An elliptical curve is a cubic curve equation whose solutions create a shape known as a torus (or a doughnut- shaped three-dimensional figure with a hole in the center and a smooth surface which could be spread out as a rectangle in two dimensions). While elliptical curves are generally thought of in terms of this mathematical construct, the use of elliptical curve equations in public key cryptography generates one of the strongest forms of encryption.
In maths, an elliptical curve can be considered a set of elements that are part of an abelian group (or numbers which have a binary operation ‘ · ’, which can be connected to an ordered pair). This goes hand in hand with five axiomatic properties which will be constant: closure, associative, identity element, inverse element, and commutative. These things together are used in concert in order to create the public key exchange. Elliptical curves can be used in conjunction with other public key cryptography algorithms, such as a Diffe-Hellman exchange. In this scenario, the key itself would be created by an exponentiation over the group (or the process of creating an exponent), where the exponentiation itself is a multiplication operation.
This particular set of algorithms is supported by NIST in Suite B, which is was originally used for both unclassified and classified information. However as of 2015, due to the concerns about quantum computing, the NSA has announced it’s intent to replace elliptical curve cryptography in use.
One well-known example of this algorithm in action is the dual elliptic curve known as PRNG (DEC PRNG), which is recommended in the ANSI standard X9.82, ISO standard 18031, NIST standard SP 800-90.