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Definition
In mathematics, the term lattice is used for two very different kinds of objects, arising, respectively, in order theory and number theory. Here, lattice always means a number-theoretical lattice. Informally speaking, a lattice is a regular infinite arrangement of points in n-dimensional space. More formally, a lattice is a discrete subgroup of \({\mathbb{R}}^{n}\).
Background
Lattices appeared in the nineteenth century in both crystallography and number theory. But in some sense, their study goes back to that of quadratic forms: Gauss [3] made a connection between quadratic forms and lattices, which was further developed by Dirichlet [2] and especially Minkowski [7]. Lattice theory is usually called geometry of numbers [1, 5, 10], a name due to Minkowski [7].
Theory
A lattice can be defined in many equivalent ways....
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Cassels JWS (1971) An introduction to the geometry of numbers. Springer, Berlin
Dirichlet JPGL (1850) Über die Reduction der positiven quadratischen Formen in drei unbestimmten ganzen Zahlen. J Reine Angew Math 40:209–227
Gauss CF (1840) Recension der “Untersuchungen über die Eigenschaften der positiven ternären quadratischen Formen von Ludwig August Seeber.” Göttingische Gelehrte Anzeigen, July 9, 1065ff, 1831. Repr. J Reine Angew Math 20:312–320. http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN23599524X&DMDID=dmdlog22
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Minkowski H (1896) Geometrie der Zahlen. Teubner, Leipzig
Nguyen PQ, Stern J (2001) The two faces of lattices in cryptology. In: Cryptography and lattices – proceedings of CALC’01, Providence. Lecture notes in computer science, vol 2146. Springer, Berlin, pp 146–180
Nguyen PQ, Vallée B (2009) The LLL algorithm: survey and applications. Information security and cryptography. Springer, Heidelberg
Siegel CL (1989) Lectures on the geometry of numbers. Springer, Berlin
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Nguyen, P. (2011). Lattice. In: van Tilborg, H.C.A., Jajodia, S. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-5906-5_456
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