





"""simple docstring"""




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

a						: Optional[int]     	=			{
    # 1536-bit
    5: {
        '''prime''': int(
            '''FFFFFFFFFFFFFFFFC90FDAA22168C234C4C6628B80DC1CD1'''
            + '''29024E088A67CC74020BBEA63B139B22514A08798E3404DD'''
            + '''EF9519B3CD3A431B302B0A6DF25F14374FE1356D6D51C245'''
            + '''E485B576625E7EC6F44C42E9A637ED6B0BFF5CB6F406B7ED'''
            + '''EE386BFB5A899FA5AE9F24117C4B1FE649286651ECE45B3D'''
            + '''C2007CB8A163BF0598DA48361C55D39A69163FA8FD24CF5F'''
            + '''83655D23DCA3AD961C62F356208552BB9ED529077096966D'''
            + '''670C354E4ABC9804F1746C08CA237327FFFFFFFFFFFFFFFF''',
            base=16,
        ),
        '''generator''': 2,
    },
    # 2048-bit
    14: {
        '''prime''': int(
            '''FFFFFFFFFFFFFFFFC90FDAA22168C234C4C6628B80DC1CD1'''
            + '''29024E088A67CC74020BBEA63B139B22514A08798E3404DD'''
            + '''EF9519B3CD3A431B302B0A6DF25F14374FE1356D6D51C245'''
            + '''E485B576625E7EC6F44C42E9A637ED6B0BFF5CB6F406B7ED'''
            + '''EE386BFB5A899FA5AE9F24117C4B1FE649286651ECE45B3D'''
            + '''C2007CB8A163BF0598DA48361C55D39A69163FA8FD24CF5F'''
            + '''83655D23DCA3AD961C62F356208552BB9ED529077096966D'''
            + '''670C354E4ABC9804F1746C08CA18217C32905E462E36CE3B'''
            + '''E39E772C180E86039B2783A2EC07A28FB5C55DF06F4C52C9'''
            + '''DE2BCBF6955817183995497CEA956AE515D2261898FA0510'''
            + '''15728E5A8AACAA68FFFFFFFFFFFFFFFF''',
            base=16,
        ),
        '''generator''': 2,
    },
    # 3072-bit
    15: {
        '''prime''': int(
            '''FFFFFFFFFFFFFFFFC90FDAA22168C234C4C6628B80DC1CD1'''
            + '''29024E088A67CC74020BBEA63B139B22514A08798E3404DD'''
            + '''EF9519B3CD3A431B302B0A6DF25F14374FE1356D6D51C245'''
            + '''E485B576625E7EC6F44C42E9A637ED6B0BFF5CB6F406B7ED'''
            + '''EE386BFB5A899FA5AE9F24117C4B1FE649286651ECE45B3D'''
            + '''C2007CB8A163BF0598DA48361C55D39A69163FA8FD24CF5F'''
            + '''83655D23DCA3AD961C62F356208552BB9ED529077096966D'''
            + '''670C354E4ABC9804F1746C08CA18217C32905E462E36CE3B'''
            + '''E39E772C180E86039B2783A2EC07A28FB5C55DF06F4C52C9'''
            + '''DE2BCBF6955817183995497CEA956AE515D2261898FA0510'''
            + '''15728E5A8AAAC42DAD33170D04507A33A85521ABDF1CBA64'''
            + '''ECFB850458DBEF0A8AEA71575D060C7DB3970F85A6E1E4C7'''
            + '''ABF5AE8CDB0933D71E8C94E04A25619DCEE3D2261AD2EE6B'''
            + '''F12FFA06D98A0864D87602733EC86A64521F2B18177B200C'''
            + '''BBE117577A615D6C770988C0BAD946E208E24FA074E5AB31'''
            + '''43DB5BFCE0FD108E4B82D120A93AD2CAFFFFFFFFFFFFFFFF''',
            base=16,
        ),
        '''generator''': 2,
    },
    # 4096-bit
    16: {
        '''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=16,
        ),
        '''generator''': 2,
    },
    # 6144-bit
    17: {
        '''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=16,
        ),
        '''generator''': 2,
    },
    # 8192-bit
    18: {
        '''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=16,
        ),
        '''generator''': 2,
    },
}


class        __UpperCamelCase				:






							def __init__(   self      ,					lowerCAmelCase__ = 14      )	-> None:
								if group not in primes:
									raise ValueError("Unsupported Group"      )
								a      :		Tuple	=				primes[group]["prime"]
								a      :		Dict	=				primes[group]["generator"]

								a      :		Dict	=				int(hexlify(urandom(32      )      )      ,					base=16      )






							def     __a   (   self      )	-> str:
								return hex(self.__private_key      )[2:]






							def     __a   (   self      )	-> str:
								a      :		List[Any]	=				pow(self.generator      ,					self.__private_key      ,					self.prime      )
								return hex(lowerCAmelCase__      )[2:]






