Public and Private Keys Explained

Every warning about crypto security traces back to one asymmetry: deriving a public key from a private key is trivial, and reversing it is impossible.

What a private key is

A Bitcoin private key is a number between 1 and roughly 2²⁵⁶. That is about 10⁷⁷ — comparable to the estimated number of atoms in the observable universe. It is generated at random, and its security rests entirely on the size of that space.

To make the scale concrete: a machine testing a trillion keys per second, running since the beginning of the universe, would have covered an immeasurably small fraction of the range. Brute force is not a practical threat. Every real-world key loss comes from poor randomness, careless storage, or the owner being tricked — never from someone guessing.

The one-way chain

The public key comes from the private key by elliptic curve multiplication. Multiplying a point on the curve by a large number is fast; working out what number was used is the elliptic curve discrete logarithm problem, for which no efficient algorithm is known.

The address comes from the public key by hashing, which is also one-way. This double layer means that even if elliptic curve cryptography were broken tomorrow, addresses whose public keys have never been revealed on-chain would still have some protection.

Signing works the same way in reverse: the private key produces a signature over a transaction, and anyone with the public key can verify the signature matches — without learning anything about the key itself.

Seed phrases and derivation paths

Managing individual private keys would be unworkable, so modern wallets use a single master secret — the seed phrase, usually 12 or 24 words from a standard 2,048-word list defined in BIP-39.

Those words encode a large random number. From it, a deterministic process (BIP-32) derives a tree of child keys, so one phrase can generate millions of addresses across multiple currencies. This is why you can restore an entire wallet on new hardware with nothing but the words.

The words also carry a checksum, which is why a phrase with a typo is usually rejected rather than silently opening an empty wallet. Different wallets sometimes use different default derivation paths, which is the usual explanation when a correct seed phrase restores an apparently empty wallet — the funds are on a different branch of the same tree.

Practical consequences

Three properties follow directly from the mathematics, and they drive every piece of security advice worth following:

1. Possession is control. There is no ownership record separate from the key. Anyone who obtains it can spend immediately, from anywhere, with no way to stop them.
2. Loss is permanent. No recovery process exists, because no one else has a copy. An estimated 20 % of all bitcoin is in addresses that have not moved in over a decade, much of it presumed lost.
3. Backups must be offline. A seed phrase in a photo, a password manager, an email draft or a cloud note is one breach away from being someone else's. Steel plates and paper in separate secure locations are the standard answer.