Below is a tentative list of topics.
The mathematical basics behind lattices:
- covolume of a lattice
- Minkowski first and second theorems
- primitive vectors and primitive sublattices
- Minkowski and Korkine-Zolotarev reduction of a lattice.
- Harder-Narasimhan filtration
- Space of lattices, Mahler compactness criterion, linear action, Haar measure
More advanced Mathematical topics:
- Hecke correspondence
- Siegel summation formula
- Application to shortest vector problem
- Rogers formula, applications to shortest vector problem
- covering radius.
- Classical computational problems on lattices
- The LLL algorithm
- Worst-case to average-case reductions
Applications to Cryptography
- Hashing and Encryption
- The Learning with Errors Problem
- Computing over Encrypted Data