Modular Exponentiation Calculator

Modular exponentiation computes base raised to an exponent, then takes the remainder modulo m, written base^exponent mod m. It keeps every intermediate value small, so the answer is exact even when base^exponent would be astronomically large.

The calculator uses the square and multiply method (binary exponentiation). It squares the base and reduces modulo m at each step, so the work grows with the number of bits in the exponent rather than its size. A million-bit exponent finishes in well under a second.

It is the core operation of public-key cryptography. RSA encrypts and decrypts with c = m^e mod n and m = c^d mod n, and Diffie-Hellman key exchange raises a shared base to secret exponents mod a prime. It also appears in primality tests like Fermat and Miller-Rabin, and in hashing.

The modular inverse of a mod m is the number x in the range 0 to m-1 with a * x = 1 mod m. It only exists when a and m share no common factor (their GCD is 1). RSA uses it to derive the private exponent from the public one.

Yes, but only when the base has a modular inverse mod m. A negative exponent a^-n mod m is computed as the inverse of a, raised to the power n. If the base and modulus are not coprime, no inverse exists and the calculator reports an error.

No practical limit. The engine uses arbitrary-precision integers (BigInt), so you can enter values with hundreds or thousands of digits. Paste large numbers directly into the fields.

No. The calculator runs entirely in your browser, so your inputs never leave your device. The optional API is for developers who want to call the same math from their own code.