Elliptic curve group¶
Definition¶
You should take a look at the definition of group here, and Elliptic curve here.
Algorithms for computing discrete logarithms¶
The fastest known classical algorithm to solve the Discrete Logarithm problem over most elliptic curve groups used in practice runs in time \(O(\sqrt{G})\). For example, a popular elliptic curve called Curve25519, which is defined over base field \(F\) of size \(2^{255} −19\), defines a cyclic group of order close to \(2^{252}\), so the time complexity is about \(O(2^{128})\).
Scalar field vs base field¶
The base field is used to represent points on a curve while the scalar field allows performing scalar multiplication on those points.