Skip to article frontmatterSkip to article content

Awesome Lattice-Based Cryptography

ZKPunk

A curated list of awesome resources for learning and using lattice-based cryptography.

Surveys & Books

TitleTimeAuthors/TeamURL
The LLL Algorithm: Survey and Applications2009Phong Q. Nguyen, Damien StehléLink
A Decade of Lattice Cryptography2016Chris PeikertLink
Fundamentals of Lattice Cryptography (Course Notes)OngoingDaniele Micciancio, Oded RegevLink
Lattice Basis Reduction: An Introduction to the LLL Algorithm and Its Applications2002Daniele Micciancio, Shafi GoldwasserPublisher Link (Book)
Complexity of Lattice Problems: A Cryptographic Perspective2002Daniele Micciancio, Shafi GoldwasserPublisher Link (Book)

Tutorials & Courses

TitleTimeAuthors/TeamURL
Basic Lattice Cryptography2025Vadim LyubashevskyLink
The Mathematics of Lattice-Based Cryptography2025Alfred MenezesLink
Kyber and Dilithium2024Alfred MenezesLink
Lattices: Algorithms, Complexity, and Cryptography2020Simons InstituteLink
Lattices: From Theory to Practice2020Simons InstituteLink
Lattice Cryptography Tutorial (Various Conferences)VariousChris PeikertExample (Crypto 2011)
Introduction to Lattice-Based Cryptography (Bar-Ilan Winter School)2015Vadim LyubashevskyLink

Blogs & Talks

TitleTimeAuthors/TeamURL
Prepping for post-quantum: a beginner’s guide to lattice cryptography2025CloudflareLink
Chris Peikert’s Blog (Lattice-Based Crypto Posts)VariousChris PeikertLink (Check publications/talks section)
Vadim Lyubashevsky’s Publications/TalksVariousVadim LyubashevskyLink
Windows on Theory (Blog posts on LWE, etc.)VariousVarious (e.g., Oded Regev)Link
Real World Crypto Symposium Talks (Search for lattice topics)AnnualVarious SpeakersLink
Zama Blog (Focus on FHE)OngoingZama TeamLink

Key Papers

Foundational & Hardness Assumptions

TitleTimeAuthors/TeamCitation
Generating hard instances of lattice problems1996Miklós AjtaiAjtai (1996)
On Lattices, Learning with Errors, Random Linear Codes, and Cryptography2005Oded RegevRegev (2005)
The Learning with Errors Problem2010Oded RegevRegev (2010)
On Ideal Lattices and Learning with Errors over Rings2010Vadim Lyubashevsky, Chris Peikert, Oded RegevLyubashevsky et al. (2010)

Public-Key Encryption (PKE) & Key Encapsulation Mechanisms (KEM)

TitleTimeAuthors/TeamCitation
NTRU: A Ring-Based Public Key Cryptosystem1998Jeffrey Hoffstein, Jill Pipher, Joseph H. SilvermanHoffstein et al. (1998)
Post-quantum key exchange - a new hope2015Erdem Alkim, Léo Ducas, Thomas Pöppelmann, Peter SchwabeAlkim et al. (2015)
CRYSTALS-Kyber: Algorithm Specifications and Supporting Documentation2017+Kyber TeamKyber Team (2017)

Digital Signatures

TitleTimeAuthors/TeamCitation
Trapdoors for Hard Lattices and New Cryptographic Constructions2008Craig Gentry, Chris Peikert, Vinod VaikuntanathanGentry et al. (2008)
Falcon: Fast-Fourier Lattice-based Compact Signatures over NTRU2017+Falcon TeamFalcon Team (2017)
CRYSTALS-Dilithium: Algorithm Specifications and Supporting Documentation2017+Dilithium TeamDilithium Team (2017)

Fully Homomorphic Encryption (FHE)

TitleTimeAuthors/TeamCitation
Fully homomorphic encryption using ideal lattices2009Craig GentryGentry (2009)
Fully Homomorphic Encryption without Bootstrapping2011Zvika Brakerski, Vinod VaikuntanathanBrakerski & Vaikuntanathan (2011)
(Leveled) fully homomorphic encryption without modulus switching2012Zvika BrakerskiBrakerski (2012)
Homomorphic Encryption from Learning with Errors: Conceptually-Simpler, Asymptotically-Faster, Attribute-Based2013Craig Gentry, Amit Sahai, Brent WatersGentry et al. (2013)
TFHE: Fast Fully Homomorphic Encryption over the Torus2016Ilaria Chillotti, Nicolas Gama, Mariya Georgieva, Malika IzabachèneChillotti et al. (2016)

