## Advanced Topics in Cryptography - Lattices (Fall 2022)

Lattices in complexity theory, cryptography, and quantum computation.

Monday 9:50 - 12:15, 6B201

Office hour: Monday 13:30 - 14:30

Email: chenyilei@mail.tsinghua.edu.cn

TAs: Zewen Fan, Xiaxi Ye

Main reference for lattice and complexity theory:

Micciancio and Goldwasser: Complexity of lattice problems: A cryptographic perspective

Micciancio and Goldwasser: Complexity of lattice problems: A cryptographic perspective

Websites/Lecture notes/Surveys related to lattices:

Damien Stehle's collection of lattice papers [ site ]

Oded Regev 2004 [ site ]

Vinod Vaikuntanathan 2015 [ site ] 2020 [ site ]

Daniele Micciancio 2016 [ site ]

TAU lattice course 2019 [ site ]

H. Lenstra: Lattices in number theory, algorithm, and applications [ link ]

A. Joux and J. Stern: Lattice reduction, a toolbox for cryptanalyst [ link ]

P. Q. Nguyen and J. Stern: The two faces of lattice in cryptology [ link ]

Damien Stehle's collection of lattice papers [ site ]

Oded Regev 2004 [ site ]

Vinod Vaikuntanathan 2015 [ site ] 2020 [ site ]

Daniele Micciancio 2016 [ site ]

TAU lattice course 2019 [ site ]

H. Lenstra: Lattices in number theory, algorithm, and applications [ link ]

A. Joux and J. Stern: Lattice reduction, a toolbox for cryptanalyst [ link ]

P. Q. Nguyen and J. Stern: The two faces of lattice in cryptology [ link ]

Main reference for cryptography:

A Course in Cryptography, Rafael Pass & abhi shelat

https://www.cs.cornell.edu/courses/cs4830/2010fa/lecnotes.pdf

A Course in Cryptography, Rafael Pass & abhi shelat

https://www.cs.cornell.edu/courses/cs4830/2010fa/lecnotes.pdf

A graduate course in applied cryptography, Dan Boneh & Victor Shoup

Foundations of Cryptography I, II, Oded Goldreich

Foundations of Cryptography I, II, Oded Goldreich

Topics:

Part 1: Introduction: Minkowski's two theorems, all what you want to know about lattices

Part 2: Algorithms for SVP and CVP: LLL and others

Part 3: Complexity: NP hardness of CVP, SVP (Ajtai, Micciancio, Khot), NP intersect coNP

Part 4: Worst-case to average-case reduction (LWE, SIS, DCP)

Part 5: The cryptographic applications of SIS and LWE: FHE, lattice trapdoor, IBE, ABE.

Part 6: Quantum and lattices

Part 7: Whatever interesting, if we have time

Last two weeks: Project presentations

Part 1: Introduction: Minkowski's two theorems, all what you want to know about lattices

Part 2: Algorithms for SVP and CVP: LLL and others

Part 3: Complexity: NP hardness of CVP, SVP (Ajtai, Micciancio, Khot), NP intersect coNP

Part 4: Worst-case to average-case reduction (LWE, SIS, DCP)

Part 5: The cryptographic applications of SIS and LWE: FHE, lattice trapdoor, IBE, ABE.

Part 6: Quantum and lattices

Part 7: Whatever interesting, if we have time

Last two weeks: Project presentations

Tentative Schedule:

09/12 Moon festival

09/19 Introduction, lattice problems

09/26 Minkowski's theorems, NP hardness of CVP, SVP is no harder than CVP

10/03 National holiday, moved to 11/19

10/10 The LLL algorithm

10/17 Short integer solution and learning with errors, q-ary lattices

10/24 Regev's quantum reduction from GapSVP to LWE

10/31 Fully homomorphic encryption, gadget matrices

11/07 Lattice trapdoor and its applications to signature

11/14 Lattice trapdoor II, identity-based encryption,

11/19 The Bonsai technique, signature without random oracle (substitution for 10/03)

11/21 Attribute-based encryption, open problems

11/28 Pseudorandom function and their advanced capabilities

12/05 *Functional encryption, witness encryption, and program obfuscation from lattices

12/12 *Advanced complexity results and algorithms, quantum and lattices

12/19 Presentation I

12/26 Presentation II

09/12 Moon festival

09/19 Introduction, lattice problems

09/26 Minkowski's theorems, NP hardness of CVP, SVP is no harder than CVP

10/03 National holiday, moved to 11/19

10/10 The LLL algorithm

10/17 Short integer solution and learning with errors, q-ary lattices

10/24 Regev's quantum reduction from GapSVP to LWE

10/31 Fully homomorphic encryption, gadget matrices

11/07 Lattice trapdoor and its applications to signature

11/14 Lattice trapdoor II, identity-based encryption,

11/19 The Bonsai technique, signature without random oracle (substitution for 10/03)

11/21 Attribute-based encryption, open problems

11/28 Pseudorandom function and their advanced capabilities

12/05 *Functional encryption, witness encryption, and program obfuscation from lattices

12/12 *Advanced complexity results and algorithms, quantum and lattices

12/19 Presentation I

12/26 Presentation II