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A private key is used to send bitcoin which was received by the corresponding public key. When bitcoin is sent to a public key, only a signature produced by the private key can spend that bitcoin. Private keys must be kept safe, as only private keys can create the signatures required to spend bitcoin. Public keys, on the other hand, can be distributed publicly.
A Bitcoin wallet is normally used to generate and safely store private keys. Wallets also help users sign transactions using the appropriate private keys.
While a private key is used to derive the public key, the public key cannot be used to reveal the private key. This kind of one-way function is what makes cryptography so useful for Bitcoin and many other fields.