
from binascii import hexlify
from hashlib import shaaaa
from os import urandom

# RFC 3526 - More Modular Exponential (MODP) Diffie-Hellman groups for
# Internet Key Exchange (IKE) https://tools.ietf.org/html/rfc3526

_SCREAMING_SNAKE_CASE    :      Union[str, Any]          =			{
    # 1536-bit
    5: {
        'prime': int(
            'FFFFFFFFFFFFFFFFC90FDAA22168C234C4C6628B80DC1CD1'
            + '29024E088A67CC74020BBEA63B139B22514A08798E3404DD'
            + 'EF9519B3CD3A431B302B0A6DF25F14374FE1356D6D51C245'
            + 'E485B576625E7EC6F44C42E9A637ED6B0BFF5CB6F406B7ED'
            + 'EE386BFB5A899FA5AE9F24117C4B1FE649286651ECE45B3D'
            + 'C2007CB8A163BF0598DA48361C55D39A69163FA8FD24CF5F'
            + '83655D23DCA3AD961C62F356208552BB9ED529077096966D'
            + '670C354E4ABC9804F1746C08CA237327FFFFFFFFFFFFFFFF',
            base=1_6,
        ),
        'generator': 2,
    },
    # 2048-bit
    1_4: {
        'prime': int(
            'FFFFFFFFFFFFFFFFC90FDAA22168C234C4C6628B80DC1CD1'
            + '29024E088A67CC74020BBEA63B139B22514A08798E3404DD'
            + 'EF9519B3CD3A431B302B0A6DF25F14374FE1356D6D51C245'
            + 'E485B576625E7EC6F44C42E9A637ED6B0BFF5CB6F406B7ED'
            + 'EE386BFB5A899FA5AE9F24117C4B1FE649286651ECE45B3D'
            + 'C2007CB8A163BF0598DA48361C55D39A69163FA8FD24CF5F'
            + '83655D23DCA3AD961C62F356208552BB9ED529077096966D'
            + '670C354E4ABC9804F1746C08CA18217C32905E462E36CE3B'
            + 'E39E772C180E86039B2783A2EC07A28FB5C55DF06F4C52C9'
            + 'DE2BCBF6955817183995497CEA956AE515D2261898FA0510'
            + '15728E5A8AACAA68FFFFFFFFFFFFFFFF',
            base=1_6,
        ),
        'generator': 2,
    },
    # 3072-bit
    1_5: {
        'prime': int(
            'FFFFFFFFFFFFFFFFC90FDAA22168C234C4C6628B80DC1CD1'
            + '29024E088A67CC74020BBEA63B139B22514A08798E3404DD'
            + 'EF9519B3CD3A431B302B0A6DF25F14374FE1356D6D51C245'
            + 'E485B576625E7EC6F44C42E9A637ED6B0BFF5CB6F406B7ED'
            + 'EE386BFB5A899FA5AE9F24117C4B1FE649286651ECE45B3D'
            + 'C2007CB8A163BF0598DA48361C55D39A69163FA8FD24CF5F'
            + '83655D23DCA3AD961C62F356208552BB9ED529077096966D'
            + '670C354E4ABC9804F1746C08CA18217C32905E462E36CE3B'
            + 'E39E772C180E86039B2783A2EC07A28FB5C55DF06F4C52C9'
            + 'DE2BCBF6955817183995497CEA956AE515D2261898FA0510'
            + '15728E5A8AAAC42DAD33170D04507A33A85521ABDF1CBA64'
            + 'ECFB850458DBEF0A8AEA71575D060C7DB3970F85A6E1E4C7'
            + 'ABF5AE8CDB0933D71E8C94E04A25619DCEE3D2261AD2EE6B'
            + 'F12FFA06D98A0864D87602733EC86A64521F2B18177B200C'
            + 'BBE117577A615D6C770988C0BAD946E208E24FA074E5AB31'
            + '43DB5BFCE0FD108E4B82D120A93AD2CAFFFFFFFFFFFFFFFF',
            base=1_6,
        ),
        'generator': 2,
    },
    # 4096-bit
    1_6: {
        'prime': int(
            'FFFFFFFFFFFFFFFFC90FDAA22168C234C4C6628B80DC1CD1'
            + '29024E088A67CC74020BBEA63B139B22514A08798E3404DD'
            + 'EF9519B3CD3A431B302B0A6DF25F14374FE1356D6D51C245'
