An elliptic curve group is the collection of points on an elliptic curve equipped with an algebraically defined group operation. Over a finite field, this collection is finite, and its arithmetic relies on field addition, subtraction, multiplication, inversion, and division. section 2.7
Over a finite field, the group is cyclic or the product of two cyclic groups. It can be written as integers modulo d₁ and d₂, with d₁ dividing d₂. Base curves usually use a prime field q and d₁ = 1, giving one cyclic group; pairing constructions instead require two separate cyclic groups. section 15.1
Implementation-relevant pieces include:
- Finite-field routines provide the arithmetic used throughout elliptic-curve calculations. section 2.7
- Point orders derive from factors of the total number of curve points; cryptographic applications seek a large prime factor. section 2.7 section 15.1
- Quadratic residues and square-root algorithms support calculations over prime fields. section 2.7
Equation 15.1
\[E_{F_q} \cong \dfrac{Z}{d_1 Z} \times \dfrac{Z}{d_2 Z} \label{pg0}\]Displays the decomposition E over Fq ≅ Z/d₁Z × Z/d₂Z used to describe elliptic-curve group structure. section 15.1
