Welcome to the Elliptic Curve Cryptosystem Classroom. This site provides an intuitive introduction to Elliptic Curves and how they are used to create a secure and powerful cryptosystem. The first three sections introduce and explain the properties of elliptic curves. A background understanding of abstract algebra is required, much of which can be found in the Background Algebra section. The next section describes the factor that makes elliptic curve groups suitable for a cryptosystem though the introduction of the Elliptic Curve Discrete Logarithm Problem (ECDLP). The last section brings the theory together and explains how elliptic curves and the ECDLP are applied in an encryption scheme. This classroom requires a JAVA enabled browser for the interactive elliptic curve experiments and animated examples.
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. Elliptic curve systems as applied to cryptography were first proposed in 1985 independently by Neal Koblitz from the University of Washington, and Victor Miller, who was then at IBM, Yorktown Heights.
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. By 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).