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Elliptic curves can be defined over any field K; the formal definition of an elliptic curve is a non-singular projective algebraic curve over K with genus 1 and endowed with a distinguished point defined over K.
For that reason we can view elliptic function as functions with the quotient group / as their domain. This quotient group, called an elliptic curve, can be visualised as a parallelogram where opposite sides are identified, which topologically is a torus. [1]
Elliptic-curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC allows smaller keys to provide equivalent security, compared to cryptosystems based on modular exponentiation in Galois fields , such as the RSA cryptosystem and ElGamal cryptosystem .
In mathematics, the rank of an elliptic curve is the rational Mordell–Weil rank of an elliptic curve defined over the field of rational numbers or more generally a number field K. Mordell's theorem (generalized to arbitrary number fields by André Weil) says the group of rational points on an elliptic curve has a finite basis. This means that ...
Parameter; CURVE: the elliptic curve field and equation used G: elliptic curve base point, a point on the curve that generates a subgroup of large prime order n: n: integer order of G, means that =, where is the identity element.
Supersingular elliptic curves have many endomorphisms over the algebraic closure ¯ in the sense that an elliptic curve is supersingular if and only if its endomorphism algebra (over ¯) is an order in a quaternion algebra.
Which smooth curve appears is described by the j-invariant in the table. Over the complex numbers, the curve with j-invariant 0 is the unique elliptic curve with automorphism group of order 6, and the curve with j-invariant 1728 is the unique elliptic curve with automorphism group of order 4. (All other elliptic curves have automorphism group ...
Elliptic curve scalar multiplication is the operation of successively adding a point along an elliptic curve to itself repeatedly. It is used in elliptic curve cryptography (ECC). The literature presents this operation as scalar multiplication , as written in Hessian form of an elliptic curve .