This appendix collects the notation, conventions, and terminology used throughout the text. Entries are organized by topic and listed in order of first appearance.
A.1 Sets and Common Number Systems
| Symbol | Meaning | Introduced |
|---|---|---|
| a ∈ S | a is an element of the set S | Def. 1.1 |
| ℕ | The natural numbers {0, 1, 2, ...} | Ex. 1.1 |
| ℤ | The integers {..., −2, −1, 0, 1, 2, ...} | Ex. 1.1 |
| ℚ | The rational numbers | Ex. 1.1 |
| ℝ | The real numbers | Ex. 1.1 |
| 𝔽ₚ | The finite field of p elements | Ch. 2 |
| ℤₙ | The integers modulo n, i.e. {0, 1, ..., n−1} | Ex. 1.1 |
| ℤₙ* | The multiplicative group of integers modulo n (elements coprime to n) | Ex. 1.8 |
A.2 Group Theory
| Symbol / Term | Meaning | Introduced |
|---|---|---|
| (G, ∗) | A group: a set G with binary operation ∗ satisfying closure, associativity, identity, and inverse | Def. 1.3 |
| e | The identity element of a group | Def. 1.3 |
| a⁻¹ | The inverse of element a | Def. 1.3 |
| Abelian | A group whose operation is commutative: a ∗ b = b ∗ a | Def. 1.4 |
| |G| | The order of a group (number of elements) | Def. 1.5 |
| ⟨g⟩ | The cyclic group generated by g | Def. 1.6 |
| gⁿ | g ∗ g ∗ ⋯ ∗ g (n times); written ng in additive notation | Def. 1.6 |
| H ≤ G | H is a subgroup of G | Def. 1.7 |
| Isomorphic | Two groups are isomorphic if there exists a bijection between them that preserves the group operation. Informally, they are structurally identical as groups—they have the same algebraic structure, differing only in the names of their elements. | Thm. 1.1 |
| log_g(h) | The discrete logarithm of h to base g | Def. 1.8 |
A.3 Elliptic Curves
| Symbol / Term | Meaning | Introduced |
|---|---|---|
| y² = x³ + ax + b | Short Weierstrass form of an elliptic curve | Def. 3.1 |
| 𝒪 | The point at infinity (identity element of the curve group) | Ch. 3 |
| Affine coordinates | Representation of curve points as (x, y) pairs, as opposed to projective or Jacobian coordinates | Ch. 3 |
A.4 Conventions
- Multiplicative vs. additive notation: When a group operation is written as ∗ or ×, we use gⁿ for repeated application. When written as +, we use ng instead. Elliptic curve groups use additive notation.
- Modular arithmetic: We write a ≡ b (mod n) to mean n divides a − b.
- Hexadecimal: Prefixed with 0x, e.g. 0xFF = 255.