            + 'E485B576625E7EC6F44C42E9A637ED6B0BFF5CB6F406B7ED'
            + 'EE386BFB5A899FA5AE9F24117C4B1FE649286651ECE45B3D'
            + 'C2007CB8A163BF0598DA48361C55D39A69163FA8FD24CF5F'
            + '83655D23DCA3AD961C62F356208552BB9ED529077096966D'
            + '670C354E4ABC9804F1746C08CA18217C32905E462E36CE3B'
            + 'E39E772C180E86039B2783A2EC07A28FB5C55DF06F4C52C9'
            + 'DE2BCBF6955817183995497CEA956AE515D2261898FA0510'
            + '15728E5A8AAAC42DAD33170D04507A33A85521ABDF1CBA64'
            + 'ECFB850458DBEF0A8AEA71575D060C7DB3970F85A6E1E4C7'
            + 'ABF5AE8CDB0933D71E8C94E04A25619DCEE3D2261AD2EE6B'
            + 'F12FFA06D98A0864D87602733EC86A64521F2B18177B200C'
            + 'BBE117577A615D6C770988C0BAD946E208E24FA074E5AB31'
            + '43DB5BFCE0FD108E4B82D120A92108011A723C12A787E6D7'
            + '88719A10BDBA5B2699C327186AF4E23C1A946834B6150BDA'
            + '2583E9CA2AD44CE8DBBBC2DB04DE8EF92E8EFC141FBECAA6'
            + '287C59474E6BC05D99B2964FA090C3A2233BA186515BE7ED'
            + '1F612970CEE2D7AFB81BDD762170481CD0069127D5B05AA9'
            + '93B4EA988D8FDDC186FFB7DC90A6C08F4DF435C934063199'
            + 'FFFFFFFFFFFFFFFF',
            base=1_6,
        ),
        'generator': 2,
    },
    # 6144-bit
    1_7: {
        'prime': int(
            'FFFFFFFFFFFFFFFFC90FDAA22168C234C4C6628B80DC1CD129024E08'
            + '8A67CC74020BBEA63B139B22514A08798E3404DDEF9519B3CD3A431B'
            + '302B0A6DF25F14374FE1356D6D51C245E485B576625E7EC6F44C42E9'
            + 'A637ED6B0BFF5CB6F406B7EDEE386BFB5A899FA5AE9F24117C4B1FE6'
            + '49286651ECE45B3DC2007CB8A163BF0598DA48361C55D39A69163FA8'
            + 'FD24CF5F83655D23DCA3AD961C62F356208552BB9ED529077096966D'
            + '670C354E4ABC9804F1746C08CA18217C32905E462E36CE3BE39E772C'
            + '180E86039B2783A2EC07A28FB5C55DF06F4C52C9DE2BCBF695581718'
            + '3995497CEA956AE515D2261898FA051015728E5A8AAAC42DAD33170D'
            + '04507A33A85521ABDF1CBA64ECFB850458DBEF0A8AEA71575D060C7D'
            + 'B3970F85A6E1E4C7ABF5AE8CDB0933D71E8C94E04A25619DCEE3D226'
            + '1AD2EE6BF12FFA06D98A0864D87602733EC86A64521F2B18177B200C'
            + 'BBE117577A615D6C770988C0BAD946E208E24FA074E5AB3143DB5BFC'
            + 'E0FD108E4B82D120A92108011A723C12A787E6D788719A10BDBA5B26'
            + '99C327186AF4E23C1A946834B6150BDA2583E9CA2AD44CE8DBBBC2DB'
            + '04DE8EF92E8EFC141FBECAA6287C59474E6BC05D99B2964FA090C3A2'
            + '233BA186515BE7ED1F612970CEE2D7AFB81BDD762170481CD0069127'
            + 'D5B05AA993B4EA988D8FDDC186FFB7DC90A6C08F4DF435C934028492'
            + '36C3FAB4D27C7026C1D4DCB2602646DEC9751E763DBA37BDF8FF9406'
            + 'AD9E530EE5DB382F413001AEB06A53ED9027D831179727B0865A8918'
            + 'DA3EDBEBCF9B14ED44CE6CBACED4BB1BDB7F1447E6CC254B33205151'
            + '2BD7AF426FB8F401378CD2BF5983CA01C64B92ECF032EA15D1721D03'
            + 'F482D7CE6E74FEF6D55E702F46980C82B5A84031900B1C9E59E7C97F'
            + 'BEC7E8F323A97A7E36CC88BE0F1D45B7FF585AC54BD407B22B4154AA'
            + 'CC8F6D7EBF48E1D814CC5ED20F8037E0A79715EEF29BE32806A1D58B'
