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**Lattice-based cryptography** is the generic term for asymmetric cryptographic primitives based on lattices.

## History[]

Lattices were first studied by mathematicians Joseph Louis Lagrange and Carl Friedrich Gauss. Lattices have been used recently in computer algorithms and in cryptanalysis. In 1996 Miklós Ajtai showed in a seminal result the use of lattices as a cryptography primitive.

## Mathematical background[]

A lattice *L* is a set of points in the *n*-dimensional Euclidean space **R**^{n} with a strong periodicity property. A basis of *L* is a set of vectors such that any element of *L* is uniquely represented as their linear combination with integer coefficients. When *n* is at least 2, each lattice has infinitely many different bases.

Mathematical problems are used to construct a cryptography system. These problems are usually hard to solve unless one has extra information. Mathematical problems based on lattices are the Shortest Vector Problem(SVP) and the Closest Vector Problem(CVP). *SVP*: Given a basis of a lattice, find the shortest vector in the lattice. *CVP*: Given a basis of a lattice and a vector not in the lattice, find the lattice vector with the least distance to the first vector.
These problems are normally hard to solve. There are algorithms to solve these problems with a good basis.

Lattice basis reduction is a transformation of an integer lattice basis in order to find a basis with short, nearly orthogonal vectors. If one can compute such a lattice basis, the CVP and SVP problems are easy to solve. A good basis reduction algorithm is the LLL algorithm.

## Lattice-based cryptosystems[]

**Encryption**

**Signature**

**Hash function**

- LASH-x (broken, see Cryptanalysis)

## See also[]

- Lattice problems
- Learning with errors

## Bibliography[]

- Oded Goldreich, Shafi Goldwasser, and Shai Halevi. "Public-key cryptosystems from lattice reduction problems". In
*CRYPTO ’97: Proceedings of the 17th Annual International Cryptology Conference on Advances in Cryptology*, pages 112–131, London, UK, 1997. Springer-Verlag. - Phong Q. Nguyen. "Cryptanalysis of the Goldreich–Goldwasser–Halevi cryptosystem from crypto ’97". In
*CRYPTO ’99: Proceedings of the 19th Annual International Cryptology Conference on Advances in Cryptology*, pages 288–304, London, UK, 1999. Springer-Verlag. - Chris Peikert, “Public-key cryptosystems from the worst-case shortest vector problem: extended abstract,” in
*Proceedings of the 41st annual ACM symposium on Theory of computing*(Bethesda, MD, USA: ACM, 2009), 333–342, http://portal.acm.org/citation.cfm?id=1536414.1536461. - Oded Regev. Lattice-based cryptography. In
*Advances in cryptology (CRYPTO)*, pages 131–141, 2006.