Clear DefinitionsGenerator Point
1 min read
Pronunciation
[jen-er-ey-ter point]
Analogy
Think of the generator point as the 'official starting line' on the elliptic curve 'racetrack'. Everyone who wants to generate a key pair starts their 'hops' (scalar multiplication) from this exact same public starting line. Your secret number of hops determines your unique finish line (public key).
Definition
In Elliptic Curve Cryptography (ECC), a pre-defined, publicly known point on a specific elliptic curve that is used as a starting point for generating public keys through scalar multiplication. It is also known as a base point.
Key Points Intro
The generator point is a standard parameter for a given elliptic curve used in cryptography.
Key Points
A fixed, specified point on a chosen elliptic curve.
Part of the domain parameters for an ECC system.
Used in conjunction with a private key (a scalar) to compute a public key via scalar multiplication (Public Key = Private Key * Generator Point).
Its order (the number of times it can be added to itself before returning to the point at infinity, multiplied by a cofactor) is a large prime number.
Example
The secp256k1 elliptic curve, used by Bitcoin and Ethereum, has a specific, standardized generator point 'G'. All users of secp256k1 use this same 'G' when deriving their public keys from their private keys.
Technical Deep Dive
The generator point 'G' is chosen such that the subgroup of points generated by repeatedly adding G to itself (i.e., {G, 2G, 3G, ... (n-1)G}, where n is the order of G) is large. The order 'n' of the generator point is a large prime, and this 'n' defines the size of the private key space (private keys are integers from 1 to n-1). The choice of 'G' and the curve parameters are critical for the security of the ECC system. These parameters are often standardized (e.g., by NIST, SECG).
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