Diffie-Hellman Key Calculator

How it is calculated

You supply the public prime p, the generator g and two private exponents a (Alice) and b (Bob). The tool computes each public value by fast modular exponentiation (square-and-multiply): A = ga mod p and B = gb mod p. Then each side raises the other's public value to its own private exponent: s = Ba mod p for Alice and Ab mod p for Bob. Both must give the same number, and the page shows both so you can check. All four inputs must be whole numbers below 2⁵³, with p above 1, g above 0 and a, b not negative. The modulus p is tested for primality (Miller-Rabin); if p is not prime, a decimal is entered, or any value is out of range, the tool shows no result. Products are computed without overflow, so large primes up to about 9 quadrillion work.

Common mistakes

• Choosing a composite modulus such as p = 24. The tool checks that p is prime and returns nothing otherwise, because the exchange is not meaningful modulo a composite number.
• Entering a decimal exponent such as 6.5, or a negative one. Only whole, non-negative private keys are accepted, so no result appears.
• Swapping the private and public values. a and b are the secret exponents; the public values A and B are outputs, not inputs.
• Picking a generator that gives a trivial exchange: g = 1 makes every public value and the shared secret equal to 1. A useful g generates many residues modulo p, so check it against p rather than choosing at random.
• Expecting real security from the result. Practical Diffie-Hellman uses primes thousands of bits long, so a calculator limited to numbers below 2⁵³ is for learning and hand-checking only.