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.venv/Lib/site-packages/rsa/prime.py

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+#  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
+#
+#      https://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.
+"""
+
+import rsa.common
+import rsa.randnum
+
+__all__ = ["getprime", "are_relatively_prime"]
+
+
+def gcd(p: int, q: int) -> int:
+    """Returns the greatest common divisor of p and q
+
+    >>> gcd(48, 180)
+    12
+    """
+
+    while q != 0:
+        (p, q) = (q, p % q)
+    return p
+
+
+def get_primality_testing_rounds(number: int) -> int:
+    """Returns minimum number of rounds for Miller-Rabing primality testing,
+    based on number bitsize.
+
+    According to NIST FIPS 186-4, Appendix C, Table C.3, minimum number of
+    rounds of M-R testing, using an error probability of 2 ** (-100), for
+    different p, q bitsizes are:
+      * p, q bitsize: 512; rounds: 7
+      * p, q bitsize: 1024; rounds: 4
+      * p, q bitsize: 1536; rounds: 3
+    See: http://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf
+    """
+
+    # Calculate number bitsize.
+    bitsize = rsa.common.bit_size(number)
+    # Set number of rounds.
+    if bitsize >= 1536:
+        return 3
+    if bitsize >= 1024:
+        return 4
+    if bitsize >= 512:
+        return 7
+    # For smaller bitsizes, set arbitrary number of rounds.
+    return 10
+
+
+def miller_rabin_primality_testing(n: int, k: int) -> bool:
+    """Calculates whether n is composite (which is always correct) or prime
+    (which theoretically is incorrect with error probability 4**-k), by
+    applying Miller-Rabin primality testing.
+
+    For reference and implementation example, see:
+    https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test
+
+    :param n: Integer to be tested for primality.
+    :type n: int
+    :param k: Number of rounds (witnesses) of Miller-Rabin testing.
+    :type k: int
+    :return: False if the number is composite, True if it's probably prime.
+    :rtype: bool
+    """
+
+    # prevent potential infinite loop when d = 0
+    if n < 2:
+        return False
+
+    # Decompose (n - 1) to write it as (2 ** r) * d
+    # While d is even, divide it by 2 and increase the exponent.
+    d = n - 1
+    r = 0
+
+    while not (d & 1):
+        r += 1
+        d >>= 1
+
+    # Test k witnesses.
+    for _ in range(k):
+        # Generate random integer a, where 2 <= a <= (n - 2)
+        a = rsa.randnum.randint(n - 3) + 1
+
+        x = pow(a, d, n)
+        if x == 1 or x == n - 1:
+            continue
+
+        for _ in range(r - 1):
+            x = pow(x, 2, n)
+            if x == 1:
+                # n is composite.
+                return False
+            if x == n - 1:
+                # Exit inner loop and continue with next witness.
+                break
+        else:
+            # If loop doesn't break, n is composite.
+            return False
+
+    return True
+
+
+def is_prime(number: int) -> bool:
+    """Returns True if the number is prime, and False otherwise.
+
+    >>> is_prime(2)
+    True
+    >>> is_prime(42)
+    False
+    >>> is_prime(41)
+    True
+    """
+
+    # Check for small numbers.
+    if number < 10:
+        return number in {2, 3, 5, 7}
+
+    # Check for even numbers.
+    if not (number & 1):
+        return False
+
+    # Calculate minimum number of rounds.
+    k = get_primality_testing_rounds(number)
+
+    # Run primality testing with (minimum + 1) rounds.
+    return miller_rabin_primality_testing(number, k + 1)
+
+
+def getprime(nbits: int) -> int:
+    """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
+    """
+
+    assert nbits > 3  # the loop will hang on too small numbers
+
+    while True:
+        integer = rsa.randnum.read_random_odd_int(nbits)
+
+        # Test for primeness
+        if is_prime(integer):
+            return integer
+
+            # Retry if not prime
+
+
+def are_relatively_prime(a: int, b: int) -> bool:
+    """Returns True if a and b are relatively prime, and False if they
+    are not.
+
+    >>> are_relatively_prime(2, 3)
+    True
+    >>> are_relatively_prime(2, 4)
+    False
+    """
+
+    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 % 100 == 0 and count:
+            print("%i times" % count)
+
+    print("Doctests done")