Let's simplify the concept of ( G ) (the base point or generator point) in the context of elliptic curve cryptography (ECC), such as with the secp256k1 curve:
- ( G ) is a specific point on the elliptic curve. Think of it as a starting point on the curve from which other points are generated.
- It has fixed coordinates (Gx, Gy). These coordinates are part of the definition of the
secp256k1curve and are the same for everyone using this curve.
- ( G ) is used as a reference point for generating public and private keys in ECC.
- It's like a "seed" from which all public keys are derived.
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Private Key: A private key is just a number. Let's call it ( d ). It's chosen randomly and is kept secret by the owner.
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Public Key Generation:
- The public key is generated from the private key using ( G ).
- The process is like saying, "Starting from ( G ), take steps on the curve ( d ) times." The point you land on is your public key ( Q ).
- Mathematically, this is written as ( Q = d \times G ), but remember, this isn't simple multiplication. It's a special operation on the curve called "elliptic curve point multiplication."
- One-way Street: It's easy (computationally) to start with ( G ) and your private key ( d ) to get your public key ( Q ). But if someone only knows ( G ) and ( Q ), it's extremely hard (computationally infeasible) for them to figure out your private key ( d ). This is the core of ECC's security.
- Universal: Everyone using
secp256k1uses the same ( G ). It's a standard starting point.
Imagine ( G ) as a specific landmark in a city. Your private key ( d ) is the number of steps you take from that landmark. The public key ( Q ) is where you end up after taking those steps. Everyone starts at the same landmark (( G )), but because everyone's number of steps (private key ( d )) is different, they all end up at different locations (public keys ( Q )). And it's designed so that if someone sees where you end up, they can't easily figure out how many steps you took to get there.