            + 'B7C5DA76F550AA3D8A1FBFF0EB19CCB1A313D55CDA56C9EC2EF29632'
            + '387FE8D76E3C0468043E8F663F4860EE12BF2D5B0B7474D6E694F91E'
            + '6DCC4024FFFFFFFFFFFFFFFF',
            base=1_6,
        ),
        'generator': 2,
    },
    # 8192-bit
    1_8: {
        'prime': int(
            'FFFFFFFFFFFFFFFFC90FDAA22168C234C4C6628B80DC1CD1'
            + '29024E088A67CC74020BBEA63B139B22514A08798E3404DD'
            + 'EF9519B3CD3A431B302B0A6DF25F14374FE1356D6D51C245'
            + 'E485B576625E7EC6F44C42E9A637ED6B0BFF5CB6F406B7ED'
            + 'EE386BFB5A899FA5AE9F24117C4B1FE649286651ECE45B3D'
            + 'C2007CB8A163BF0598DA48361C55D39A69163FA8FD24CF5F'
            + '83655D23DCA3AD961C62F356208552BB9ED529077096966D'
            + '670C354E4ABC9804F1746C08CA18217C32905E462E36CE3B'
            + 'E39E772C180E86039B2783A2EC07A28FB5C55DF06F4C52C9'
            + 'DE2BCBF6955817183995497CEA956AE515D2261898FA0510'
            + '15728E5A8AAAC42DAD33170D04507A33A85521ABDF1CBA64'
            + 'ECFB850458DBEF0A8AEA71575D060C7DB3970F85A6E1E4C7'
            + 'ABF5AE8CDB0933D71E8C94E04A25619DCEE3D2261AD2EE6B'
            + 'F12FFA06D98A0864D87602733EC86A64521F2B18177B200C'
            + 'BBE117577A615D6C770988C0BAD946E208E24FA074E5AB31'
            + '43DB5BFCE0FD108E4B82D120A92108011A723C12A787E6D7'
            + '88719A10BDBA5B2699C327186AF4E23C1A946834B6150BDA'
            + '2583E9CA2AD44CE8DBBBC2DB04DE8EF92E8EFC141FBECAA6'
            + '287C59474E6BC05D99B2964FA090C3A2233BA186515BE7ED'
            + '1F612970CEE2D7AFB81BDD762170481CD0069127D5B05AA9'
            + '93B4EA988D8FDDC186FFB7DC90A6C08F4DF435C934028492'
            + '36C3FAB4D27C7026C1D4DCB2602646DEC9751E763DBA37BD'
            + 'F8FF9406AD9E530EE5DB382F413001AEB06A53ED9027D831'
            + '179727B0865A8918DA3EDBEBCF9B14ED44CE6CBACED4BB1B'
            + 'DB7F1447E6CC254B332051512BD7AF426FB8F401378CD2BF'
            + '5983CA01C64B92ECF032EA15D1721D03F482D7CE6E74FEF6'
            + 'D55E702F46980C82B5A84031900B1C9E59E7C97FBEC7E8F3'
            + '23A97A7E36CC88BE0F1D45B7FF585AC54BD407B22B4154AA'
            + 'CC8F6D7EBF48E1D814CC5ED20F8037E0A79715EEF29BE328'
            + '06A1D58BB7C5DA76F550AA3D8A1FBFF0EB19CCB1A313D55C'
            + 'DA56C9EC2EF29632387FE8D76E3C0468043E8F663F4860EE'
            + '12BF2D5B0B7474D6E694F91E6DBE115974A3926F12FEE5E4'
            + '38777CB6A932DF8CD8BEC4D073B931BA3BC832B68D9DD300'
            + '741FA7BF8AFC47ED2576F6936BA424663AAB639C5AE4F568'
            + '3423B4742BF1C978238F16CBE39D652DE3FDB8BEFC848AD9'
            + '22222E04A4037C0713EB57A81A23F0C73473FC646CEA306B'
            + '4BCBC8862F8385DDFA9D4B7FA2C087E879683303ED5BDD3A'
            + '062B3CF5B3A278A66D2A13F83F44F82DDF310EE074AB6A36'
            + '4597E899A0255DC164F31CC50846851DF9AB48195DED7EA1'
            + 'B1D510BD7EE74D73FAF36BC31ECFA268359046F4EB879F92'
            + '4009438B481C6CD7889A002ED5EE382BC9190DA6FC026E47'
            + '9558E4475677E9AA9E3050E2765694DFC81F56E880B96E71'
            + '60C980DD98EDD3DFFFFFFFFFFFFFFFFF',
            base=1_6,
        ),
        'generator': 2,
    },
}







