Modular exponentiation calculator (a^b mod m)
Enter the base, exponent and modulus to get the remainder of a^b divided by m without computing the full power. Useful for number theory and cryptography like RSA.
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How to use it
- Enter base a, exponent b and modulus m.
- The result appears straight away and updates as you edit.
- Copy the result, or copy a link that keeps the same values to share it.
How it is calculated
Square and multiply: read the bits of b from the lowest, square the base modulo m at each step, and multiply it into the result when the bit is 1.
Example
Inputs: Base a = 4, Exponent b = 13, Modulus m = 497. Result: a^b mod m: 445.
Limits and caveats
- Base and exponent run from 0 to 10^15, the modulus from 1 to 10^15.
- We take 0^0 = 1, as most number theory texts and programming languages do.
Questions about this tool
Why not compute a^b and then divide?
The full power can have billions of digits. Square and multiply keeps every step below m and needs only about log₂(b) steps.
What is 4^13 mod 497?
445, a common textbook example of the method.
How does it relate to Fermat's little theorem?
If p is prime and does not divide a, then a^(p-1) mod p = 1, for example 2^1000000006 mod 1000000007 = 1.