What is modular exponentiation?

Modular exponentiation is the operation of computing base^exponent mod modulus — raising a number to a power and taking the remainder when divided by the modulus. It is the core operation in public-key cryptography algorithms such as RSA and Diffie-Hellman.

Why is the square-and-multiply algorithm used?

Naive exponentiation (multiplying the base by itself exponent times) is impractical for large exponents. Square-and-multiply reduces the number of operations to O(log n) by squaring the running total and multiplying only when the corresponding bit of the exponent is 1.

How large can the numbers be?

All arithmetic is done with JavaScript BigInt, so there is no fixed upper limit — numbers with hundreds of digits are handled correctly. Very large inputs may generate a long step table; the trace is capped at 200 rows to keep the page responsive.

What happens when the modulus is 1?

Any integer mod 1 equals 0, so the result is always 0 when modulus is 1. This edge case is handled and displayed correctly.

Can the exponent be zero?

Yes. By convention, any non-zero base raised to the power 0 equals 1, so base^0 mod modulus = 1 mod modulus (which is 0 when modulus is 1, otherwise 1).

Is this tool safe for cryptographic use?

This tool is intended for learning and verification. All computation runs locally in your browser with no data sent to any server. For production cryptographic systems, use a vetted library.

Modular Exponentiation Calculator

Calculate base^exponent mod modulus instantly using the fast square-and-multiply algorithm, with a full step-by-step breakdown.

Inputs

Base

Exponent

Modulus

Quick Examples

Enter values and click Calculate

Step-by-Step Trace

Step	Bit	Operation	Current value (mod n)

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Summary

This modular exponentiation calculator computes base^exponent mod modulus using the binary square-and-multiply (fast exponentiation) method. It supports arbitrarily large integers through JavaScript BigInt, making it suitable for cryptographic problems, RSA key verification, and number theory exercises. Enter any three non-negative integers to get the result and a detailed trace capped at the first 200 steps.

How it works

Enter the base, exponent, and modulus in the input fields.
Click "Calculate" (or press Enter) to run the algorithm.
The calculator converts the exponent to binary and processes each bit.
For each bit, the current value is squared and, if the bit is 1, multiplied by the base — all mod the modulus.
The step-by-step trace table shows up to the first 200 intermediate results.
The final result appears at the top alongside the total step count.

Use cases

Verify RSA encryption and decryption results by hand.
Check Diffie-Hellman key exchange intermediate values.
Solve discrete logarithm and modular arithmetic homework problems.
Validate Miller-Rabin primality test computations.
Explore Euler's theorem and Fermat's little theorem examples.
Debug cryptographic implementations during development.
Teaching aid for university number theory and cryptography courses.
Quick sanity-check for large-exponent modular arithmetic.

Frequently Asked Questions

Last updated: 2026-05-29 · Reviewed by Nham Vu