class  UpperCamelCase__		:







	'''simple docstring'''





	def __init__(			self				,       UpperCamelCase__ = 14		):
					if group not in primes:
									raise ValueError('''Unsupported Group'''		)
					A__  :	List[str]         =							primes[group]['''prime''']
					A__  :	Tuple         =							primes[group]['''generator''']

					A__  :	Optional[int]         =							int(hexlify(urandom(32		)		)				,       base=16		)





	def   __snake_case				(			self		):
					return hex(self.__private_key		)[2:]





	def   __snake_case				(			self		):
					A__  :	Optional[Any]         =							pow(self.generator				,       self.__private_key				,       self.prime		)
					return hex(UpperCamelCase__		)[2:]





	def   __snake_case				(			self				,       UpperCamelCase__		):
					# check if the other public key is valid based on NIST SP800-56
					return (
					    2 <= key <= self.prime - 2
					    and pow(UpperCamelCase__				,       (self.prime - 1) // 2				,       self.prime		) == 1
					)





	def   __snake_case				(			self				,       UpperCamelCase__		):
					A__  :	str         =							int(UpperCamelCase__				,       base=16		)
					if not self.is_valid_public_key(UpperCamelCase__		):
									raise ValueError('''Invalid public key'''		)
					A__  :	List[Any]         =							pow(UpperCamelCase__				,       self.__private_key				,       self.prime		)
					return shaaaa(str(UpperCamelCase__		).encode()		).hexdigest()





	@staticmethod
	def   __snake_case				(			UpperCamelCase__				,       UpperCamelCase__		):
					# check if the other public key is valid based on NIST SP800-56
					return (
					    2 <= remote_public_key_str <= prime - 2
					    and pow(UpperCamelCase__				,       (prime - 1) // 2				,       UpperCamelCase__		) == 1
					)





	@staticmethod
	def   __snake_case				(			UpperCamelCase__				,       UpperCamelCase__				,       UpperCamelCase__ = 14		):
					A__  :	Optional[Any]         =							int(UpperCamelCase__				,       base=16		)
					A__  :	Any         =							int(UpperCamelCase__				,       base=16		)
					A__  :	int         =							primes[group]['''prime''']
					if not DiffieHellman.is_valid_public_key_static(UpperCamelCase__				,       UpperCamelCase__		):
									raise ValueError('''Invalid public key'''		)
					A__  :	Union[str, Any]         =							pow(UpperCamelCase__				,       UpperCamelCase__				,       UpperCamelCase__		)
					return shaaaa(str(UpperCamelCase__		).encode()		).hexdigest()


if __name__ == "__main__":
					import doctest

					doctest.testmod()

def 				SCREAMING_SNAKE_CASE     (      __UpperCamelCase :						int       ) ->       List[Any]:

