Cryptography

2.

Mathematical Foundations

2.1.

Number Theory Basics

2.1.1.

2.1.1.1.

Definition and Properties

2.1.1.2.

Prime Factorization

2.1.1.3.

Primality Testing

2.1.1.4.

Prime Generation

2.1.2.

Modular Arithmetic

2.1.2.1.

Congruence Relations

2.1.2.2.

Modular Addition and Multiplication

2.1.2.3.

Modular Inverses

2.1.2.4.

Extended Euclidean Algorithm

2.1.3.

Greatest Common Divisor

2.1.3.1.

Euclidean Algorithm

2.1.3.2.

Bézout's Identity

2.1.4.

Chinese Remainder Theorem

2.1.4.1.

Statement and Applications

2.1.4.2.

Constructive Proof

2.1.5.

Euler's Theorem

2.1.5.1.

Euler's Totient Function

2.1.5.2.

Applications in Cryptography

2.1.6.

Fermat's Little Theorem

2.1.6.1.

Statement and Proof

2.1.6.2.

Cryptographic Applications

2.2.

2.2.1.

2.2.1.1.

Definition and Properties

2.2.1.2.

2.2.1.3.

2.2.2.

2.2.2.1.

Field Properties

2.2.2.2.

2.2.2.3.

Polynomial Arithmetic

2.2.3.

Discrete Logarithm Problem

2.2.3.1.

Computational Difficulty

2.3.

Elliptic Curve Mathematics

2.3.1.

Elliptic Curve Definition

2.3.2.

2.3.3.

Scalar Multiplication

2.3.4.

Elliptic Curve Groups

2.3.5.

Elliptic Curve Discrete Logarithm Problem

2.4.

Probability and Information Theory

2.4.1.

Basic Probability

2.4.2.

Random Variables

2.4.3.

2.4.4.

Perfect Secrecy

2.4.5.

Computational Indistinguishability

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1. Foundations of Cryptography

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3. Symmetric Key Cryptography