Ideal Lattice Cryptography

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Ideal lattices are a special class of lattices and a generalization of cyclic lattices. 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. These cryptosystems are provably secure under the assumption that the Shortest Vector Problem (SVP) is hard in these ideal lattices.

quantum computer attack generalization cryptography

References

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. http://cseweb.ucsd.edu/users/vlyubash/papers/idlatticeconf.pdf
Vadim Lyubashevsky, Chris Peikert and Oded Regev. On Ideal Lattices and Learning with Errors over Rings. In Eurocrypt 2010, Lecture Notes in Computer Science, 2010. http://www.springerlink.com/content/p0k0124216567122/
Jintai Ding and Richard Lindner. Identifying Ideal Lattices. In Cryptology ePrint Archive, Report 2007/322, 2007. http://eprint.iacr.org/2007/322.pdf
Lyubashevsky, V., Micciancio, D. Generalized compact knapsacks are collision resistant.. In CBugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4052, pp. 144–155. Springer, Heidelberg (2006). http://cseweb.ucsd.edu/users/vlyubash/papers/generalknapsackfull.pdf
Micciancio, D. Generalized compact knapsacks, cyclic lattices, and efficient oneway functions.. In Computational Complexity 16(4), 365–411 (2007). http://www.springerlink.com/content/g11573q628x12970/fulltext.pdf
Peikert, C., Rosen, A. Efficient collision-resistant hashing from worst-case assumptions on cyclic lattices.. In Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 145–166. Springer, Heidelberg (2006). http://www.cc.gatech.edu/~cpeikert/pubs/cyclic-crh.pdf
Vadim Lyubashevsky and Daniele Micciancio. Asymptotically efficient lattice-based digital signatures. In Proceedings of the 5th conference on Theory of cryptography, 2008. https://www.iacr.org/archive/tcc2008/49480032/49480032.pdf
Ding, Jintai; Xie, Xiang; Lin, Xiaodong (2012). A Simple Provably Secure Key Exchange Scheme Based on the Learning with Errors Problem. https://eprint.iacr.org/2012/688.pdf.
Peikert, Chris (2014-10-01). Mosca, Michele. ed. Lattice Cryptography for the Internet. Lecture Notes in Computer Science. Springer International Publishing. pp. 197–219. ISBN 978-3-319-11658-7. https://link.springer.com/chapter/10.1007/978-3-319-11659-4_12.
Lyubashevsky, Vadim (29 Jul 2012). "Lattice Signatures Without Trapdoors". IACR. https://eprint.iacr.org/2011/537.pdf. Retrieved 21 June 2014.
Daniele Micciancio, Oded Regev Lattice-based Cryptography. In POST-QUANTUM CRYPTOGRAPHY, 2009. http://www.cs.tau.ac.il/~odedr/papers/pqc.pdf
Oded Regev. On lattices, learning with errors, random linear codes, and cryptography . In Journal of the ACM, 2009. http://www.cs.tau.ac.il/~odedr/papers/qcrypto.pdf
"http://www.di.ens.fr/~lyubash/papers/signaturechess.pdf". http://www.di.ens.fr/~lyubash/papers/signaturechess.pdf. Retrieved 2015-06-28.
Singh, Vikram (2015). A Practical Key Exchange for the Internet using Lattice Cryptography. http://eprint.iacr.org/2015/138.
Damien Stehlé, Ron Steinfeld, Keisuke Tanaka and Keita Xagawa. Efficient public key encryption based on ideal lattices. In Lecture Notes in Computer Science, 2009. http://eprint.iacr.org/2009/285.pdf
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Craig Gentry. Fully Homomorphic Encryption Using Ideal Lattices. In the 41st ACM Symposium on Theory of Computing (STOC), 2009. http://portal.acm.org/citation.cfm?id=1536414.1536440

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