Diffie-Hellman Key Exchange

I. Overview link

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flowchart LR
    A["Public channel\n(insecure channel)"] -- "Key derivation via discrete logarithm" --> B["Shared secret key"]
    style A fill:#f9f9f9,stroke:#333,stroke-width:3px
    style B fill:#e1f5fe,stroke:#01579b,stroke-width:3px

Definition: A cryptographic algorithm that, based on the computational difficulty of the discrete logarithm problem, allows communicating parties to derive a common secret key over a public channel without ever sharing their private keys.

Key Features:
( Solves the Key Distribution Problem ) A session key can be derived securely without needing to share a key in advance
( The First Public-Key-Based Scheme ) Published in 1976, it was the first public-key cryptographic algorithm and the starting point of modern cryptography
( Based on the Discrete Logarithm ) Relies on the mathematical principle that reversing exponentiation over a finite field is computationally hard

II. Mechanism & Components link

A. Key Exchange Process (Alice & Bob) link

sequenceDiagram
    participant A as "Alice"
    participant B as "Bob"

    Note over A,B: "① Agree on common parameters (public)<br/>choose prime p, generator g"
    A->>A: "② Generate private key a (secret)<br/>compute A = g^a mod p"
    B->>B: "② Generate private key b (secret)<br/>compute B = g^b mod p"
    A->>B: "③ Send public value A"
    B->>A: "③ Send public value B"
    A->>A: "④ Compute K = B^a mod p"
    B->>B: "④ Compute K = A^b mod p"
    Note over A,B: "Shared secret key K = g^ab mod p is derived"

Step-by-step description:

1. Agree on common parameters: A large prime p and a generator g are chosen publicly
2. Generate private keys: Alice generates a random private key a, and Bob generates b (both secret)
3. Compute and exchange public values:
   • Alice computes A = g^a mod p and sends it to Bob
   • Bob computes B = g^b mod p and sends it to Alice
4. Derive the shared secret key:
   • Alice computes K = B^a mod p = (g^b)^a mod p
   • Bob computes K = A^b mod p = (g^a)^b mod p
   • As a result, both sides hold the identical K = g^ab mod p

B. Security Characteristics link

III. Advanced Topics & Comparison link

flowchart LR
    DH["Basic DH\n(Diffie-Hellman)"] -->|"Key reuse problem"| DHE["DHE\n(DH Ephemeral)\none-time key per session"]
    DHE -->|"Computation cost problem"| ECDHE["ECDHE\n(Elliptic Curve DHE)\nshort keys · fast computation"]

    DH -.->|"Vulnerable to MitM"| WARN["No authentication\n→ must be combined with certificates (PKI)"]
    ECDHE -->|"Modern TLS standard"| TLS["TLS 1.3\ndefault key exchange algorithm"]

Key point: ECDHE delivers both high security strength and PFS with a short key length (256-bit ≈ RSA 3072-bit), which is why it was adopted as the default key exchange method in TLS 1.3.