Modular Exponentiation Calculator (a^b mod m)

Real-world use cases

• Hand-verify an RSA encryption or decryption step

  You are learning RSA and want to confirm a textbook example by hand. With p, q, e and d chosen, encryption is c = m^e mod n and decryption is m = c^d mod n. Drop in the message as the base, the public exponent e as the exponent, and n as the modulus, and read c directly. Then feed c back with the private exponent d and watch the original message come out. Because everything is BigInt, a realistic 1024-bit n works exactly, not just the tiny toy moduli most homework examples are limited to.

• Check a Diffie-Hellman key exchange by hand

  Diffie-Hellman has each party compute g^secret mod p. To verify a worked example, put g as the base, your secret exponent as the exponent, and the prime p as the modulus to get your public value. Then raise the other side's public value to your secret, also mod p, and confirm both sides land on the same shared key. The square-and-multiply view makes it obvious why the exchange is cheap to compute forward yet hard to reverse without the secret.

• Debug a modPow implementation in your own code

  You wrote a fast-exponentiation routine in C, Python or Go and your test vectors disagree. Paste the same base, exponent and modulus here to get the known-good answer from a BigInt reference, then turn on the step view to compare your intermediate squarings against the tool's. A common bug is forgetting to reduce mod m after the multiply step, which only shows up on large inputs, and the step list pinpoints exactly where your values diverge.

• Teach or study fast exponentiation in a number theory class

  When introducing square-and-multiply, the gap between b multiplications and log2(b) squarings is hard to feel from a formula. Set a small base and an exponent like 13 or 100, turn on the step view, and the bit-by-bit breakdown shows precisely which squares get multiplied in. Students can change the exponent and watch the step count grow with the number of bits, not with the size of the number, which is the whole insight the algorithm is built on.

Common pitfalls

• Computing a^b first and then taking mod m. For any real cryptographic size this overflows or freezes, because a^b can have more digits than the universe has atoms. Always reduce mod m at every multiply, which is exactly what square-and-multiply does and what this tool does internally.

• Expecting a negative exponent to always work. a^(-b) mod m only exists when the base is coprime to the modulus, because it needs the modular inverse. If gcd(base, m) is not 1 there is no inverse, so 4^(-1) mod 6 is undefined and the tool reports an error rather than inventing a value.

• Passing a modulus of 0 or 1 without thinking. Modulus 0 is invalid because the remainder is undefined. Modulus 1 is valid but always returns 0, since every integer is congruent to 0 mod 1, so a^b mod 1 = 0 even for a^0, which can surprise you if you expected 1.

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