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155 lines
3.4 KiB
Python
155 lines
3.4 KiB
Python
from typing import Tuple
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def modular_division(a: int, b: int, n: int) -> int:
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"""
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Modular Division :
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An efficient algorithm for dividing b by a modulo n.
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GCD ( Greatest Common Divisor ) or HCF ( Highest Common Factor )
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Given three integers a, b, and n, such that gcd(a,n)=1 and n>1, the algorithm should
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return an integer x such that 0≤x≤n−1, and b/a=x(modn) (that is, b=ax(modn)).
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Theorem:
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a has a multiplicative inverse modulo n iff gcd(a,n) = 1
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This find x = b*a^(-1) mod n
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Uses ExtendedEuclid to find the inverse of a
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>>> modular_division(4,8,5)
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2
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>>> modular_division(3,8,5)
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1
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>>> modular_division(4, 11, 5)
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4
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"""
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assert n > 1 and a > 0 and greatest_common_divisor(a, n) == 1
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(d, t, s) = extended_gcd(n, a) # Implemented below
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x = (b * s) % n
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return x
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def invert_modulo(a: int, n: int) -> int:
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"""
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This function find the inverses of a i.e., a^(-1)
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>>> invert_modulo(2, 5)
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3
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>>> invert_modulo(8,7)
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1
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"""
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(b, x) = extended_euclid(a, n) # Implemented below
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if b < 0:
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b = (b % n + n) % n
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return b
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# ------------------ Finding Modular division using invert_modulo -------------------
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def modular_division2(a: int, b: int, n: int) -> int:
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"""
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This function used the above inversion of a to find x = (b*a^(-1))mod n
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>>> modular_division2(4,8,5)
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2
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>>> modular_division2(3,8,5)
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1
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>>> modular_division2(4, 11, 5)
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4
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"""
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s = invert_modulo(a, n)
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x = (b * s) % n
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return x
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def extended_gcd(a: int, b: int) -> Tuple[int, int, int]:
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"""
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Extended Euclid's Algorithm : If d divides a and b and d = a*x + b*y for integers x
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and y, then d = gcd(a,b)
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>>> extended_gcd(10, 6)
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(2, -1, 2)
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>>> extended_gcd(7, 5)
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(1, -2, 3)
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** extended_gcd function is used when d = gcd(a,b) is required in output
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"""
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assert a >= 0 and b >= 0
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if b == 0:
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d, x, y = a, 1, 0
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else:
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(d, p, q) = extended_gcd(b, a % b)
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x = q
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y = p - q * (a // b)
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assert a % d == 0 and b % d == 0
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assert d == a * x + b * y
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return (d, x, y)
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def extended_euclid(a: int, b: int) -> Tuple[int, int]:
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"""
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Extended Euclid
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>>> extended_euclid(10, 6)
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(-1, 2)
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>>> extended_euclid(7, 5)
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(-2, 3)
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"""
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if b == 0:
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return (1, 0)
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(x, y) = extended_euclid(b, a % b)
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k = a // b
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return (y, x - k * y)
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def greatest_common_divisor(a: int, b: int) -> int:
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"""
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Euclid's Lemma : d divides a and b, if and only if d divides a-b and b
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Euclid's Algorithm
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>>> greatest_common_divisor(7,5)
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1
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Note : In number theory, two integers a and b are said to be relatively prime,
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mutually prime, or co-prime if the only positive integer (factor) that divides
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both of them is 1 i.e., gcd(a,b) = 1.
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>>> greatest_common_divisor(121, 11)
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11
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"""
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if a < b:
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a, b = b, a
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while a % b != 0:
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a, b = b, a % b
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return b
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if __name__ == "__main__":
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from doctest import testmod
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testmod(name="modular_division", verbose=True)
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testmod(name="modular_division2", verbose=True)
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testmod(name="invert_modulo", verbose=True)
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testmod(name="extended_gcd", verbose=True)
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testmod(name="extended_euclid", verbose=True)
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testmod(name="greatest_common_divisor", verbose=True)
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