How does it handle huge exponents without overflowing?

It uses the square-and-multiply method (binary exponentiation), reducing modulo the modulus at every step. The intermediate values never exceed the modulus squared, so even thousand-bit exponents compute quickly and exactly.

What is modular exponentiation used for?

It is the core operation of RSA, Diffie-Hellman and many other public-key cryptography schemes, as well as primality testing and hashing.

What is 4^13 mod 497?

4^13 mod 497 = 445. The calculator reaches this without ever computing the full 67-million-digit-free value of 4^13, by reducing modulo 497 at each step.

What inputs are allowed?

Three non-negative whole numbers — base, exponent and modulus — with the modulus greater than 0. Arbitrary-precision integers mean there is no practical size limit.

Why not just compute base^exponent and then take the modulus?

For cryptographic sizes, base^exponent has thousands of digits and is far too large to store. Square-and-multiply keeps every intermediate value below the modulus squared, so it stays fast and exact.

Is anything sent to a server?

No — the calculation runs entirely in your browser using arbitrary-precision integers, and nothing is uploaded.

Modular Exponentiation Calculator

This calculator computes (base ^ exponent) mod modulus — the core operation of RSA, Diffie-Hellman and other public-key cryptography. It is built for students and developers who need an exact result for large inputs that ordinary calculators cannot handle.

How it works

It uses the square-and-multiply (binary exponentiation) algorithm with arbitrary-precision integers. The exponent is read bit by bit: at each step the running base is squared, and whenever the current bit is 1 the result is multiplied in. Crucially, the value is reduced modulo the modulus after every multiplication, so intermediate numbers never exceed the modulus squared. This means even thousand-bit exponents compute almost instantly and exactly.

Example

To compute 4¹³ mod 497:

The exponent 13 is 1101 in binary. Square-and-multiply walks those bits, reducing mod 497 each step, giving 445 — without ever forming the full 8-digit value of 4¹³ (67,108,864).

base	exponent	modulus	result
4	13	497	445
7	256	13	9
2	10	1000	24

All arithmetic uses arbitrary-precision integers and runs locally in your browser; nothing is uploaded.