Python/blockchain/modular_division.py

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# Modular Division :
# An efficient algorithm for dividing b by a modulo n.
# GCD ( Greatest Common Divisor ) or HCF ( Highest Common Factor )
# Given three integers a, b, and n, such that gcd(a,n)=1 and n>1, the algorithm should
# return an integer x such that 0≤x≤n1, and b/a=x(modn) (that is, b=ax(modn)).
# Theorem:
# a has a multiplicative inverse modulo n iff gcd(a,n) = 1
# This find x = b*a^(-1) mod n
# Uses ExtendedEuclid to find the inverse of a
def modular_division(a, b, n):
"""
>>> modular_division(4,8,5)
2
>>> modular_division(3,8,5)
1
>>> modular_division(4, 11, 5)
4
"""
assert n > 1 and a > 0 and greatest_common_divisor(a, n) == 1
(d, t, s) = extended_gcd(n, a) # Implemented below
x = (b * s) % n
return x
# This function find the inverses of a i.e., a^(-1)
def invert_modulo(a, n):
"""
>>> invert_modulo(2, 5)
3
>>> invert_modulo(8,7)
1
"""
(b, x) = extended_euclid(a, n) # Implemented below
if b < 0:
b = (b % n + n) % n
return b
# ------------------ Finding Modular division using invert_modulo -------------------
# This function used the above inversion of a to find x = (b*a^(-1))mod n
def modular_division2(a, b, n):
"""
>>> modular_division2(4,8,5)
2
>>> modular_division2(3,8,5)
1
>>> modular_division2(4, 11, 5)
4
"""
s = invert_modulo(a, n)
x = (b * s) % n
return x
# Extended Euclid's Algorithm : If d divides a and b and d = a*x + b*y for integers x
# and y, then d = gcd(a,b)
def extended_gcd(a, b):
"""
>>> extended_gcd(10, 6)
(2, -1, 2)
>>> extended_gcd(7, 5)
(1, -2, 3)
** extended_gcd function is used when d = gcd(a,b) is required in output
"""
assert a >= 0 and b >= 0
if b == 0:
d, x, y = a, 1, 0
else:
(d, p, q) = extended_gcd(b, a % b)
x = q
y = p - q * (a // b)
assert a % d == 0 and b % d == 0
assert d == a * x + b * y
return (d, x, y)
# Extended Euclid
def extended_euclid(a, b):
"""
>>> extended_euclid(10, 6)
(-1, 2)
>>> extended_euclid(7, 5)
(-2, 3)
"""
if b == 0:
return (1, 0)
(x, y) = extended_euclid(b, a % b)
k = a // b
return (y, x - k * y)
# Euclid's Lemma : d divides a and b, if and only if d divides a-b and b
# Euclid's Algorithm
def greatest_common_divisor(a, b):
"""
>>> greatest_common_divisor(7,5)
1
Note : In number theory, two integers a and b are said to be relatively prime,
mutually prime, or co-prime if the only positive integer (factor) that divides
both of them is 1 i.e., gcd(a,b) = 1.
>>> greatest_common_divisor(121, 11)
11
"""
if a < b:
a, b = b, a
while a % b != 0:
a, b = b, a % b
return b
if __name__ == "__main__":
from doctest import testmod
testmod(name="modular_division", verbose=True)
testmod(name="modular_division2", verbose=True)
testmod(name="invert_modulo", verbose=True)
testmod(name="extended_gcd", verbose=True)
testmod(name="extended_euclid", verbose=True)
testmod(name="greatest_common_divisor", verbose=True)