Modular Exponentiation Calculator

Work out modular exponentiation instantly with clear inputs, formula shown and shareable results.

About Modular Exponentiation Calculator

Repeated squaring computes bᵉ mod m in about log₂ e multiplications, reducing modulo m at each step so the numbers never grow. Computing bᵉ directly would be astronomically large.

Formula

Fast exponentiation

b^(2k) = (b^k)², b^(2k+1) = b·(b^k)², reducing mod m at every step

Worked example

With the default values below, Modular Exponentiation Calculator returns base^exponent mod m of 460. Change any input above and every figure here is recalculated in your browser.

Inputs used

Base: 7
Exponentminimum 0: 128
Modulusminimum 2: 561

Results

base^exponent mod m: 460
Squaring steps used: 8
Base reduced mod m: 7
Digits the direct power would have: 108

Step by step

1Values used
2Fast exponentiation
3base^exponent mod m= 460
4Squaring steps used= 8
5Base reduced mod m= 7
6Digits the direct power would have= 108

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to whole numbers, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

What is 7¹²⁸ mod 561?

460. Since 561 = 3 × 11 × 17, the residues 1, 9 and 1 combine by the Chinese remainder theorem to give 460 — reached in just eight squarings.

Why does cryptography need this?

RSA and Diffie-Hellman raise numbers to exponents with hundreds of digits; only modular exponentiation makes that feasible.