






from __future__ import annotations

from math import gcd






def   UpperCamelCase  (       snake_case__     :   int	,snake_case__     :   int = 2	,snake_case__     :   int = 1	,snake_case__     :   int = 3	,):




     '''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(snake_case__	,2  ) + step) % modulus

     for _ in range(snake_case__  ):
          # These track the position within the cycle detection logic.
          __snake_case			:Optional[Any]          =						seed
          __snake_case			:Tuple          =						seed

          while True:
               # At each iteration, the tortoise moves one step and the hare moves two.
               __snake_case			:str          =						rand_fn(snake_case__	,snake_case__	,snake_case__  )
               __snake_case			:Optional[Any]          =						rand_fn(snake_case__	,snake_case__	,snake_case__  )
               __snake_case			:Tuple          =						rand_fn(snake_case__	,snake_case__	,snake_case__  )

               # 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``.
               __snake_case			:Dict          =						gcd(hare - tortoise	,snake_case__  )

               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.
          __snake_case			:Any          =						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

    lowerCamelCase__							=     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""",
    )
    lowerCamelCase__							=     parser.parse_args()

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










def   UpperCamelCase  (       snake_case__     :   str	,snake_case__     :   int  ):




     '''simple docstring'''






     __snake_case			:list[list[str]]          =						[[] for _ in range(snake_case__  )]
     __snake_case			:Union[str, Any]          =						key - 1

     if key <= 0:
          raise ValueError("""Height of grid can't be 0 or negative"""  )
     if key == 1 or len(snake_case__  ) <= key:
          return input_string

     for position, character in enumerate(snake_case__  ):
          __snake_case			:Any          =						position % (lowest * 2)  # puts it in bounds
          __snake_case			:Optional[int]          =						min(snake_case__	,lowest * 2 - num  )  # creates zigzag pattern
          temp_grid[num].append(snake_case__  )
     __snake_case			:List[Any]          =						["""""".join(snake_case__  ) for row in temp_grid]
     __snake_case			:Any          =						"""""".join(snake_case__  )

     return output_string






def   UpperCamelCase  (       snake_case__     :   str	,snake_case__     :   int  ):




     '''simple docstring'''






     __snake_case			:Union[str, Any]          =						[]
     __snake_case			:Optional[int]          =						key - 1

     if key <= 0:
          raise ValueError("""Height of grid can't be 0 or negative"""  )
     if key == 1:
          return input_string

     __snake_case			:list[list[str]]          =						[[] for _ in range(snake_case__  )]  # generates template
     for position in range(len(snake_case__  )  ):
          __snake_case			:List[str]          =						position % (lowest * 2)  # puts it in bounds
          __snake_case			:int          =						min(snake_case__	,lowest * 2 - num  )  # creates zigzag pattern
          temp_grid[num].append("""*"""  )

     __snake_case			:str          =						0
     for row in temp_grid:  # fills in the characters
          __snake_case			:str          =						input_string[counter : counter + len(snake_case__  )]
          grid.append(list(snake_case__  )  )
          counter += len(snake_case__  )

     __snake_case			:Any          =						""""""  # reads as zigzag
     for position in range(len(snake_case__  )  ):
          __snake_case			:Optional[int]          =						position % (lowest * 2)  # puts it in bounds
          __snake_case			:int          =						min(snake_case__	,lowest * 2 - num  )  # creates zigzag pattern
          output_string += grid[num][0]
          grid[num].pop(0  )
     return output_string






def   UpperCamelCase  (       snake_case__     :   str  ):




     '''simple docstring'''






     __snake_case			:Optional[Any]          =						{}
     for key_guess in range(1	,len(snake_case__  )  ):  # tries every key
          __snake_case			:Dict          =						decrypt(snake_case__	,snake_case__  )
     return results


if __name__ == "__main__":
     import doctest

     doctest.testmod()


