1d28bdfdddb96ac1c1fba6602db3354428
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+# -*- coding: utf-8 -*-
+#
+#  Copyright 2011 Sybren A. Stüvel <sybren@stuvel.eu>
+#
+#  Licensed under the Apache License, Version 2.0 (the "License");
+#  you may not use this file except in compliance with the License.
+#  You may obtain a copy of the License at
+#
+#      http://www.apache.org/licenses/LICENSE-2.0
+#
+#  Unless required by applicable law or agreed to in writing, software
+#  distributed under the License is distributed on an "AS IS" BASIS,
+#  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+#  See the License for the specific language governing permissions and
+#  limitations under the License.
+
+'''Numerical functions related to primes.
+
+Implementation based on the book Algorithm Design by Michael T. Goodrich and
+Roberto Tamassia, 2002.
+'''
+
+__all__ = [ 'getprime', 'are_relatively_prime']
+
+import rsa.randnum
+
+def gcd(p, q):
+    '''Returns the greatest common divisor of p and q
+
+    >>> gcd(48, 180)
+    12
+    '''
+
+    while q != 0:
+        if p < q: (p,q) = (q,p)
+        (p,q) = (q, p % q)
+    return p
+    
+
+def jacobi(a, b):
+    '''Calculates the value of the Jacobi symbol (a/b) where both a and b are
+    positive integers, and b is odd
+
+    :returns: -1, 0 or 1
+    '''
+
+    assert a > 0
+    assert b > 0
+
+    if a == 0: return 0
+    result = 1
+    while a > 1:
+        if a & 1:
+            if ((a-1)*(b-1) >> 2) & 1:
+                result = -result
+            a, b = b % a, a
+        else:
+            if (((b * b) - 1) >> 3) & 1:
+                result = -result
+            a >>= 1
+    if a == 0: return 0
+    return result
+
+def jacobi_witness(x, n):
+    '''Returns False if n is an Euler pseudo-prime with base x, and
+    True otherwise.
+    '''
+
+    j = jacobi(x, n) % n
+
+    f = pow(x, n >> 1, n)
+
+    if j == f: return False
+    return True
+
+def randomized_primality_testing(n, k):
+    '''Calculates whether n is composite (which is always correct) or
+    prime (which is incorrect with error probability 2**-k)
+
+    Returns False if the number is composite, and True if it's
+    probably prime.
+    '''
+
+    # 50% of Jacobi-witnesses can report compositness of non-prime numbers
+
+    # The implemented algorithm using the Jacobi witness function has error
+    # probability q <= 0.5, according to Goodrich et. al
+    #
+    # q = 0.5
+    # t = int(math.ceil(k / log(1 / q, 2)))
+    # So t = k / log(2, 2) = k / 1 = k
+    # this means we can use range(k) rather than range(t)
+
+    for _ in range(k):
+        x = rsa.randnum.randint(n-1)
+        if jacobi_witness(x, n): return False
+    
+    return True
+
+def is_prime(number):
+    '''Returns True if the number is prime, and False otherwise.
+
+    >>> is_prime(42)
+    False
+    >>> is_prime(41)
+    True
+    '''
+
+    return randomized_primality_testing(number, 6)
+
+def getprime(nbits):
+    '''Returns a prime number that can be stored in 'nbits' bits.
+
+    >>> p = getprime(128)
+    >>> is_prime(p-1)
+    False
+    >>> is_prime(p)
+    True
+    >>> is_prime(p+1)
+    False
+    
+    >>> from rsa import common
+    >>> common.bit_size(p) == 128
+    True
+    
+    '''
+
+    while True:
+        integer = rsa.randnum.read_random_int(nbits)
+
+        # Make sure it's odd
+        integer |= 1
+
+        # Test for primeness
+        if is_prime(integer):
+            return integer
+
+        # Retry if not prime
+
+
+def are_relatively_prime(a, b):
+    '''Returns True if a and b are relatively prime, and False if they
+    are not.
+
+    >>> are_relatively_prime(2, 3)
+    1
+    >>> are_relatively_prime(2, 4)
+    0
+    '''
+
+    d = gcd(a, b)
+    return (d == 1)
+    
+if __name__ == '__main__':
+    print('Running doctests 1000x or until failure')
+    import doctest
+    
+    for count in range(1000):
+        (failures, tests) = doctest.testmod()
+        if failures:
+            break
+        
+        if count and count % 100 == 0:
+            print('%i times' % count)
+    
+    print('Doctests done')