Groups and Diffie-Hellman
6 hours ago
- #Discrete Logarithm
- #Cryptography
- #Diffie-Hellman
- Diffie-Hellman key exchange solves how two parties (Alice and Bob) can establish a shared secret key over a public channel using group theory.
- The protocol relies on the discrete logarithm problem being hard: given a group element h and g = h^n, finding n is computationally difficult, while computing h^n is easy.
- A common implementation uses the multiplicative group of integers modulo a prime p, (ℤ/pℤ)×, which is cyclic and has a large order, making brute-force attacks infeasible.
- Despite (ℤ/pℤ)× being isomorphic to ℤ/(p-1)ℤ, the discrete logarithm problem remains hard due to computational representation differences; no efficient isomorphism computation is known.
- Diffie-Hellman is vulnerable to man-in-the-middle attacks without additional authentication and is often replaced by RSA for scenarios like e-commerce, though it remains a foundational cryptographic tool.