Elliptic Curves and Mathematical Cryptography

Elliptic curves are the basis for Elliptic Curve Cryptography (ECC) that utilise points on a curve for its public and private keys. A special kind of addition, point addition, adds two points on the curve to get a third point on the same curve. In order for this to be cryptographically useful the addition takes place over a finite field. Elliptic Curve Cryptography has various uses, maybe most notably in the use of blockchains. ECC has the advantage of offering higher security for shorter keys compared to for example RSA. To motivate the existence of this point relation we explore algebraic curves, points and lines at infinity as well as intersections between curves.