Group Element

A group element is a point on the elliptic curve that underlies Aleo's cryptography. In Leo, the group type represents these points.

Group elements show up in cryptographic operations that need elliptic curve points: key derivation, commitment schemes, signature verification. They appear less often than field or address types in typical application code but are necessary for building custom cryptographic primitives.

let point: group = group::GEN;  // Generator point
let doubled: group = point + point;  // Point addition
let scaled: group = 5scalar * point;  // Scalar multiplication

• field: Elements of the finite field the curve is defined over. Most arithmetic happens in the field.
• scalar: Elements of the scalar field (the curve group's order). Used to multiply group elements.
• address: Derived from group elements. An Aleo address is an encoded public key, which is a group element.

• Group elements support addition and scalar multiplication, but not multiplication of two group elements together.
• Security rests on the discrete logarithm problem: given P = s * G, recovering s from P is computationally infeasible.

• Field Element
• Scalar
• Zero-Knowledge Proof