Fast Modular Exponentiation Calculator

What is Fast Modular Exponentiation?

Fast modular exponentiation is an algorithm used to efficiently compute b^e mod m, where b is the base, e is the exponent, and m is the modulus. This operation is particularly useful in:

• Cryptographic algorithms (like RSA and Diffie-Hellman)
• Number theory and cryptography
• Computing large powers in efficient time
• Algorithm design and optimization

How to Use the Calculator

1. Enter the base (b) - the number to be raised to a power
2. Enter the exponent (e) - the power to raise the base to
3. Enter the modulus (m) - the number to take the remainder after exponentiation
4. Click the "Calculate" button to compute the result

Example

Suppose you want to compute 3¹⁰ mod 7:

Enter 3 as the base, 10 as the exponent, and 7 as the modulus. The calculator will show you that 3¹⁰ = 59049, and 59049 mod 7 = 4.

Working Principle

The calculator uses the fast modular exponentiation algorithm, which works by breaking down the exponentiation process into smaller steps using the binary representation of the exponent. This reduces the number of multiplications needed, making it much more efficient for large numbers:

For each bit in the exponent (starting from the least significant bit):

• square the current result
• if the current bit is 1, multiply by the base
• take the modulus at each step to keep numbers manageable