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






							def     __a   (   self      ,					lowerCAmelCase__      )	-> str:
								a      :		Tuple	=				int(lowerCAmelCase__      ,					base=16      )
								if not self.is_valid_public_key(lowerCAmelCase__      ):
									raise ValueError("Invalid public key"      )
								a      :		Any	=				pow(lowerCAmelCase__      ,					self.__private_key      ,					self.prime      )
								return shaaaa(str(lowerCAmelCase__      ).encode()      ).hexdigest()






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






							@staticmethod
							def     __a   (   lowerCAmelCase__      ,					lowerCAmelCase__      ,					lowerCAmelCase__ = 14      )	-> str:
								a      :		Dict	=				int(lowerCAmelCase__      ,					base=16      )
								a      :		Any	=				int(lowerCAmelCase__      ,					base=16      )
								a      :		Dict	=				primes[group]["prime"]
								if not DiffieHellman.is_valid_public_key_static(lowerCAmelCase__      ,					lowerCAmelCase__      ):
									raise ValueError("Invalid public key"      )
								a      :		int	=				pow(lowerCAmelCase__      ,					lowerCAmelCase__      ,					lowerCAmelCase__      )
								return shaaaa(str(lowerCAmelCase__      ).encode()      ).hexdigest()


if __name__ == "__main__":
		import doctest

		doctest.testmod()









"""simple docstring"""




import heapq as hq
import math
from collections.abc import Iterator


class        __UpperCamelCase				:






							def __init__(   self      ,					lowerCAmelCase__      )	-> Optional[int]:
								a      :		List[str]	=				str(id_      )
								a      :		Optional[Any]	=				None
								a      :		Tuple	=				None
								a      :		str	=				[]
								a      :		Any	=				{}  # {vertex:distance}






							def __lt__(   self      ,					lowerCAmelCase__      )	-> Any:
								return self.key < other.key






							def __repr__(   self      )	-> Optional[Any]:
								return self.id






							def     __a   (   self      ,					lowerCAmelCase__      )	-> Any:
								self.neighbors.append(lowerCAmelCase__      )






							def     __a   (   self      ,					lowerCAmelCase__      ,					lowerCAmelCase__      )	-> Tuple:
								a      :		Optional[Any]	=				weight






def       _SCREAMING_SNAKE_CASE		(	_lowercase						:    int    ,       _lowercase						:    int    ,       _lowercase						:    int    ,       _lowercase						:    Union[str, Any]					) ->str:






	'''simple docstring'''






	graph[a - 1].add_neighbor(graph[b - 1]					)
	graph[b - 1].add_neighbor(graph[a - 1]					)
	# add the edges:
	graph[a - 1].add_edge(graph[b - 1]    ,       _lowercase					)
	graph[b - 1].add_edge(graph[a - 1]    ,       _lowercase					)






def       _SCREAMING_SNAKE_CASE		(	_lowercase						:    list    ,       _lowercase						:    Vertex					) ->list:






	'''simple docstring'''






	a      :		int	=				[]
	for u in graph:
		a      :		List[str]	=				math.inf
		a      :		int	=				None
	a      :		str	=				0
	a      :		Union[str, Any]	=				graph[:]
	while q:
		a      :		List[Any]	=				min(_lowercase					)
		q.remove(_lowercase					)
		for v in u.neighbors:
			if (v in q) and (u.edges[v.id] < v.key):
				a      :		List[Any]	=				u
				a      :		Optional[int]	=				u.edges[v.id]
	for i in range(1    ,       len(_lowercase					)					):
		a.append((int(graph[i].id					) + 1, int(graph[i].pi.id					) + 1)					)
	return a






def       _SCREAMING_SNAKE_CASE		(	_lowercase						:    list    ,       _lowercase						:    Vertex					) ->Iterator[tuple]:






	'''simple docstring'''






	for u in graph:
		a      :		str	=				math.inf
		a      :		Dict	=				None
	a      :		Dict	=				0

	a      :		List[Any]	=				list(_lowercase					)
	hq.heapify(_lowercase					)

	while h:
		a      :		Dict	=				hq.heappop(_lowercase					)
		for v in u.neighbors:
			if (v in h) and (u.edges[v.id] < v.key):
				a      :		Dict	=				u
				a      :		Optional[Any]	=				u.edges[v.id]
				hq.heapify(_lowercase					)

	for i in range(1    ,       len(_lowercase					)					):
		yield (int(graph[i].id					) + 1, int(graph[i].pi.id					) + 1)






def       _SCREAMING_SNAKE_CASE		(	) ->None:






	'''simple docstring'''








if __name__ == "__main__":
		import doctest

		doctest.testmod()