				"""simple docstring"""







				A__  :	Optional[Any]         =							0
				A__  :	Optional[Any]         =							len(__UpperCamelCase       )

				for i in range(n - 1       ):
								for j in range(i + 1							,						__UpperCamelCase       ):
												if arr[i] > arr[j]:
																num_inversions += 1

				return num_inversions

def 				SCREAMING_SNAKE_CASE     (      __UpperCamelCase :						int       ) ->       Tuple:

				"""simple docstring"""







				if len(__UpperCamelCase       ) <= 1:
								return arr, 0
				A__  :	Optional[int]         =							len(__UpperCamelCase       ) // 2
				A__  :	List[str]         =							arr[0:mid]
				A__  :	Union[str, Any]         =							arr[mid:]

				A__       ,							A__  :	List[Any]         =							count_inversions_recursive(__UpperCamelCase       )
				A__       ,							A__  :	int         =							count_inversions_recursive(__UpperCamelCase       )
				A__       ,							A__  :	Dict         =							_count_cross_inversions(__UpperCamelCase							,						__UpperCamelCase       )

				A__  :	Any         =							inversion_p + inversions_q + cross_inversions
				return c, num_inversions

def 				SCREAMING_SNAKE_CASE     (      __UpperCamelCase :						List[str]							,						__UpperCamelCase :						List[Any]       ) ->       Dict:

				"""simple docstring"""







				A__  :	str         =							[]
				A__  :	Tuple         =							0
				while i < len(__UpperCamelCase       ) and j < len(__UpperCamelCase       ):
								if p[i] > q[j]:
												# if P[1] > Q[j], then P[k] > Q[k] for all  i < k <= len(P)
												# These are all inversions. The claim emerges from the
												# property that P is sorted.
												num_inversion += len(__UpperCamelCase       ) - i
												r.append(q[j]       )
												j += 1
								else:
												r.append(p[i]       )
												i += 1

				if i < len(__UpperCamelCase       ):
								r.extend(p[i:]       )
				else:
								r.extend(q[j:]       )

				return r, num_inversion

def 				SCREAMING_SNAKE_CASE     (      ) ->       Tuple:

				"""simple docstring"""







				A__  :	List[str]         =							[10, 2, 1, 5, 5, 2, 11]

				# this arr has 8 inversions:
				# (10, 2), (10, 1), (10, 5), (10, 5), (10, 2), (2, 1), (5, 2), (5, 2)

				A__  :	int         =							count_inversions_bf(__UpperCamelCase       )
				A__       ,							A__  :	int         =							count_inversions_recursive(__UpperCamelCase       )

				assert num_inversions_bf == num_inversions_recursive == 8

				print('''number of inversions = '''							,						__UpperCamelCase       )

				# testing an array with zero inversion (a sorted arr_1)

				arr_a.sort()
				A__  :	Optional[Any]         =							count_inversions_bf(__UpperCamelCase       )
				A__       ,							A__  :	Dict         =							count_inversions_recursive(__UpperCamelCase       )

				assert num_inversions_bf == num_inversions_recursive == 0
				print('''number of inversions = '''							,						__UpperCamelCase       )

				# an empty list should also have zero inversions
				A__  :	Union[str, Any]         =							[]
				A__  :	Union[str, Any]         =							count_inversions_bf(__UpperCamelCase       )
				A__       ,							A__  :	Any         =							count_inversions_recursive(__UpperCamelCase       )

				assert num_inversions_bf == num_inversions_recursive == 0
				print('''number of inversions = '''							,						__UpperCamelCase       )


if __name__ == "__main__":
					main()