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Cryptology Academy · Lesson

Diffie-Hellman Key Exchange Math

Step through the DH protocol and the discrete-log hardness assumption.

Lesson 2 of 413 steps

Diffie-Hellman Key Exchange Math is a free Cryptology Academy lesson on CoddyKit. This is lesson 2 of 4. You can read the complete lesson below for free — then practise it hands-on in the browser with a built-in code editor and a 24/7 AI tutor. It is part of the Cryptology Academy learning path, and your progress syncs across the web and the CoddyKit app. The Cryptology Academy course includes 4 lessons in total.

Welcome

In this lesson we trace the Diffie-Hellman key exchange step by step, from the public parameters through Alice and Bob's private exponents to the shared secret.

Public Parameters

Alice and Bob agree publicly on: - p: a large prime (2048+ bits in production) - g: a generator of Z*p (usually g=2 or g=5) Eve can see p and g. Knowing them does NOT break the protocol.

Alice's Private Key

Alice generates a random secret: a (1 < a < p-1) Alice computes her public value: A = g^a mod p Alice sends A to Bob. Eve sees A. Eve cannot find a from A — discrete logarithm problem.

Bob's Private Key

Bob generates a random secret: b (1 < b < p-1) Bob computes his public value: B = g^b mod p Bob sends B to Alice. Eve sees B. Eve cannot find b from B.

Shared Secret Computation

Alice: S = B^a mod p = (g^b)^a mod p = g^(ab) mod p Bob: S = A^b mod p = (g^a)^b mod p = g^(ab) mod p Both compute the same S = g^(ab) mod p without ever transmitting it.

Small Number Example

p=23, g=5 Alice: a=6, A = 5^6 mod 23 = 8 Bob: b=15, B = 5^15 mod 23 = 19 Alice: S = 19^6 mod 23 = 2 Bob: S = 8^15 mod 23 = 2 ✓ Eve sees: 23, 5, 8, 19 but cannot find 2 easily.

Discrete Logarithm Problem

Given A = g^a mod p, find a. For large p (2048 bits), the best known algorithm (Number Field Sieve) takes sub-exponential but still astronomical time: ~2^100 operations for 2048-bit DH.

Man-in-the-Middle Attack

DH is vulnerable to MITM: Mallory intercepts A and B, substitutes her own values. She shares one secret with Alice, another with Bob. Solution: authenticate the exchange (certificates or pre-shared auth).

Safe Primes

p must be chosen carefully. A safe prime p=2q+1 (where q is prime) ensures the DH group has no small subgroups that could be exploited in small-subgroup attacks. NIST specifies approved DH groups.

DH Parameter Sizes

NIST recommendations: - 2048-bit: minimum for new deployments - 3072-bit: equivalent to RSA-3072 - 4096-bit: high security Never use pre-generated Oakley Group 1/2 (768/1024-bit) — broken by state actors (Logjam attack).

Logjam Attack (2015)

Logjam showed many TLS servers used the same 1024-bit DH prime. State actors could precompute the discrete log for these specific primes, allowing real-time decryption. Solution: use 2048-bit+ unique primes.

Quick Check

In DH with p=23, g=5, a=4, b=3: what is the shared secret?
Answer it in the interactive lesson

Recap

DH key exchange math is clear. Next we apply the same concept on elliptic curves to get ECDH — same security, much smaller keys.
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Frequently Asked Questions

Is the “Diffie-Hellman Key Exchange Math” lesson free?

Yes — the full text of “Diffie-Hellman Key Exchange Math” is free to read here on the web. To practise it interactively (a built-in code editor and a 24/7 AI tutor) and unlock the rest of the Cryptology Academy course, upgrade to CoddyKit PRO. The Cryptology Academy course includes 4 lessons in total.

What will I learn in “Diffie-Hellman Key Exchange Math”?

Step through the DH protocol and the discrete-log hardness assumption. You practise Cryptology Academy with hands-on code you run directly in the browser, and a 24/7 AI tutor answers your questions as you work through the lesson.

Do I need any experience to start Cryptology Academy?

No prior experience is required. Cryptology Academy on CoddyKit is structured for beginners through advanced learners, so you can start here or from the beginning and move at your own pace. This is lesson 2 of 4.

How long does the “Diffie-Hellman Key Exchange Math” lesson take?

Most CoddyKit lessons take about 5–10 minutes. Each one is bite-sized and interactive, so you make steady progress and pick up exactly where you left off across the web and the app.

Can I write and run code in this Cryptology Academy lesson?

Yes. Every Cryptology Academy lesson includes a built-in code editor, so you write and run real code right in your browser and get instant AI feedback — no local setup required.

All lessons in this course

  1. The Key Distribution Problem
  2. Diffie-Hellman Key Exchange Math
  3. ECDH: Elliptic Curves for Key Exchange
  4. Forward Secrecy & Ephemeral Key Exchange
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