Advanced Diffie Hellman Calculator

Calculator Input

Enter a prime modulus, a generator, and private exponents for Alice and Bob. The calculator returns public keys, shared secrets, and validation notes.

Prime modulus (p)

Use a positive prime number greater than 2.

Generator (g)

For standard use, choose 1 < g < p.

Alice private key (a)

Keep this value secret in a real exchange.

Bob private key (b)

Use a different private key for Bob.

Alice public key formula

Computed after submission.

Bob public key formula

Computed after submission.

Example data table

Prime modulus (p)	Generator (g)	Alice private key (a)	Bob private key (b)	Alice public key (A)	Bob public key (B)	Shared secret
23	5	6	15	8	19	2
29	2	11	7	18	12	17
31	3	9	14	29	14	16

Formula used

Public key for Alice: A = g^a mod p

Public key for Bob: B = g^b mod p

Shared secret for Alice: S = B^a mod p

Shared secret for Bob: S = A^b mod p = g^ab mod p

Diffie Hellman relies on modular exponentiation. Alice and Bob publish A and B, then each side raises the received public value using its own private exponent.

Because B^a mod p and A^b mod p both reduce to g^ab mod p, both parties arrive at the same secret.

For standard classroom style setups, use a prime modulus p and a suitable generator g. A primitive root gives the widest nonzero residue coverage.

How to use this calculator

Enter a prime modulus p.
Choose a generator g smaller than p.
Enter Alice private key a.
Enter Bob private key b.
Press Submit to compute the exchange.
Review public keys, shared secret, and configuration checks.
Use the chart to inspect modular cycles.
Download the result as CSV or PDF.

Frequently asked questions

1. What does this calculator compute?

It computes Alice and Bob public keys, both shared secret values, generator checks, modulus status, multiplicative order, and a modular exponentiation graph.

2. Why should the modulus be prime?

A prime modulus supports the standard finite field version of Diffie Hellman. It makes generator analysis cleaner and helps create predictable subgroup behavior.

3. What is a primitive root in this context?

A primitive root is a generator whose powers produce every nonzero residue modulo a prime. It gives the largest possible cycle for the exchange.

4. Why do both parties get the same secret?

Alice computes B^a mod p and Bob computes A^b mod p. Both expressions equal g^ab mod p, so the final secret matches.

5. Can I use large real-world cryptographic values here?

This page is designed for educational and moderate integer examples. Very large production scale parameters usually need specialized big integer libraries and hardened implementations.

6. What does multiplicative order mean?

It is the smallest positive exponent k for which g^k mod p equals 1. It describes the cycle length generated by g.

7. Why is the graph useful?

The graph shows how powers of the generator move through modular residues. It helps visualize cycles, repeats, and whether the chosen generator spreads broadly.

8. What should I keep private in a real exchange?

The private exponents must remain secret. Only the public keys are shared. Revealing the private values breaks the confidentiality of the derived secret.