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In the geometry of numbers, a lattice algorithm works with a Euclidean lattice, usually specified by linearly independent vectors $b_1,\dots ,b_m$ whose integer linear combinations formThe basis of a Euclidean lattice is not unique. For example, $(1,0)$ and $(100,1)$ generate the same Euclidean lattice as $(1,0)$ and $(0,1)$: replace the second vector by itself minus 100 times the first. This integer linear combination is reversible, so it changes the representation without changing the set of Euclidean lattice points.

Lattice reduction seeks a more useful basis of a Euclidean lattice, with shorter and less nearly dependent vectors. The Lenstra-Lenstra-Lovász lattice basis reduction algorithm combines integer basis changes with Gram-Schmidt orthogonalization and comparisons controlling the quality of the result. It is a polynomial-time example, but it does not generally return an exactly shortest vector.^[ref] Better bases make geometric searches and lattice-based cryptography analysis more practical. The geometry and the sizes of the integer coordinates both matter: a small number of arithmetic steps need not imply a small bit complexity.

Lattice-based cryptography constructs security mechanisms from computational problems associated with Euclidean lattices and related noisy linear equations. A lattice is easy to describe by a basis, while finding particular short vectors or recovering hidden information from noisy equations can be difficult in suitably chosen dimensions and parameter ranges. Different constructions rely on different problems and reductions.

For the schematic learning with errors relation $b=As+e(\mathrm{mod}q)$, $A$ and $b$ are public, while the secret vector $s$ is masked by a small error vector $e$. With no error, ordinary modular linear algebra may recover $s$ efficiently when the system has suitable rank. The noise changes the problem. Real security depends on error distribution, dimension, modulus, available samples and attack model, rather than merely making the displayed equation look complicated. ML-KEM and ML-DSA illustrate encapsulation and signature applications; they use structured module arithmetic to obtain practical sizes and computation costs.

Léo Ducas is a computer scientist whose work concerns lattice-based cryptography and lattice algorithms. His research connects the geometry and computation of Euclidean lattices with the construction of cryptographic protocols.^[ref]

He is a co-designer of CRYSTALS-Kyber and CRYSTALS-Dilithium, the schemes that were standardized as ML-KEM and ML-DSA. They serve different purposes: a key encapsulation mechanism establishes a shared secret through a public-key operation, while a digital signature authenticates a message under a signing key.^[ref]

Homepage: homepages.cwi.nl/~ducas/

Oded Regev introduced learning with errors as a foundation for lattice-based cryptography. His paper connects appropriately parameterized noisy modular linear equations to worst-case lattice problems through a quantum reduction, and constructs a classical public-key encryption scheme from that learning assumption.^[ref]

The distinction between a quantum reduction and a classical cryptosystem matters: the encryption and decryption procedures need not run on a quantum computer, even though the reduction establishing the hardness connection uses quantum computation. The work also makes the error distribution part of the problem definition. Removing the errors gives ordinary modular linear equations and changes the computational task fundamentally. Regev's research homepage identifies this lattice-based work among his contributions.^[ref]