Key-Exchange on Le Site de François

Diffie-Hellman (DH / ECDH / X25519)

Mon, 01 Jan 0001 00:00:00 +0000

Diffie-Hellman (DH) is a key exchange protocol published by Whitfield Diffie and Martin Hellman in 1976 — the first public description of asymmetric cryptography and one of the most consequential cryptographic publications in history. Its fundamental contribution is solving the key establishment problem: two parties who have never communicated before, communicating over a channel that an adversary can fully observe, can nonetheless agree on a shared secret that the adversary cannot determine. The security of finite-field DH rests on the discrete logarithm problem: given g^a mod p and g^b mod p (the public values exchanged), computing g^ab mod p (the shared secret) requires solving for either a or b, which is computationally infeasible for sufficiently large groups. The 1976 original uses multiplicative groups of integers modulo a prime p; the security level is determined by the size of p (currently 2048-bit minimum, 3072-bit recommended) and the group’s structure. Finite-field DH is still deployed in TLS 1.2 DHE cipher suites and legacy IPsec configurations, but has been supplanted in new deployments by Elliptic Curve Diffie-Hellman (ECDH) and specifically by X25519, which provide equivalent security at dramatically smaller key sizes.

ECC (Elliptic Curve Cryptography)

Mon, 01 Jan 0001 00:00:00 +0000

Elliptic Curve Cryptography (ECC) is a family of public-key cryptographic algorithms built on the mathematics of elliptic curves over finite fields. Its security rests on the Elliptic Curve Discrete Logarithm Problem (ECDLP): given a public point Q = k × G on a curve (where G is a fixed base point and k is the private key scalar), recovering k from Q and G is computationally infeasible on classical computers. The practical advantage over RSA is dramatic key size efficiency: a 256-bit ECC key provides roughly the same classical security as a 3072-bit RSA key, because the best known classical algorithms for ECDLP (Pollard’s rho) are exponential whereas the best RSA algorithms (GNFS) are sub-exponential. This size difference has compounding benefits — smaller keys mean faster operations, smaller certificates, smaller TLS handshake messages, and lower power consumption on constrained devices. ECC is now the dominant choice for all new asymmetric cryptography deployments: TLS 1.3 mandates ECDHE for key exchange, and ECDSA or EdDSA for authentication; SSH defaults to Ed25519; code signing infrastructure increasingly uses ECDSA P-256 or Ed25519.

ML-KEM (Module-Lattice-Based Key Encapsulation Mechanism)

Mon, 01 Jan 0001 00:00:00 +0000

ML-KEM (Module-Lattice-Based Key Encapsulation Mechanism), standardised as NIST FIPS 203 in August 2024, is the primary post-quantum replacement for key encapsulation and key exchange. It replaces the role of ECDH (X25519, P-256) and RSA key transport in TLS handshakes, IPsec IKEv2 negotiations, and any other protocol that needs two parties to establish a shared secret without prior key material. ML-KEM is derived from CRYSTALS-Kyber, the submission that won NIST’s lattice-based KEM selection, and its security rests on the Module Learning With Errors (MLWE) problem: distinguishing a structured noisy linear system from a random one is computationally hard, and no efficient quantum algorithm for this problem is known. The “module” qualifier means the construction uses polynomial rings structured in a way that allows a good balance between security and efficiency, contrasting with pure LWE (larger keys, simpler structure) and NTRU (smaller keys, different structure).

RSA (Rivest–Shamir–Adleman)

Mon, 01 Jan 0001 00:00:00 +0000

RSA (Rivest–Shamir–Adleman), published in 1977, was the first widely adopted public-key cryptosystem and for decades the most deployed asymmetric algorithm in existence. Its security rests on the integer factorisation problem: given a public modulus n = p × q (the product of two large primes), recovering p and q is computationally infeasible on classical computers for sufficiently large n. The public key is the pair (n, e) and the private key is (n, d), where e and d are related by the modular arithmetic of Euler’s totient function. RSA enables two operations: encryption (the sender uses the public key to encrypt a message that only the private key holder can decrypt) and signing (the private key holder produces a signature that anyone with the public key can verify). In practice, RSA encryption is used almost exclusively for key encapsulation — encrypting a randomly generated symmetric key — rather than encrypting arbitrary data directly, both because RSA is slow and because direct RSA encryption of large messages requires padding schemes that are historically error-prone.