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"""

Binary Exponentiation

This is a method to find a^b in O(log b) time complexity and is one of the most commonly

used methods of exponentiation. The method is also useful for modular exponentiation,

when the solution to (a^b) % c is required.

To calculate a^b:

- If b is even, then a^b = (a * a)^(b / 2)

- If b is odd, then a^b = a * a^(b - 1)

Repeat until b = 1 or b = 0

For modular exponentiation, we use the fact that (a * b) % c = ((a % c) * (b % c)) % c

"""

def binary_exp_recursive(base: float, exponent: int) -> float:

"""

Computes a^b recursively, where a is the base and b is the exponent

>>> binary_exp_recursive(3, 5)

243

>>> binary_exp_recursive(11, 13)

34522712143931

>>> binary_exp_recursive(-1, 3)

-1

>>> binary_exp_recursive(0, 5)

>>> binary_exp_recursive(3, 1)

>>> binary_exp_recursive(3, 0)

>>> binary_exp_recursive(1.5, 4)

5.0625

>>> binary_exp_recursive(3, -1)

Traceback (most recent call last):

...

ValueError: Exponent must be a non-negative integer

"""

if exponent < 0:

raise ValueError("Exponent must be a non-negative integer")

if exponent == 0:

return 1

if exponent % 2 == 1:

return binary_exp_recursive(base, exponent - 1) * base

b = binary_exp_recursive(base, exponent // 2)

return b * b

def binary_exp_iterative(base: float, exponent: int) -> float:

"""

Computes a^b iteratively, where a is the base and b is the exponent

>>> binary_exp_iterative(3, 5)

243

>>> binary_exp_iterative(11, 13)

34522712143931

>>> binary_exp_iterative(-1, 3)

-1

>>> binary_exp_iterative(0, 5)

>>> binary_exp_iterative(3, 1)

>>> binary_exp_iterative(3, 0)

>>> binary_exp_iterative(1.5, 4)

5.0625

>>> binary_exp_iterative(3, -1)

Traceback (most recent call last):

...

ValueError: Exponent must be a non-negative integer

"""

if exponent < 0:

raise ValueError("Exponent must be a non-negative integer")

res: int | float = 1

while exponent > 0:

if exponent & 1:

res *= base

base *= base

exponent >>= 1

return res

def binary_exp_mod_recursive(base: float, exponent: int, modulus: int) -> float:

"""

Computes a^b % c recursively, where a is the base, b is the exponent, and c is the

modulus

>>> binary_exp_mod_recursive(3, 4, 5)

>>> binary_exp_mod_recursive(11, 13, 7)

>>> binary_exp_mod_recursive(1.5, 4, 3)

2.0625

>>> binary_exp_mod_recursive(7, -1, 10)

Traceback (most recent call last):

...

ValueError: Exponent must be a non-negative integer

>>> binary_exp_mod_recursive(7, 13, 0)

Traceback (most recent call last):

...

ValueError: Modulus must be a positive integer

"""

if exponent < 0:

raise ValueError("Exponent must be a non-negative integer")

if modulus <= 0:

raise ValueError("Modulus must be a positive integer")

if exponent == 0:

return 1

if exponent % 2 == 1:

return (binary_exp_mod_recursive(base, exponent - 1, modulus) * base) % modulus

r = binary_exp_mod_recursive(base, exponent // 2, modulus)

return (r * r) % modulus

def binary_exp_mod_iterative(base: float, exponent: int, modulus: int) -> float:

"""

Computes a^b % c iteratively, where a is the base, b is the exponent, and c is the

modulus

>>> binary_exp_mod_iterative(3, 4, 5)

>>> binary_exp_mod_iterative(11, 13, 7)

>>> binary_exp_mod_iterative(1.5, 4, 3)

2.0625

>>> binary_exp_mod_iterative(7, -1, 10)

Traceback (most recent call last):

...

ValueError: Exponent must be a non-negative integer

>>> binary_exp_mod_iterative(7, 13, 0)

Traceback (most recent call last):

...

ValueError: Modulus must be a positive integer

"""

if exponent < 0:

raise ValueError("Exponent must be a non-negative integer")

if modulus <= 0:

raise ValueError("Modulus must be a positive integer")

res: int | float = 1

while exponent > 0:

if exponent & 1:

res = ((res % modulus) * (base % modulus)) % modulus

base *= base

exponent >>= 1

return res

if __name__ == "__main__":

from timeit import timeit

a = 1269380576

b = 374

c = 34

runs = 100_000

print(

timeit(

f"binary_exp_recursive({a}, {b})",

setup="from __main__ import binary_exp_recursive",

number=runs,

)

print(

timeit(

f"binary_exp_iterative({a}, {b})",

setup="from __main__ import binary_exp_iterative",

number=runs,

)

print(

timeit(

f"binary_exp_mod_recursive({a}, {b}, {c})",

setup="from __main__ import binary_exp_mod_recursive",

number=runs,

)

print(

timeit(

f"binary_exp_mod_iterative({a}, {b}, {c})",

setup="from __main__ import binary_exp_mod_iterative",

number=runs,

)

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