Lattice-Based Cryptography

Lattice-based cryptography is a family of cryptographic constructions whose security rests on the assumed hardness of computational problems in high-dimensional lattices — most importantly the Shortest Vector Problem (SVP) and Learning With Errors (LWE). These problems have resisted decades of classical and quantum attack; no sub-exponential quantum algorithm is known for them, in contrast to the factoring and discrete-logarithm problems that Shor's Algorithm eliminates.

A lattice is a regular grid of points in n-dimensional space, generated by a basis of linearly independent vectors. Finding the shortest non-zero vector in such a lattice (SVP) is believed to be hard even for quantum computers; the best known algorithms require time exponential in the dimension n. Learning With Errors adds Gaussian noise to a linear system over a finite field, creating a problem that is provably as hard as SVP in the worst case.

The NIST PQC standards selected CRYSTALS-Kyber and CRYSTALS-Dilithium — both lattice-based — as the primary key encapsulation and signature algorithms. Lattice cryptography is not merely a stopgap; it is the mathematically deepest branch of algorithmic hardness theory currently producing deployable systems.