Diffie-Hellman Key Exchange

Diffie-Hellman key exchange is a method that allows two parties to securely share a secret key over a public channel. This shared secret key can be used for encrypted communication using Symmetric Cryptography.

Key exchange process

1. Public parameters
   • Two numbers are agreed upon by both, which is known in the public:
     • A large prime number $n$ (a prime number is the one only divisible by 1 or itself)
     • A large prime root module $g$
2. Private key selection
   • Alice choose a private key $x$, a random number between 1 and $n-1$
   • Bob choose a private key $y$, a random number between 1 and $n-1$
3. Public key computation
   • Alice computes her public key A as: $A=g^x\mathrm{mod}p$
   • Bob computes his public key B as: $B=g^y\mathrm{mod}p$
4. Public key exchange
   • Alice sends her public key A to Bob
   • Bob sends his public key B to Alice
5. Shared secret key computation
   • Alice computes the shared secret key: $K=B^x\mathrm{mod}n$
   • Bob computes the shared secret key: $K=A^y\mathrm{mod}n$

Security

• Discrete Logarithm Problem: The security of the Diffie-Hellman key exchange relies on the difficulty of the discrete logarithm problem. Given $g$, $p$, and $g^x\mathrm{mod}p$, it is computationally infeasible to determine aaa if $p$ is a large enough prime and $g$ is a suitable primitive root.
• Eavesdroppers: An eavesdropper who intercepts $g$, $p$, A, and B cannot determine the shared secret $K$ without solving the discrete logarithm problem, which is considered computationally hard.

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