Welcome to the Certicom Elliptic Curve Tutorial. This tutorial requires a JAVA enabled browser for the interactive elliptic curve experiments and animated examples.
Sections 2.0 and 3.0 introduce and explain the properties of elliptic curves. A background understanding of abstract algebra is required.
Section 4.0 describes the factor that makes elliptic curve groups suitable for a cryptosystem though the introduction of the Elliptic Curve Discrete Logarithm Problem (ECDLP).
Section 5.0 brings the theory together and explains how elliptic curves and the ECDLP are applied in an encryption scheme.
Elliptic curves as algebraic/geometric entities have been studied extensively for the past 150 years, and from these studies has emerged a rich and deep theory.
Many cryptosystems often require the use of algebraic groups. Elliptic curves may be used to form elliptic curve groups. A group is a set of elements with custom-defined arithmetic operations on those elements. For elliptic curve groups, these specific operations are defined geometrically. Introducing more stringent properties to the elements of a group, such as limiting the number of points on such a curve, creates an underlying field for an elliptic curve group. In this classroom, elliptic curves are first examined over real numbers in order to illustrate the geometrical properties of elliptic curve groups. Thereafter, elliptic curves groups are examined with the underlying fields of Fp (where p is a prime) and F2m (a binary representation with 2m elements).