Long before Bitcoin, in 1982, three computer scientists — Leslie Lamport, Robert Shostak and Marshall Pease — told a riddle. Several generals surround a city. They must all attack together, or all retreat. But they can speak only by messenger, and some among them may be traitors.

How can honest generals agree on one plan, when any message might be a lie? This was the Byzantine Generals Problem, and for decades it had no good answer for a network open to strangers. Satoshi's proof of work offered one: make every message costly to forge.

Imagine a machine that swallows any book, letter or list, and prints a short fingerprint for it — always the same fingerprint for the same input. Change a single comma, and out comes a fingerprint that looks nothing like the first. Nobody can run the machine backwards.

That machine is a hash function. Bitcoin uses one called SHA-256, published in 2001. Each block carries the fingerprint of the one before it, and miners hunt for a block whose fingerprint starts with enough zeros. It is how the notebook knows when a page has been tampered with.

In 1976, Whitfield Diffie and Martin Hellman published an idea that changed secrecy forever: a lock with two keys. One key can be shared with the whole world; the other must be kept hidden. RSA followed in 1977.

Think of a village postbox. Anyone can drop a letter through the slot — that is your public key, or your address. Only you hold the key that opens the door — your private key. With it, you can also sign a message in a way no one can forge, and anyone can check. Every bitcoin payment is such a digital signature.

A private key is a very long number, hard for any human to remember. So in 2013 builders agreed on a kinder form: a seed phrase of twelve or twenty-four ordinary words, from which every key in a wallet can be regrown.

Lose the words, and the coins are lost for good — there is no bank to call. In 2013 a Welsh engineer, James Howells, threw away an old hard drive holding the keys to about 8,000 bitcoin. It lies somewhere in a landfill still. Researchers think millions of bitcoin may be locked away forever by forgotten keys.

Keep the words on paper or steel, never in a photo or a message. Anyone who asks for them is a thief.

Picture a ring-shaped cave with a magic door at the back. Peggy says she knows the password. Victor waits outside while she walks in, then calls out which side she must return from. If she truly knows the word, she can always pass the door and obey. Repeat it twenty times, and Victor is convinced — yet he never hears the password. This is a zero-knowledge proof, first described by Goldwasser, Micali and Rackoff in 1985.

In October 2016, a coin called Zcash used such proofs to let people pay privately while still proving that no coin was forged. Today the same mathematics powers ZK-rollups, which prove to Ethereum that thousands of transactions were done correctly — with a receipt small enough to fit on a single page.