Paper 2015/939

A Decade of Lattice Cryptography

Chris Peikert


\emph{Lattice-based cryptography} is the use of conjectured hard problems on point lattices in~$\R^{n}$ as the foundation for secure cryptographic systems. Attractive features of lattice cryptography include apparent resistance to \emph{quantum} attacks (in contrast with most number-theoretic cryptography), high asymptotic efficiency and parallelism, security under \emph{worst-case} intractability assumptions, and solutions to long-standing open problems in cryptography. This work surveys most of the major developments in lattice cryptography over the past ten years. The main focus is on the foundational \emph{short integer solution}~(SIS) and \emph{learning with errors}~(LWE) problems (and their more efficient ring-based variants), their provable hardness assuming the worst-case intractability of standard lattice problems, and their many cryptographic applications.

Note: Added details to some proof outlines; minor typographical edits.

Available format(s)
Publication info
Preprint. MINOR revision.
latticessurveyshort integer solutionlearning with errors
Contact author(s)
cpeikert @ alum mit edu
2016-02-18: revised
2015-09-28: received
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      author = {Chris Peikert},
      title = {A Decade of Lattice Cryptography},
      howpublished = {Cryptology ePrint Archive, Paper 2015/939},
      year = {2015},
      note = {\url{}},
      url = {}
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