Software Libraries

TitleLanguageAuthors/TeamDescriptionURL
PALISADE Homomorphic Encryption LibraryC++PALISADE Team (NJIT, Duality Technologies, et al.)Homomorphic Encryption LibraryLink
Microsoft SEAL (Simple Encrypted Arithmetic Library)C++Microsoft ResearchSimple Encrypted Arithmetic LibraryLink
HElib (Homomorphic Encryption Library)C++IBM ResearchHomomorphic Encryption LibraryLink
NFLlib (Number Field Lattice Library)C++Various ContributorsNumber Field Lattice LibraryLink
Open Quantum Safe (liboqs)COQS ProjectIncludes implementations of NIST PQC candidatesLink
FHEW / TFHEC++/CVarious ContributorsFast FHE libraries based on GSW/torusLink
LattigoGoTune InsightGo library for lattice crypto, including HELink

Standardization Efforts

TitleTimeAuthors/TeamURL
NIST Post-Quantum Cryptography (PQC) Standardization2016-PresentNIST & CommunityLink
HomomorphicEncryption.org StandardsOngoingLink

Tools

TitleDescriptionURL
Lattice ChallengeChallenge problems and benchmarks for lattice cryptographyLink
Lattice EstimatorTool for estimating the security of lattice-based cryptographic schemesLink

Contributions welcome! Please open an issue or pull request to suggest additions or corrections.

References
  1. Micciancio, D., & Goldwasser, S. (2002). Lattice Basis Reduction: An Introduction to the LLL Algorithm and Its Applications. Springer. 10.1007/978-1-4615-0715-9
  2. Micciancio, D., & Goldwasser, S. (2002). Complexity of Lattice Problems: A Cryptographic Perspective. Springer. 10.1007/978-1-4615-0897-7
  3. Ajtai, M. (1996). Generating hard instances of lattice problems. STOC ’96: Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, 99–108. 10.1145/237814.237838
  4. Regev, O. (2005). On Lattices, Learning with Errors, Random Linear Codes, and Cryptography. STOC ’05: Proceedings of the Thirty-Seventh Annual ACM Symposium on Theory of Computing, 84–93. 10.1145/1060590.1060603
  5. Regev, O. (2010). The Learning with Errors Problem. \\urlhttps://cims.nyu.edu/ regev/papers/lwesurvey.pdf.
  6. Lyubashevsky, V., Peikert, C., & Regev, O. (2010). On Ideal Lattices and Learning with Errors over Rings. In H. Gilbert (Ed.), Advances in Cryptology – EUROCRYPT 2010 (Vol. 6110, pp. 1–23). Springer. 10.1007/978-3-642-13190-5_1
  7. Hoffstein, J., Pipher, J., & Silverman, J. H. (1998). NTRU: A Ring-Based Public Key Cryptosystem. In J. P. Buhler (Ed.), Algorithmic Number Theory, ANTS-III (Vol. 1423, pp. 267–288). Springer. 10.1007/BFb0054868
  8. Alkim, E., Ducas, L., P\\"oppelmann, T., & Schwabe, P. (2015). Post-quantum key exchange - a new hope. Cryptology ePrint Archive, Report 2015/1092. https://eprint.iacr.org/2015/1092
  9. Kyber Team. (2017–Present). CRYSTALS-Kyber: Algorithm Specifications and Supporting Documentation. \\urlhttps://pq-crystals.org/kyber/.
  10. Gentry, C., Peikert, C., & Vaikuntanathan, V. (2008). Trapdoors for Hard Lattices and New Cryptographic Constructions. STOC ’08: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, 197–206. 10.1145/1374376.1374407
  11. Falcon Team. (2017–Present). Falcon: Fast-Fourier Lattice-based Compact Signatures over NTRU. \\urlhttps://falcon-sign.info/.
  12. Dilithium Team. (2017–Present). CRYSTALS-Dilithium: Algorithm Specifications and Supporting Documentation. \\urlhttps://pq-crystals.org/dilithium/.
  13. Gentry, C. (2009). Fully homomorphic encryption using ideal lattices. STOC ’09: Proceedings of the 41st Annual ACM Symposium on Theory of Computing, 169–178. 10.1145/1536414.1536440
  14. Brakerski, Z., & Vaikuntanathan, V. (2011). Fully Homomorphic Encryption without Bootstrapping. Cryptology ePrint Archive, Report 2011/277. https://eprint.iacr.org/2011/277
  15. Brakerski, Z. (2012). (Leveled) fully homomorphic encryption without modulus switching. Cryptology ePrint Archive, Report 2011/344. https://eprint.iacr.org/2011/344
LaWiki
LaWiki
Math & Assumptions
CVP