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Encyclopedia of Cryptography and Security pp 713–715Cite as

Lattice-Based Cryptography

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Related Concepts

Closest Vector Problem; Lattice; Lattice Reduction; NTRU; Post-quantum Cryptography; Public Key Cryptography; Shortest Vector Problem

Definition

Lattice-based cryptography is a generic term used to encompass a wide range of cryptographic functions whose security is based on the conjectured intractability of Lattice problems, like (variants of) the Shortest Vector Problem and the Closest Vector Problems.

For applications of lattices in cryptanalysis, Lattice Reduction.

Background

The study of lattice-based cryptography was pioneered by Ajtai in 1996 [1], who proved that certain variants of the knapsack cryptographic schemes are at least as hard to break on the average as approximating(the length estimation variant of) theShortest Vector Problem(GapSVP)within factors that grow only polynomially in the dimensionn of thelattice. Two distinguishing features of Ajtai’s result, and lattice-basedcryptography in general, are that:

  • Breaking the cryptographic functions is...

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Recommended Reading

  1. Ajtai M (1996) Generating hard instances of lattice problems (extended abstract). In: Proceedings of the twenty-eighth annual ACM Symposium on the Theory of Computing (STOC’96), Philadelphia, 22–24 May 1996. ACM Press, New York, pp 99–108

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  2. Ajtai M, Dwork C (1997) A public-key cryptosystem with worstcase/average-case equivalence. In: Proceedings of the twenty-ninth annual ACM Symposium on Theory of Computing (STOC ’97), E1 Paso, 4–6 May 1997. ACM Press, New York, pp 284–293

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  3. Gentry C, Peikert C, Vaikuntanathan V (2008) Trapdoors for hard lattices and new cryptographic constructions. In: Proceedings of STOC ’08, Victoria, 17–20 May 2008. ACM Press, New York, pp 197–206

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  4. Lyubashevsky V, Micciancio D (2008) Asymptotically efficient lattice-based digital signatures. In: Proceedings of TCC ’08, New York, 19–21 March 2008. Lecture notes in computer science, vol 4948. Springer, Berlin, pp 37–54

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  5. Lyubashevsky V, Micciancio D (2009) On bounded distance decoding, unique shortest vectors, and the minimum distance problem. In: Proceedings of CRYPTO 2009, Santa Barbara, 16–20 August 2009. Lecture notes in computer science, vol 5677. Springer, Berlin, pp 577–594

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  6. Lyubashevsky V, Micciancio D, Peikert C, Rosen A (2008) Swifft: a modest proposal for FFT hashing. In: Proceedings of FSE ’08, Lausanne, 10–13 February 2008. Lecture notes in computer science, vol 5086. Springer, Berlin, pp 54–72

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  7. Micciancio D (2007) Generalized compact knapsacks, cyclic lattices, and efficient one-way functions. Comput Complex 16(4):365–411

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  8. Micciancio D, Regev O (2007) Worst-case to average-case reductions based on Gaussian measure. SIAM J Comput 37(1):267–302

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  9. Micciancio D, Regev O (2008) Lattice-based cryptography. In: Bernstein DJ, Buchmann J, Dahmén E (eds) Post-quantum cryptography. Springer, Berlin

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  10. Peikert C, Vaikuntanathan V (2008) Noninteractive statistical zeroknowledge proofs for lattice problems. In: Proceedings of CRYPTO ’08, Santa Barbara, 17–21 August 2008. Lecture notes in computer science, vol 5157. Springer, Berlin, pp 536–553

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Author information

Authors and Affiliations

  1. Department of Computer Science & Engineering, University of California, 9500 Gilman Drive, La Jolla, 92093-0404, San Diego, CA, USA

    Daniele Micciancio Prof.

Authors
  1. Daniele Micciancio Prof.
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Editor information

Editors and Affiliations

  1. Department of Mathematics and Computing Science, Eindhoven University of Technology, 5600 MB, Eindhoven, The Netherlands

    Henk C. A. van Tilborg

  2. Center for Secure Information Systems, George Mason University, Fairfax, VA, 22030-4422, USA

    Sushil Jajodia

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Micciancio, D. (2011). Lattice-Based Cryptography. 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_417

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