What are elliptic curves used for in cryptography?
In cryptography, elliptic curves are used for creating secure public key cryptographic systems. These include encryption, digital signatures, and key agreement protocols. Their efficiency and high level of security for a smaller key size make them particularly valuable in resource-constrained environments.
How do you define an elliptic curve over a finite field?
An elliptic curve over a finite field F_p is defined by an equation of the form y^2 = x^3 + ax + b, where a, b are elements in F_p, and the discriminant 4a^3 + 27b^2 is non-zero in F_p. This ensures the curve is non-singular, that is, without cusps or self-intersections.
What is the difference between elliptic curves over real numbers and those over finite fields?
Elliptic curves over real numbers are plotted in the continuous xy-plane, displaying a smooth curve. In contrast, elliptic curves over finite fields, such as those used in cryptography, consist of a discrete set of points following the same types of equations, but calculated within a finite field.
What are the basic properties of elliptic curves?
Elliptic curves are smooth, non-singular plane curves defined by a cubic equation, featuring a unique point at infinity. They possess a group structure, allowing points on the curve to be added or multiplied. These curves are symmetrical about the x-axis and every non-vertical line intersects them at three points (accounting for multiplicity).
What is the rank of an elliptic curve and why is it important?
The rank of an elliptic curve is the maximum number of independent rational points that generate its group, excluding the point at infinity. It's crucial as it reflects the curve's complexity and influences the density of rational solutions, bearing significance in areas like cryptography and number theory.