How is a crypto address made from a private key?

A private key is a random number

Mastering Bitcoin puts it simply: a private key is just a number, picked at random.¹ On Ethereum, making one comes down to picking a number between 1 and 2^256.³ The Ethereum Yellow Paper, a technical description of the protocol, describes it as a randomly selected positive integer, stored as 32 bytes.⁴

2^256 is about 10^77, a number with 78 digits. For comparison, Mastering Ethereum notes that the visible universe is estimated to contain about 10^80 atoms.³

ethereum.org says a private key is usually written as 64 hexadecimal ("hex") characters.² Hex uses the digits 0 to 9 and the letters a to f, so each character stands for 4 bits, and 64 × 4 = 256 bits.^2,3

An SEC investor bulletin calls the private key a passcode that lets you approve transactions. It can't be changed or replaced once it is made.⁵

Private key to public key: one way only

The public key is worked out from the private key using a kind of maths called elliptic curve multiplication.³ Bitcoin and Ethereum use the same curve, called secp256k1.³

Mastering Ethereum explains why this matters. The calculation is easy in one direction and impossible in the other, because there is no way to "divide" it back.³ So you can share a public key without giving away the private key.^1,2

The bulletin compares a public key to the email address of your wallet. It lets others check your transactions and send you crypto, but it can't approve transactions.⁵

Public key to Ethereum address

An Ethereum address is made from the public key in four steps.^2,4

1. Start with the public key. The Yellow Paper treats it as 64 bytes: two numbers joined together.⁴
2. Hash it with Keccak-256. A hash turns any input into a fixed-size fingerprint (Module 2 covers hashes).^2,3
3. Keep the last 20 bytes. The Yellow Paper calls this the rightmost 160 bits of the hash.^4,2
4. Write it in hex and add 0x. 20 bytes make 40 hex characters, so the address is 42 characters long.²

A smart contract's address is made differently. It usually comes from the creator's address and a counter of the creator's transactions.² Contract accounts have no private key at all (Module 4 covers them).²

Bitcoin addresses

Bitcoin has several kinds of address. You can tell them apart by how they start (Module 3 covers them).¹

The EIP-55 checksum

A checksum is extra information that lets software spot a typing mistake.^6,3 Mastering Ethereum says Ethereum addresses first had none, and that this led to lost funds from mistyped addresses.³

A 2016 standard, EIP-55, added a checksum without changing the address. It works by choosing which letters (a to f) are written as capitals, based on a hash of the address itself.⁶ The address stays 40 characters long, and software that ignores capitals still reads it.⁶

EIP-55 says that, on average, a mistyped address will pass its check by accident only 0.0247% of the time.⁶ Wallets that support EIP-55 check the capitals when an address mixes capitals and lowercase letters.³ An address pasted all in lowercase still works, but a wallet may have nothing to check, so a typo in it can go unnoticed.^6,3

What a checksum can't do

A checksum catches typos. It can't tell you who owns an address.^6,7 ethereum.org says to make sure the address you send to exactly matches the one you want.⁷

That matters because of a scam called address poisoning (Module 10 covers this scam). MetaMask, a wallet maker, says scammers send you a tiny transaction from an address that looks like one you've used, hoping you'll copy it later.⁸ MetaMask says to check the middle characters, not just the start and the end.⁸