Ring theory, via coding theory and cryptography

In these notes on ring theory, I tried to cover enough material to get a feeling for basic ring theory, via cyclic codes and ring-based cryptosystems such as NTRU. Here’s a list of the topics.

1 Introduction to rings
1.1 Definition of a ring
1.2 Integral domains and fields
1.3 Ring homomorphisms and ideals
1.4 Quotient rings
1.5 UFDs
1.6 Polynomial rings
1.6.1 Application: Shamir’s Secret Sharing Scheme
1.6.2 Application: NTRU
1.6.3 Application: Modified NTRU
1.6.4 Application to LFSRs

2 Structure of finite fields
2.1 Cyclic multiplicative group
2.2 Extension fields
2.3 Back to the LFSR

3 Error-correcting codes
3.1 The communication model
3.2 Basic definitions
3.3 Binary Hamming codes
3.4 Coset leaders and the covering radius
3.5 Reed-Solomon codes as polynomial codes
3.6 Cyclic codes as polynomial codes
3.6.1 Reed-Solomon codes as cyclic codes
3.6.2 Quadratic residue codes
3.6.3 BCH bound for cyclic codes
3.6.4 Decoding cyclic codes
3.6.5 Cyclic codes and LFSRs

4 Lattices
4.1 Basic definitions
4.2 The shortest vector problem
4.2.1 Application to a congruential PKC
4.3 LLL and a reduced lattice basis
4.4 Hermite normal form
4.5 NTRU as a lattice cryptosystem

Calculus on graphs

In these notes, I tried to cover enough material to get a feeling for “calculus on graphs”, with applications to sports rankings and the Friendship Theorem. Here’s a list of the topics.

1 . Introduction
2. Examples
3. Basic definitions
3.1 Diameter, radius, and all that
3.2 Treks, trails, paths
3.3 Maps between graphs
3.4 Colorings
3.5 Transitivity
4. Adjacency matrix
4.1 Definition
4.2 Basic results
4.3 The spectrum
4.4 Application to the Friendship Theorem
4.5 Eigenvector centrality and the Keener ranking
4.6 Strongly regular graphs
4.7  Orientation on a graph
5. Incidence matrix
5.1 The unsigned incidence matrix
5.2 The oriented case
5.3 Cycle space and cut space
6. Laplacian matrix
6.1 The Laplacian spectrum
7  Hodge decomposition for graphs
7.1 Abstract simplicial complexes
7.2 The Bjorner complex and the Riemann hypothesis
7.3 Homology groups
8. Comparison graphs
8.1 Comparison matrices
8.2 HodgeRank
8.3 HodgeRank example