What is modular exponentiation?

Modular exponentiation computes a^b mod m — the remainder when a raised to the power b is divided by m. It is the fundamental operation in RSA, Diffie-Hellman, and many other public-key systems.

Why use the square-and-multiply algorithm?

Naively multiplying a by itself b times requires O(b) multiplications. Square-and-multiply only needs O(log b) steps by processing the binary representation of the exponent, making very large exponents practical.

What are the valid input ranges?

All three values must be positive integers. The modulus m must be at least 2. This calculator uses JavaScript BigInt, so arbitrarily large values work without precision loss.

How is this used in RSA?

RSA encryption is c = m^e mod n and decryption is m = c^d mod n. Both are modular exponentiations, which is exactly what this calculator computes.

What does each column in the step table mean?

Step = iteration number. Bit = current binary digit of the exponent (MSB first). Square = running result after squaring mod m. Multiply = result after multiplying by the base (only when bit = 1). Result = value carried into the next step.

Modular Exponentiation Calculator

Enter base, exponent, and modulus to compute a^b mod m with step-by-step square-and-multiply work shown.

Inputs

Base (a)

Exponent (b)

Modulus (m)

Quick examples

Result

2¹⁰ mod 1000

Answer

Exponent bits: 1010 Steps: 4

Square-and-Multiply Steps

MSB first

Step	Bit	Operation	Running result

How square-and-multiply works

Convert the exponent to binary. Start with result = 1. For each bit from most-significant to least-significant: always square the result mod m; if the bit is 1, also multiply by the base mod m. This keeps numbers small at every step and runs in O(log b) multiplications.

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Summary

This tool computes a^b mod m using the square-and-multiply (binary exponentiation) algorithm — the same method used inside RSA and Diffie-Hellman. Enter any positive integers for base, exponent, and modulus, and the calculator returns the result instantly while showing every intermediate squaring and multiplication step so you can follow the algorithm exactly.

How it works

Enter the base (a), exponent (b), and modulus (m) in the input fields.
The calculator converts the exponent to binary.
It iterates each bit using square-and-multiply: square the running result, then multiply by the base when the bit is 1.
Every intermediate step is shown so you can follow the algorithm.
The final result is the value of a^b mod m.

Use cases

Verify RSA encryption/decryption steps by hand.
Understand Diffie-Hellman public key derivation.
Check modular exponentiation homework problems.
Learn the square-and-multiply algorithm for cryptography courses.
Debug custom RSA or ElGamal implementations.
Quickly compute large power-mod values without a full CAS tool.

Frequently Asked Questions

Last updated: 2026-10-05 · Reviewed by Nham Vu