An elliptic curve group is the algebraic structure formed by the points on an elliptic curve defined over a finite field, together with a geometrically defined point-addition operation that satisfies the group axioms. The apparent difficulty of the discrete logarithm problem within this group, recovering a scalar multiplier from a known point and its scalar multiple, is the hard mathematical problem that elliptic curve cryptography relies on for security. Elliptic curve groups underpin cryptographic primitives such as Schnorr signatures and Pedersen commitments used in blockchain systems.