Elliptic Curves

Related Concepts

Hyperelliptic Curves

Definition

This section introduces elliptic curves and associated group operations, along with basic structural properties of particular interest in cryptography.

Background

Although classical, elliptic curves have been used in integer factoring algorithms and in primality proving algorithms, and also for designing public-key cryptographic schemes. In cryptography the attractiveness is due, in part, to the apparent resistance (at a given key size) to discrete logarithm attacks compared to methods where security is based on the intractability of integer factorization or finding logarithms in finite fields, especially as security requirements increase.

Theory

An elliptic curveE over a field F is defined by a Weierstrass equation

$$ E/F : {y}^{2} + {a}_{ 1}xy + {a}_{3}y = {x}^{3} + {a}_{ 2}{x}^{2} + {a}_{ 4}x + {a}_{6} $$

(1)

with a ₁, a ₂, a ₃, a ₄, a ₆ ∈ F and Δ≠0, where Δ is the discriminant of E and is defined as follows:

$$ \begin{array}{rcl}...