
"""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

lowerCamelCase__ :       Dict						     =			{
    # 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 				lowercase__:
	'''simple docstring'''

	def __init__(						self    :Optional[Any]		, lowerCamelCase_    :int = 14					)							->			None:







			'''simple docstring'''




			if group not in primes:
					raise ValueError('''Unsupported Group'''					)
			SCREAMING_SNAKE_CASE       :     Tuple				= primes[group]['''prime''']
			SCREAMING_SNAKE_CASE       :     Any				= primes[group]['''generator''']

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





	def   __lowerCAmelCase (						self    :Optional[Any]					)							->			str:







			'''simple docstring'''




			return hex(self.__private_key					)[2:]





	def   __lowerCAmelCase (						self    :Optional[Any]					)							->			str:







			'''simple docstring'''




			SCREAMING_SNAKE_CASE       :     Any				= pow(self.generator		, self.__private_key		, self.prime					)
			return hex(lowerCamelCase_					)[2:]





	def   __lowerCAmelCase (						self    :List[Any]		, lowerCamelCase_    :int					)							->			bool:







			'''simple docstring'''




			return (
			    2 <= key <= self.prime - 2
			    and pow(lowerCamelCase_		, (self.prime - 1) // 2		, self.prime					) == 1
			)





	def   __lowerCAmelCase (						self    :List[str]		, lowerCamelCase_    :str					)							->			str:







			'''simple docstring'''




			SCREAMING_SNAKE_CASE       :     Tuple				= int(lowerCamelCase_		, base=16					)
			if not self.is_valid_public_key(lowerCamelCase_					):
					raise ValueError('''Invalid public key'''					)
			SCREAMING_SNAKE_CASE       :     Union[str, Any]				= pow(lowerCamelCase_		, self.__private_key		, self.prime					)
			return shaaaa(str(lowerCamelCase_					).encode()					).hexdigest()





	@staticmethod
	def   __lowerCAmelCase (						lowerCamelCase_    :int		, lowerCamelCase_    :int					)							->			bool:







			'''simple docstring'''




			return (
			    2 <= remote_public_key_str <= prime - 2
			    and pow(lowerCamelCase_		, (prime - 1) // 2		, lowerCamelCase_					) == 1
			)







	@staticmethod
	def   __lowerCAmelCase (						lowerCamelCase_    :str		, lowerCamelCase_    :str		, lowerCamelCase_    :int = 14					)							->			str:







			'''simple docstring'''




			SCREAMING_SNAKE_CASE       :     List[str]				= int(lowerCamelCase_		, base=16					)
			SCREAMING_SNAKE_CASE       :     Union[str, Any]				= int(lowerCamelCase_		, base=16					)
			SCREAMING_SNAKE_CASE       :     Union[str, Any]				= primes[group]['''prime''']
			if not DiffieHellman.is_valid_public_key_static(lowerCamelCase_		, lowerCamelCase_					):
					raise ValueError('''Invalid public key'''					)
			SCREAMING_SNAKE_CASE       :     Tuple				= pow(lowerCamelCase_		, lowerCamelCase_		, lowerCamelCase_					)
			return shaaaa(str(lowerCamelCase_					).encode()					).hexdigest()


if __name__ == "__main__":
					import doctest

					doctest.testmod()







"""simple docstring"""






from __future__ import annotations

from numpy import array, cos, cross, floataa, radians, sin
from numpy.typing import NDArray





def __A						(   a_ :							float		,		a_ :							float		,		a_ :							bool = False     )->							list[float]:




		'''simple docstring'''






		if radian_mode:
				return [magnitude * cos(a_     ), magnitude * sin(a_     )]
		return [magnitude * cos(radians(a_     )     ), magnitude * sin(radians(a_     )     )]





def __A						(   a_ :							NDArray[floataa]		,		a_ :							NDArray[floataa]		,		a_ :							float = 10**-1     )->							bool:




		'''simple docstring'''






		SCREAMING_SNAKE_CASE       :     NDArray[floataa]				= cross(a_		,		a_     )
		SCREAMING_SNAKE_CASE       :     float				= sum(a_     )
		return abs(a_     ) < eps


if __name__ == "__main__":
					# Test to check if it works
					lowerCamelCase__ :       Optional[Any]						     =			array(
					    [
					        polar_force(7_1_8.4, 180 - 30),
					        polar_force(8_7_9.5_4, 45),
					        polar_force(100, -90),
					    ]
					)

					lowerCamelCase__ :       NDArray[floataa]						     =			array([[0, 0], [0, 0], [0, 0]])

					assert in_static_equilibrium(forces, location)

					# Problem 1 in image_data/2D_problems.jpg
					lowerCamelCase__ :       Union[str, Any]						     =			array(
					    [
					        polar_force(30 * 9.8_1, 15),
					        polar_force(215, 180 - 45),
					        polar_force(264, 90 - 30),
					    ]
					)

					lowerCamelCase__ :       Any						     =			array([[0, 0], [0, 0], [0, 0]])

					assert in_static_equilibrium(forces, location)

					# Problem in image_data/2D_problems_1.jpg
					lowerCamelCase__ :       Union[str, Any]						     =			array([[0, -2000], [0, -1200], [0, 15600], [0, -12400]])

					lowerCamelCase__ :       Optional[int]						     =			array([[0, 0], [6, 0], [10, 0], [12, 0]])

					assert in_static_equilibrium(forces, location)

					import doctest

					doctest.testmod()





