Diffie-Hellman Key Exchange Calculator

What is the Diffie-Hellman key exchange?

Diffie–Hellman, published in 1976, lets two people agree on a secret number over a public channel without ever sending the secret itself. Alice and Bob publicly agree on a prime p and a base g. Alice chooses a private number a and sends A = g^a mod p; Bob chooses b and sends B = g^b mod p.

Alice then computes B^a mod p and Bob computes A^b mod p. Both equal g^ab mod p, so they now share a secret that an eavesdropper, who sees only p, g, A and B, cannot easily compute.

Why it is secure (and when it is not)

To find the secret, an attacker would need a from A = g^a mod p. This is the discrete logarithm problem, which has no known fast algorithm for well-chosen large primes. With the tiny numbers used in lessons, however, the attacker can simply try every exponent. Real deployments use primes of at least 2048 bits or elliptic curves, and they authenticate the messages, because the plain exchange does not protect against a man in the middle who impersonates both parties.

Worked example

Take p = 23 and g = 5, with a = 6 and b = 15. Alice sends A = 5⁶ mod 23 = 8 and Bob sends B = 5¹⁵ mod 23 = 19. Alice computes 19⁶ mod 23 = 2 and Bob computes 8¹⁵ mod 23 = 2. The shared secret is 2. Choosing g as a primitive root of p, as 5 is for 23, means the powers of g reach every non-zero residue, which gives the largest possible space of secrets.

What the calculator checks

The calculator verifies that p is prime, that g and both secret numbers lie in the valid range 2 ≤ g ≤ p − 2 and 1 ≤ a, b ≤ p − 2, and that the two computed secrets agree. For primes up to 10¹⁵ it also computes the order of g and tells you whether g is a primitive root, which is the standard textbook requirement.