ModPow Calculator

Number a (Base)
Number b (Exponent)
Number n (Modulus)

Description

Modular Exponentiation: In computing ( a^b mod n ), a method that efficiently avoids the immense computation of ( a^b ) is utilized. In this operation, the exponent ( b ) can be extremely large, making direct calculation of ( a^b ) impractical for computers. The goal of modular exponentiation is to compute ( a^b mod n ) effectively, meaning to compute the result of raising ( a ) to the power of ( b ) and then taking the modulus ( n ).
A common approach involves using the Fast Exponentiation algorithm (Exponentiation by Squaring), which achieves computation in O(log b) time complexity. The basic idea of this algorithm is to leverage the binary representation of the exponent to reduce the number of multiplication operations, thereby improving efficiency. The steps are as follows:
  • 1. Convert the exponent ( b ) into its binary form.
  • 2. Process each bit from the highest to the lowest: If the current bit is 1, multiply by the current base ( a ). After each multiplication, take the modulus ( n ) to prevent overflow. Right shift the exponent (i.e., divide by 2), square the base.
This method allows for the computation of ( a^b mod n ) in a short amount of time, suitable for scenarios involving large numbers and exponents, such as in cryptographic algorithms like RSA and Diffie-Hellman key exchange.