Modular Exponentiation Calculator (aᵇ mod m)

For large exponents a^b has astronomically many digits and cannot be formed in practice. Square-and-multiply keeps every intermediate value below m², so it stays fast and exact regardless of the exponent's size.

It is the number x such that a·x ≡ 1 (mod m). It exists only when the base and modulus share no common factor (gcd = 1); otherwise the tool shows that no inverse exists. This is the value used when 'dividing' in modular arithmetic.

a^0 mod m is 1 for any base (and 0 when m = 1). Exponents must be non-negative whole numbers; the base may be negative and is normalised into the range 0…m−1 first.

No. The arithmetic is correct, but a real implementation also has to be constant-time: a straightforward square-and-multiply takes a different amount of time depending on the bits of the exponent, and that timing difference has been used to recover private keys from live systems. Use this to understand and to check results, and use a vetted library for anything that touches a secret.

Reducing modulo m at every step, so the numbers never grow past the size of the modulus, and squaring rather than multiplying repeatedly — which turns an exponent of a thousand bits into about a thousand squarings instead of an astronomically long chain. That is why the exponent's size affects the time roughly linearly while its value does not appear anywhere in the cost.