> For the complete documentation index, see [llms.txt](https://cryptohack.gitbook.io/cryptobook/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://cryptohack.gitbook.io/cryptobook/lattices.md).

# Lattices

- [Introduction](https://cryptohack.gitbook.io/cryptobook/lattices/definitions.md)
- [LLL reduction](https://cryptohack.gitbook.io/cryptobook/lattices/lll-reduction.md)
- [Gram-Schmidt Orthogonalization](https://cryptohack.gitbook.io/cryptobook/lattices/lll-reduction/gram-schmidt-orthogonalization.md)
- [Lagrange's algorithm](https://cryptohack.gitbook.io/cryptobook/lattices/lll-reduction/gaussian-reduction.md)
- [LLL reduction](https://cryptohack.gitbook.io/cryptobook/lattices/lll-reduction/lll-reduced-basis.md)
- [Lattice reduction](https://cryptohack.gitbook.io/cryptobook/lattices/shortest-vector-problem.md)
- [Minkowski reduced](https://cryptohack.gitbook.io/cryptobook/lattices/shortest-vector-problem/minkowski-reduced.md)
- [HKZ reduced](https://cryptohack.gitbook.io/cryptobook/lattices/shortest-vector-problem/hkz-reduced.md)
- [LLL reduced](https://cryptohack.gitbook.io/cryptobook/lattices/shortest-vector-problem/lll-reduced.md)
- [Applications](https://cryptohack.gitbook.io/cryptobook/lattices/applications.md)
- [Coppersmith algorithm](https://cryptohack.gitbook.io/cryptobook/lattices/applications/coppersmith-algorithm.md)
- [Extensions of Coppersmith algorithm](https://cryptohack.gitbook.io/cryptobook/lattices/applications/extensions-of-coppersmith-algorithm.md)
- [Hard lattice problems](https://cryptohack.gitbook.io/cryptobook/lattices/hard-lattice-problems.md)
- [Lattices of interest](https://cryptohack.gitbook.io/cryptobook/lattices/lattices-of-interest.md)
- [Cryptographic lattice problems](https://cryptohack.gitbook.io/cryptobook/lattices/cryptographic-lattice-problems.md)
- [Short integer solutions (SIS)](https://cryptohack.gitbook.io/cryptobook/lattices/cryptographic-lattice-problems/short-integer-solutions-sis.md)
- [Learning with errors (LWE)](https://cryptohack.gitbook.io/cryptobook/lattices/cryptographic-lattice-problems/learning-with-errors-lwe.md)
- [Ring-LWE](https://cryptohack.gitbook.io/cryptobook/lattices/cryptographic-lattice-problems/ring-lwe.md)
- [NTRU](https://cryptohack.gitbook.io/cryptobook/lattices/cryptographic-lattice-problems/ntru.md)
- [Interactive fun](https://cryptohack.gitbook.io/cryptobook/lattices/interactive-fun.md)
- [Resources and notations](https://cryptohack.gitbook.io/cryptobook/lattices/resources-and-notations.md)