Syllabus

Topics (Lecture Notes)

• Why lattices?

• Definitions and computational problems

References

• A Decade of Lattice Cryptography, by Chris Peikert (Chapter 2)

Topics (Lecture Notes)

• Complexity and algorithms for lattice problems

• The short integer solutions (SIS) problem

References

• A Decade of Lattice Cryptography, by Chris Peikert (Chapter 4.1)

Topics (Lecture Notes)

• Collision-resistant hash functions from SIS

• The leftover hash lemma (LHL)

• Commitments from SIS

References

• A Decade of Lattice Cryptography, by Chris Peikert (Chapter 5.1)

Topics (Lecture Notes)

• Inhomogeneous SIS and gadget trapdoors

• Lattice-based signatures in the random oracle model

References

• A Decade of Lattice Cryptography, by Chris Peikert (Chapter 5.4-5.5)

• Trapdoors for Lattices: Simpler, Tighter, Faster, Smaller, by Daniele Micciancio and Chris Peikert

Topics (Lecture Notes)

• Preimage-sampleable trapdoor functions

• Discrete Gaussians over lattices

References

• A Decade of Lattice Cryptography, by Chris Peikert (Chapter 5.4-5.5)

• How to Use a Short Basis: Trapdoors for Hard Lattices and New Cryptographic Constructions, by Craig Gentry, Chris Peikert, and Vinod Vaikuntanathan

• Trapdoors for Lattices: Simpler, Tighter, Faster, Smaller, by Daniele Micciancio and Chris Peikert

Topics (Lecture Notes)

• The smoothing parameter

• Rounded Gaussians vs. discrete Gaussians

References

• A Decade of Lattice Cryptography, by Chris Peikert (Chapter 5.4-5.5)

• How to Use a Short Basis: Trapdoors for Hard Lattices and New Cryptographic Constructions, by Craig Gentry, Chris Peikert, and Vinod Vaikuntanathan

• Trapdoors for Lattices: Simpler, Tighter, Faster, Smaller, by Daniele Micciancio and Chris Peikert

Topics (Lecture Notes)

• Discrete Gaussian sampling over the integers

• Discrete Gaussian sampling over an arbitrary lattice

References

• A Decade of Lattice Cryptography, by Chris Peikert (Chapter 5.4-5.5)

• How to Use a Short Basis: Trapdoors for Hard Lattices and New Cryptographic Constructions, by Craig Gentry, Chris Peikert, and Vinod Vaikuntanathan

• Trapdoors for Lattices: Simpler, Tighter, Faster, Smaller, by Daniele Micciancio and Chris Peikert

• An Efficient and Parallel Gaussian Sampler for Lattices, by Chris Peikert

Topics (Lecture Notes)

• The learning with errors (LWE) assumption

• Public-key encryption from LWE

• Key exchange from LWE

References

• A Decade of Lattice Cryptography, by Chris Peikert (Chapter 4.2, 5.2)

Topics (Lecture Notes)

• Somewhat homomorphic encryption (SWHE)

• The Gentry-Sahai-Waters (GSW) FHE scheme

References

• A Decade of Lattice Cryptography, by Chris Peikert (Chapter 6.1)

• Homomorphic Encryption from Learning with Errors: Conceptually-Simpler, Asymptotically-Faster, Attribute-Based by Craig Gentry, Amit Sahai, and Brent Waters

Topics (Lecture Notes)

• Bootstrapping SWHE to FHE

• Bootstrapping the GSW encryption scheme

References

• A Decade of Lattice Cryptography, by Chris Peikert (Chapter 6.1)

• Lattice-Based FHE as Secure as PKE, by Zvika Brakerski and Vinod Vaikuntanathan

Topics (Lecture Notes)

• Key exchange from LWE

References

• Frodo: Take off the ring! Practical, Quantum-Secure Key Exchange from LWE by Joppe Bos, et al.

Topics (Lecture Notes)

• Homomorphic signatures

• Lattice homomorphisms

References

• A Decade of Lattice Cryptography, by Chris Peikert (Chapter 6.1)

• Leveled Fully Homomorphic Signatures from Standard Lattices, by Sergey Gorbunov, Vinod Vaikuntanathan, and Daniel Wichs

Topics (Lecture Notes)

• Context-hiding for homomorphic signatures

• Homomorphic commitments

References

• A Decade of Lattice Cryptography, by Chris Peikert (Chapter 6.1)

• Leveled Fully Homomorphic Signatures from Standard Lattices, by Sergey Gorbunov, Vinod Vaikuntanathan, and Daniel Wichs

Topics (Lecture Notes)

