Discrete Mathematics

  1. Number Theory
    1. Divisibility Theory
      1. Division Algorithm
        1. Statement and Proof
          1. Quotient and Remainder
          2. Divisibility Properties
            1. Basic Properties
              1. Divisibility Rules
                1. Transitivity of Divisibility
                2. Prime Numbers
                  1. Definition of Prime Numbers
                    1. Composite Numbers
                      1. Fundamental Theorem of Arithmetic
                        1. Prime Factorization
                          1. Infinitude of Primes
                          2. Greatest Common Divisor
                            1. Definition and Properties
                              1. Euclidean Algorithm
                                1. Extended Euclidean Algorithm
                                  1. Bézout's Identity
                                  2. Least Common Multiple
                                    1. Definition and Properties
                                      1. Relationship with GCD
                                    2. Modular Arithmetic
                                      1. Congruence Relations
                                        1. Definition of Congruence
                                          1. Properties of Congruence
                                            1. Congruence Classes
                                            2. Arithmetic with Congruences
                                              1. Addition and Subtraction
                                                1. Multiplication
                                                  1. Division and Multiplicative Inverses
                                                  2. Linear Congruences
                                                    1. Solving Linear Congruences
                                                      1. Existence and Uniqueness
                                                        1. Systems of Linear Congruences
                                                        2. Chinese Remainder Theorem
                                                          1. Statement and Proof
                                                            1. Constructive Algorithm
                                                          2. Special Topics in Number Theory
                                                            1. Euler's Totient Function
                                                              1. Definition and Calculation
                                                                1. Properties and Formulas
                                                                  1. Multiplicative Property
                                                                  2. Fermat's Little Theorem
                                                                    1. Statement and Proof
                                                                    2. Euler's Theorem
                                                                      1. Statement and Proof
                                                                        1. Generalization of Fermat's Little Theorem
                                                                        2. Wilson's Theorem
                                                                          1. Statement and Applications
                                                                          2. Quadratic Residues
                                                                            1. Definition and Properties
                                                                              1. Legendre Symbol
                                                                            2. Cryptographic Applications
                                                                              1. Classical Cryptography
                                                                                1. Caesar Cipher
                                                                                  1. Affine Cipher
                                                                                    1. Cryptanalysis Methods
                                                                                    2. Modern Cryptography Foundations
                                                                                      1. One-Way Functions
                                                                                        1. Trapdoor Functions
                                                                                          1. Public Key Concepts
                                                                                          2. RSA Cryptosystem
                                                                                            1. Key Generation
                                                                                              1. Encryption Process
                                                                                                1. Decryption Process
                                                                                                  1. Security Considerations

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                                                                                              3. Combinatorics

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                                                                                              5. Graph Theory