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 — elliptical_curves [2021/07/26 17:53] (current)spencer created 2021/07/26 17:53 spencer created 2021/07/26 17:53 spencer created Line 1: Line 1: + ====== Elliptical Curves ====== + ===== INTRO ===== + + + 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. + ===== A MORE MATHEMATICAL EXPLANATION ===== + + 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. + ===== QUANTUM CONCERNS ===== + + 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. + ===== 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. 