In discrete mathematics, ideal lattices are a special class of lattices and a generalization of cyclic lattices.[1] Ideal lattices naturally occur in many parts of number theory, but also in other areas. In particular, they have a significant place in cryptography. Micciancio defined a generalization of cyclic lattices as ideal lattices. They can be used in cryptosystems to decrease by a square root the number of parameters necessary to describe a lattice, making them more efficient. Ideal lattices are a new concept, but similar lattice classes have been used for a long time. For example, cyclic lattices, a special case of ideal lattices, are used in NTRUEncrypt and NTRUSign.
Ideal lattices also form the basis for quantum computer attack resistant cryptography based on the Ring Learning with Errors.[2] These cryptosystems are provably secure under the assumption that the shortest vector problem (SVP) is hard in these ideal lattices.
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Vadim Lyubashevsky. Lattice-Based Identification Schemes Secure Under Active Attacks. In Proceedings of the Practice and theory in public key cryptography, 11th international conference on Public key cryptography, 2008.
^Lyubashevsky, Vadim; Peikert, Chris; Regev, Oded (2010). "On ideal lattices and learning with errors over rings". In Proc. Of EUROCRYPT, Volume 6110 of LNCS: 1–23. CiteSeerX 10.1.1.297.6108.
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