How are a private key and a public key related?
Correct answer
Answer C: The public key is derived from the private key.
Quick explanation
The private key is kept secret and allows payments to be authorized. The public key is derived from it.
Detailed explanation
A private key is a very large secret number. A defined mathematical operation on the secp256k1 curve calculates the corresponding public key. This direction is efficient: a wallet can always derive the same public key from its private key. The reverse is intended to be practically impossible. Someone knowing only the public key cannot recover the private key with realistic effort.
The private key is used to create a digital signature and thereby satisfy a spending condition. Using public data, other nodes can check the signature without learning the private key. Modern hierarchical deterministic wallets derive many key pairs from a common starting value. Extended public keys can even generate many receiving keys without being able to sign spends. They are therefore not secret like private keys but can reveal substantial information about addresses, payments and balances. Private keys remain strictly secret; public keys and derived receiving information serve other purposes.
Key takeaway
Public keys are derived from private keys, not the other way around.
What the sources support
NIST explains the mathematical relationship between public and private keys in general. Bitcoin sources show the concrete application: the private key remains secret while public keys and wallet structures are derived from it.