Keys, addresses and signatures
Two numbers, mathematically linked. Share one freely, guard the other absolutely. Ownership in this entire field reduces to that.
A key pair is two numbers generated together. The private key is a very large random number. The public key is derived from it by a calculation that is easy forwards and infeasible to reverse — the same asymmetry as hashing, applied differently.
What each one does
- Private key. Produces signatures. Anyone holding it can spend the associated balance. There is no second factor, no recovery, no appeal.
- Public key. Verifies signatures. Safe to publish.
- Address. Derived from the public key by hashing. Shorter, includes an error-detecting checksum, and is what you actually hand to someone paying you.
How a signature works, conceptually
To spend, you construct a transaction and sign it with the private key. The signature is a number derived from both the key and the exact contents of that transaction.
That the signer held the private key, and that the message was not altered afterwards. Change a single digit of the recipient address or the amount and the signature no longer verifies. This is why a transaction cannot be modified in transit by anyone relaying it — and why there is no way to ask for it back once it verifies.
The scale of a private key
A Bitcoin private key is a 256-bit number, so there are 2256 possibilities — a figure with 78 digits, comparable to estimates of the number of atoms in the observable universe. Generating a key at random and colliding with someone else's is not a practical concern. Every real loss has come from keys being stolen, badly generated, or lost — never from being guessed.
Which is where the danger actually is
- Bad randomness. A key generated from a weak source can be reproduced by an attacker. Several early wallet tools had exactly this flaw, and the balances were swept.
- Exposure. A private key typed into a website, photographed, stored in cloud notes or pasted into a chat is compromised from that moment, whether or not anything happens immediately.
- Loss. No key, no access, no exception. A meaningful share of all bitcoin ever mined is believed to sit at addresses whose keys no longer exist anywhere.
In practice you handle a seed phrase
Modern wallets do not ask you to record raw keys. They generate a master secret, display it as twelve or twenty-four ordinary words, and derive every key you will ever need from it deterministically. Those words are the private key, in a form humans can copy without error. See seed phrases.
In a bank, your money is an entry the bank maintains and your password is a way of asking them to change it. Here, the key is the money in every meaningful sense: whoever holds it can move the balance, and nobody who does not hold it can. There is no separate record of who ought to own it.