1. What is Modular Exponentiation?

Modular exponentiation is a type of exponentiation performed over a modulus. It is particularly useful in computer science, especially in the field of cryptography.

2. How Does the Calculator Work?

The calculator uses the modular exponentiation formula:

Where:

• \( base \) — The base number
• \( exp \) — The exponent
• \( modulus \) — The modulus (must be positive)

Explanation: The calculation raises the base to the power of exponent, then finds the remainder when divided by the modulus.

3. Applications of Modular Exponentiation

Details: Modular exponentiation is widely used in public-key cryptography systems like RSA, Diffie-Hellman key exchange, and in primality testing algorithms.

4. Using the Calculator

Tips: Enter the base, exponent, and modulus values. The modulus must be a positive number. The calculator will compute the result of (base^exp) mod modulus.

5. Frequently Asked Questions (FAQ)

Q1: Why is modular exponentiation important in cryptography?
A: It allows computation of very large exponents modulo a number efficiently, which is fundamental to many encryption algorithms.

Q2: What happens if the modulus is 1?
A: The result will always be 0 since any number modulo 1 is 0.

Q3: Can this calculator handle negative exponents?
A: The current implementation doesn't support negative exponents as they would require modular inverses.

Q4: Is there a limit to the size of numbers I can input?
A: The calculator is limited by PHP's floating point number handling. For very large numbers, specialized algorithms would be needed.

Q5: How is this different from regular exponentiation?
A: Regular exponentiation gives the full result, while modular exponentiation gives only the remainder after division by the modulus.