5.7.1 Quantum-Resistant Cryptographic Families
Most post-quantum research focuses on five main mathematical families:
- Lattice-based cryptography. Security is based on the hardness of lattice problems such as the shortest vector problem (SVP) or learning with errors (LWE). Lattice schemes are efficient, well-understood, and suited to both software and hardware implementation. The CRYSTALS (Cryptographic Suite for Algebraic Lattices) project produced CRYSTALS-Kyber (key encapsulation) and CRYSTALS-Dilithium (digital signatures), two closely related lattice-based schemes that share common security foundations in the LWE problem, both of which were selected by NIST in July 2022 for standardization as part of the Post-Quantum Cryptography (PQC) Project.
- Code-based cryptography. These systems, derived from error-correcting-code theory, rely on the difficulty of decoding a general linear code. The Classic McEliece scheme, first proposed in 1978, remains secure against all known classical and quantum attacks but requires large public keys.
- Multivariate-quadratic (MQ) cryptography. Security depends on the hardness of solving systems of nonlinear quadratic equations over finite fields. MQ schemes such as Rainbow offer compact signatures and fast verification but have suffered from structural vulnerabilities discovered during the NIST evaluation.
- Hash-based signatures. These rely only on the one-way nature of cryptographic hash functions and are thus among the most conservative post-quantum options. Schemes such as XMSS (eXtended Merkle Signature Scheme) and SPHINCS+ provide strong, stateless digital signatures and are already standardized for certain applications. SPHINCS+ is a stateless hash-based signature scheme which is slower than Dilithium but offers security grounded purely in hash-function hardness, making it the most conservative option. It’s especially suitable for applications requiring long-term security assurance (e.g., firmware signing, archival data).
- Isogeny-based cryptography. Built on the mathematics of elliptic-curve isogenies, these schemes—such as supersingular isogeny key encapsulation (SIKE)—offer compact key sizes and strong theoretical elegance, although several have recently been broken by classical attacks.