Dual lattice
Derive the inverse-transpose basis and check the integer inner-product definition in two dimensions.
Start with vectors and basesDeep dives
Short, focused explanations for concepts that connect lattice geometry to cryptography.
These guides build on the introductory chapters. Each takes one idea, works through a concrete example, and shows where it appears in cryptographic constructions. Read them in order, or open the concept you need now.
q-ary lattices
Solve a modular equation by hand and see how its solutions form the lattices behind SIS and LWE.
Start with notable latticesDiscrete Gaussian sampling
Calculate the probability of integer samples and understand how sampling appears in cryptographic proofs.
Start with lattice metricsLattice attacks on ECDSA
Turn a few leaked nonce bits into a hidden number problem and recover the private key with lattice reduction.
Start with the LLL algorithmMinkowski's convex body theorem
When a symmetric convex set is guaranteed to contain a nonzero lattice point: the exact statement, the proof and a worked example.
Start with geometry of numbersMinkowski's second theorem
Successive minima and the two-sided bound on their product, checked on a two-dimensional example.
Start with Minkowski's convex body theoremRing-LWE and Module-LWE
How one polynomial replaces a whole LWE matrix, with a small worked example, and why ML-KEM and ML-DSA use module lattices.
Start with LWE