





"""simple docstring"""



from __future__ import annotations

from math import gcd







def 		__lowercase					(       snake_case_  :  int    ,snake_case_  :  int = 2    ,snake_case_  :  int = 1    ,snake_case_  :  int = 3    ,)       ->Optional[Any]:



    '''simple docstring'''







    if num < 2:
        raise ValueError('''The input value cannot be less than 2'''						)

    # Because of the relationship between ``f(f(x))`` and ``f(x)``, this
    # algorithm struggles to find factors that are divisible by two.
    # As a workaround, we specifically check for two and even inputs.
    #   See: https://math.stackexchange.com/a/2856214/165820
    if num > 2 and num % 2 == 0:
        return 2

    # Pollard's Rho algorithm requires a function that returns pseudorandom
    # values between 0 <= X < ``num``.  It doesn't need to be random in the
    # sense that the output value is cryptographically secure or difficult
    # to calculate, it only needs to be random in the sense that all output
    # values should be equally likely to appear.
    # For this reason, Pollard suggested using ``f(x) = (x**2 - 1) % num``
    # However, the success of Pollard's algorithm isn't guaranteed and is
    # determined in part by the initial seed and the chosen random function.
    # To make retries easier, we will instead use ``f(x) = (x**2 + C) % num``
    # where ``C`` is a value that we can modify between each attempt.
    def rand_fn(snake_case_  :  int    ,snake_case_  :  int    ,snake_case_  :  int						) -> int:
        return (pow(a__    ,2						) + step) % modulus

    for _ in range(a__						):
        # These track the position within the cycle detection logic.
        __A   :						Tuple									=					seed
        __A   :						Any									=					seed

        while True:
            # At each iteration, the tortoise moves one step and the hare moves two.
            __A   :						int									=					rand_fn(a__    ,a__    ,a__						)
            __A   :						int									=					rand_fn(a__    ,a__    ,a__						)
            __A   :						Any									=					rand_fn(a__    ,a__    ,a__						)

            # At some point both the tortoise and the hare will enter a cycle whose
            # length ``p`` is a divisor of ``num``.  Once in that cycle, at some point
            # the tortoise and hare will end up on the same value modulo ``p``.
            # We can detect when this happens because the position difference between
            # the tortoise and the hare will share a common divisor with ``num``.
            __A   :						List[Any]									=					gcd(hare - tortoise    ,a__						)

            if divisor == 1:
                # No common divisor yet, just keep searching.
                continue
            else:
                # We found a common divisor!
                if divisor == num:
                    # Unfortunately, the divisor is ``num`` itself and is useless.
                    break
                else:
                    # The divisor is a nontrivial factor of ``num``!
                    return divisor

        # If we made it here, then this attempt failed.
        # We need to pick a new starting seed for the tortoise and hare
        # in addition to a new step value for the random function.
        # To keep this example implementation deterministic, the
        # new values will be generated based on currently available
        # values instead of using something like ``random.randint``.

        # We can use the hare's position as the new seed.
        # This is actually what Richard Brent's the "optimized" variant does.
        __A   :						Dict									=					hare

        # The new step value for the random function can just be incremented.
        # At first the results will be similar to what the old function would
        # have produced, but the value will quickly diverge after a bit.
        step += 1

    # We haven't found a divisor within the requested number of attempts.
    # We were unlucky or ``num`` itself is actually prime.
    return None


if __name__ == "__main__":
       import argparse

       a_			=      argparse.ArgumentParser()
       parser.add_argument(
           """num""",
           type=int,
           help="""The value to find a divisor of""",
       )
       parser.add_argument(
           """--attempts""",
           type=int,
           default=3,
           help="""The number of attempts before giving up""",
       )
       a_			=      parser.parse_args()

       a_			=      pollard_rho(args.num, attempts=args.attempts)
       if divisor is None:
              print(f'''{args.num} is probably prime''')
       else:
              a_			=      args.num // divisor
              print(f'''{args.num} = {divisor} * {quotient}''')











"""simple docstring"""



import string

# frequency taken from https://en.wikipedia.org/wiki/Letter_frequency
a_			=      {
    """E""": 12.70,
    """T""": 9.06,
    """A""": 8.17,
    """O""": 7.51,
    """I""": 6.97,
    """N""": 6.75,
    """S""": 6.33,
    """H""": 6.09,
    """R""": 5.99,
    """D""": 4.25,
    """L""": 4.03,
    """C""": 2.78,
    """U""": 2.76,
    """M""": 2.41,
    """W""": 2.36,
    """F""": 2.23,
    """G""": 2.02,
    """Y""": 1.97,
    """P""": 1.93,
    """B""": 1.29,
    """V""": 0.98,
    """K""": 0.77,
    """J""": 0.15,
    """X""": 0.15,
    """Q""": 0.10,
    """Z""": 0.07,
}
a_			=      """ETAOINSHRDLCUMWFGYPBVKJXQZ"""
a_			=      """ABCDEFGHIJKLMNOPQRSTUVWXYZ"""







def 		__lowercase					(       snake_case_  :  str						)       ->dict[str, int]:



    '''simple docstring'''







    __A   :						Any									=					{letter: 0 for letter in string.ascii_uppercase}
    for letter in message.upper():
        if letter in LETTERS:
            letter_count[letter] += 1

    return letter_count







def 		__lowercase					(       snake_case_  :  tuple						)       ->str:



    '''simple docstring'''







    return x[0]







def 		__lowercase					(       snake_case_  :  str						)       ->str:



    '''simple docstring'''







    __A   :						Union[str, Any]									=					get_letter_count(snake_case_						)
    __A   :						dict[int, list[str]]									=					{
        freq: [] for letter, freq in letter_to_freq.items()
    }
    for letter in LETTERS:
        freq_to_letter[letter_to_freq[letter]].append(snake_case_						)

    __A   :						dict[int, str]									=					{}

    for freq in freq_to_letter:
        freq_to_letter[freq].sort(key=ETAOIN.find    ,reverse=snake_case_						)
        __A   :						Optional[int]									=					''''''.join(freq_to_letter[freq]						)

    __A   :						str									=					list(freq_to_letter_str.items()						)
    freq_pairs.sort(key=snake_case_    ,reverse=snake_case_						)

    __A   :						list[str]									=					[freq_pair[1] for freq_pair in freq_pairs]

    return "".join(snake_case_						)







def 		__lowercase					(       snake_case_  :  str						)       ->int:



    '''simple docstring'''







    __A   :						Any									=					get_frequency_order(snake_case_						)
    __A   :						str									=					0
    for common_letter in ETAOIN[:6]:
        if common_letter in freq_order[:6]:
            match_score += 1

    for uncommon_letter in ETAOIN[-6:]:
        if uncommon_letter in freq_order[-6:]:
            match_score += 1

    return match_score


if __name__ == "__main__":
       import doctest

       doctest.testmod()