• Dual-mode homomorphic commitments

• Designated-prover NIZKs

References

• A Decade of Lattice Cryptography, by Chris Peikert (Chapter 6.1)

• Leveled Fully Homomorphic Signatures from Standard Lattices, by Sergey Gorbunov, Vinod Vaikuntanathan, and Daniel Wichs

• Multi-Theorem Preprocessing NIZKs from Lattices, by Sam Kim and David J. Wu

Topics (Lecture Notes)

• Dual Regev encryption

• Attribute-based encryption (ABE)

References

• A Decade of Lattice Cryptography, by Chris Peikert (Chapter 6.2)

• Fully Key-Homomorphic Encryption, Arithmetic Circuit ABE, and Compact Garbled Circuits, by Dan Boneh, et al.

Topics (Lecture Notes)

• Attribute-based encryption

• Punctured trapdoors

References

• A Decade of Lattice Cryptography, by Chris Peikert (Chapter 6.2)

• Fully Key-Homomorphic Encryption, Arithmetic Circuit ABE, and Compact Garbled Circuits, by Dan Boneh, et al.

Topics (Lecture Notes)

• Predicate encryption

• The dual-use approach

References

• Predicate Encryption for Circuits from LWE, by Sergey Gorbunov, Vinod Vaikuntanathan, and Hoeteck Wee

• Private Constrained PRFs (and more) from LWE, by Zvika Brakerski, Rotem Tsabary, Vinod Vaikuntanathan, and Hoeteck Wee

Topics (Lecture Notes)

• Functional encryption

• Garbled circuits

References

• Functional Encryption: Definitions and Challenges, by Dan Boneh, Amit Sahai, and Brent Waters

• Evans-Kolesnikov-Rosulek (Chapter 3.1)

Topics (Lecture Notes)

• Functional encryption from public-key encryption

• Succinct FE from garbled circuits, ABE and FHE

References

• Worry-free encryption: Functional Encryption with Public Keys, by Amit Sahai and Hakan A. Seyalioglu

• Reusable Garbled Circuits and Succinct Functional Encryption, by Shafi Goldwasser, Yael Kalai, Raluca Ada Popa, Vinod Vaikuntanathan, and Nickolai Zeldovich

Topics (Lecture Notes)

• Blum's protocol for graph Hamiltonicity

• One-time NIZK from PKE

References

• Construction of a Non-Malleable Encryption Scheme from Any Semantically Secure One, by Rafael Pass, abhi shelat, and Vinod Vaikuntanathan

Topics (Lecture Notes)

• Reusable designated-verifier NIZKs from ABE

References

• New Constructions of Reusable Designated-Verifier NIZKs, by Alex Lombardi, Willy Quach, Ron D. Rothblum, Daniel Wichs, and David J. Wu

Topics (Lecture Notes)

• Correlation-intractable hash functions

• NIZKs from circular-secure FHE

References

• Fiat-Shamir From Simpler Assumptions, by Ran Canetti, Yilei Chen, Justin Holmgren, Alex Lombardi, Guy N. Rothblum, and Ron D. Rothblum

• Fiat-Shamir: From Practice to Theory, Part II, by Ran Canetti, Alex Lombardi, and Daniel Wichs

Topics (Lecture Notes)

• Correlation-intractable hash functions from LWE

• NIZKs from LWE

References

• Noninteractive Zero Knowledge for NP from (Plain) Learning With Errors, by Chris Peikert and Sina Shiehian

Topics (Lecture Notes)

• Multi-key FHE

References

• Two Round Multiparty Computation via Multi-Key FHE, by Pratyay Mukherjee and Daniel Wichs

Topics (Lecture Notes)

• Secret sharing

• Homomorphic secret sharing

References

• Function Secret Sharing and Homomorphic Secret Sharing, by Elette Boyle

• Spooky Encryption and its Applications, by Yevgeniy Dodis, Shai Halevi, Ron D. Rothblum, and Daniel Wichs

Topics (Lecture Notes)

• Function secret sharing

• Distributed point functions

• DPFs from one-way functions

• Using DPFs for private database queries

References

• Function Secret Sharing and Homomorphic Secret Sharing, by Elette Boyle

• Distributed Point Functions and their Applications, by Niv Gilboa and Yuval Ishai

Topics (Lecture Notes)

• Private information retrieval (PIR)

• Single-server PIR from homomorphic encryption

• Ring learning with errors (RLWE)

References

• On Ideal Lattices and Learning with Errors Over Rings, by Vadim Lyubashevsky, Chris Peikert, and Oded Regev

• Spiral: Fast, High-Rate Single-Server PIR via FHE Composition, by Samir Jordan Menon and David J. Wu