DiffieHellman's Key exchange

Diffie-Hellman Key Exchange

Introduction

The Diffie-Hellman Key Exchange was invented in 1976 by Whitfield Diffie and Martin Hellman. It is a cryptographic protocol that allows two parties to create a shared secret over an insecure channel without directly exchanging the secret itself.

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Color Analogy for Intuitive Understanding

To understand the concept visually, imagine the process of mixing colors:

1. Shared Public Colors

• Alice and Bob publicly agree on:
  • A clear container → represents the prime number $p$.
  • Yellow paint → represents the generator $g$.
• These public colors can be seen by everyone, including an attacker.

2. Private Colors

• Alice secretly chooses red paint → her private key $a$.
• Bob secretly chooses blue paint → his private key $b$.
• These private colors are never shared.

3. Public Key Generation

• Alice mixes her red paint with the yellow paint → creating orange.
• Bob mixes his blue paint with the yellow paint → creating green.
• They exchange these mixed colors publicly.

4. Shared Secret Generation

• Alice receives Bob’s green and mixes it with her red → brown.
• Bob receives Alice’s orange and mixes it with his blue → brown.
• Both end up with the same brown color, representing the shared secret.

🔥 Why is it secure?

• An attacker can see the orange and green but cannot determine the original red and blue.
• Since color mixing cannot be easily reversed, the attacker cannot recreate the shared brown color without the original private colors.

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Mathematical Steps

1. Parameter Agreement

Alice and Bob publicly agree on:

• A large prime number $p$ → the container.
• A generator $g$ → the yellow paint.

These values are public and can be shared openly.

sequenceDiagram
    participant Alice
    participant Bob
    participant Public

    Alice->>Public: Send prime $p$ and base $g$
    Bob->>Public: Acknowledges $p$ and $g$
    Note right of Public: $p$ and $g$ are public parameters

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2. Private Key Generation

• Alice selects a private key $a$.
• Bob selects a private key $b$.
• These values remain secret and are never shared.

sequenceDiagram
    participant Alice
    participant Bob

    Alice->>Alice: Chooses private key $a$
    Bob->>Bob: Chooses private key $b$
    Note right of Alice: $a$ is kept secret
    Note left of Bob: $b$ is kept secret

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3. Public Key Calculation

• Alice computes her public key:

$$A=g^a\mathrm{mod}p$$

• Bob computes his public key:

$$B=g^b\mathrm{mod}p$$

• Both send their public keys to each other.

sequenceDiagram
    participant Alice
    participant Bob
    participant Public

    Alice->>Public: Send $A = g^a \mod p$
    Bob->>Public: Send $B = g^b \mod p$
    Note right of Public: Public keys $A$ and $B$ are exchanged

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4. Shared Secret Calculation

• Alice uses Bob’s public key $B$ and her private key $a$:

$$S=B^a\mathrm{mod}p=(g^b{)}^a\mathrm{mod}p$$

• Bob uses Alice’s public key $A$ and his private key $b$:

$$S=A^b\mathrm{mod}p=(g^a{)}^b\mathrm{mod}p$$

• Both end up with the same shared secret $S$.

sequenceDiagram
    participant Alice
    participant Bob

    Alice->>Alice: Compute $S = B^a \mod p$
    Bob->>Bob: Compute $S = A^b \mod p$
    Note right of Alice: Shared secret $S$
    Note left of Bob: Shared secret $S$

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Example with Small Numbers

Let’s go through a simple example:

• $p=23$ (public)
• $g=5$ (public)
• Alice’s private key: $a=6$
• Bob’s private key: $b=15$

Public Key Generation:

• Alice computes:

$$A=5^6\mathrm{mod}23=8$$

• Bob computes:

$$B=5^{15}\mathrm{mod}23=19$$

Shared Secret Calculation:

• Alice computes:

$$S=19^6\mathrm{mod}23=2$$

• Bob computes:

$$S=8^{15}\mathrm{mod}23=2$$

Thus, Alice and Bob now share the secret key $S=2$, which is never directly exchanged.

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Key Takeaway

The Diffie-Hellman Key Exchange enables two parties to securely generate a shared secret over an insecure channel.

• The security relies on the difficulty of solving the discrete logarithm problem.
• Even if an attacker intercepts all public information, they cannot determine the shared secret without solving the discrete logarithm, which is infeasible with large prime numbers.