Discrete Mathematics

Set partitions, equivalence relations, graph theory connectivity, recurrence relations, and finite group cryptography.

1Set Partitions, Equivalence Relations & Quotient Rings

Reflexive, symmetric, and transitive binary relations, partition equivalence classes, and modular congruence arithmetic.

Equivalence Relations & Modulo Partition Rings

Axiomatic set theory relations, quotient sets Z/nZ, and modular congruence applications.

Equivalence Axioms, Quotient Set Partitions & Modular Arithmetic Rings

Formal proof of the Fundamental Partition Theorem for equivalence relations and construction of integers modulo n.

2Combinatorics, Generating Functions & Asymptotic Counting

Permutations, combinations, Pigeonhole Principle, Inclusion-Exclusion, and Ordinary Generating Functions.

Combinatorial Enumeration, Recurrence Relations & Formal Power Series

Bijection principles, inclusion-exclusion derangements, and solving linear recurrences via generating functions.

Combinatorics, Inclusion-Exclusion, Derangements & Generating Functions

Explore fundamental enumeration principles, the generalized Pigeonhole Principle, the Principle of Inclusion-Exclusion, derangements, and Ordinary Generating Functions (OGFs).

3Graph Theory, Spectral Invariants & Network Flow Optimization

Graph axioms, articulation points, planarity, Graph Laplacians, and the Max-Flow Min-Cut theorem.

Spectral Graph Theory & Maximum Flow Minimum Cut

Algebraic connectivity via Graph Laplacians, Edmonds-Karp residual flow, and cut duality.

Graph Theory, Spectral Laplacians & Edmonds-Karp Network Flow

Explore rigorous graph theory: vertex degrees, Euler's planar formula, the Graph Laplacian matrix, spectral partitioning via the Fiedler vector, and the Edmonds-Karp Max-Flow Min-Cut theorem.

4Abstract Algebra, Galois Fields & Cryptographic Group Foundations

Group axioms, Lagrange's coset theorem, finite fields GF(2^8), polynomial arithmetic, and cryptographic S-box design.

Galois Fields GF(2^8), Modular Polynomials & S-Box Construction

Finite field arithmetic, irreducible polynomials, multiplicative inverses via extended GCD, and non-linear cryptographic mappings.

Abstract Algebra, Galois Fields $GF(2^8)$ & Cryptographic S-Box Inverses

Master abstract algebraic structures: Groups, Rings, Fields, Lagrange's Theorem, construction of Galois Fields $GF(2^8)$ using irreducible polynomials, and multiplicative inversion in AES/Rijndael.