Step by Step solution about How to resolve the algorithm Modular exponentiation step by step in the Python programming language

Explanation of the Code:

This Python code calculates the result of raising a large number a to the power of another large number b modulo a given modulus m. It provides both a solution using the built-in pow() function and a custom implementation without using built-in functions.

Built-In Power Calculation:

• pow(a, b, m) calculates the result of a raised to the power of b modulo m. It uses the fast exponentiation algorithm to efficiently compute the result while handling the modulo operation.

Custom Power Calculation (Without Built-In Functions):

• The power_mod function is a custom implementation of modulo exponentiation. It uses a loop to repeatedly square b and update the result x based on whether e (the exponent) is odd or even.

Detailed Breakdown:

• Line 1:

  • a and b are assigned large integer values.
  • m is assigned the modulus value (10^40).
  • The result of a raised to the power of b modulo m is calculated using the pow() function and printed.

• Line 6-14:

  • The power_mod function is defined to calculate the result of modulo exponentiation.

• Line 7:

  • It initializes x to 1. x will store the intermediate result.

• Line 8:

  • The while loop continues until e becomes 0.

• Line 9:

  • b is squared, and e is halved. This is the key step in fast exponentiation.

• Line 10-11:

  • If e is odd, x is updated by multiplying it with the current b. Otherwise, x bleibt unverändert.

• Line 15-18:

  • power_mod is called to calculate the result of a raised to the power of b modulo m. The result is printed.

a = 2988348162058574136915891421498819466320163312926952423791023078876139
b = 2351399303373464486466122544523690094744975233415544072992656881240319
m = 10 ** 40
print(pow(a, b, m))


def power_mod(b, e, m):
    " Without using builtin function "
    x = 1
    while e > 0:
        b, e, x = (
            b * b % m,
            e // 2,
            b * x % m if e % 2 else x
        )

    return x


a = 2988348162058574136915891421498819466320163312926952423791023078876139
b = 2351399303373464486466122544523690094744975233415544072992656881240319
m = 10 ** 40
print(power_mod(a, b, m))