Hierarchical determinism: how Bitcoin's HD wallets are born

In my explainer on mnemonic seeds, we explored how recovery phrases emerge from pure entropy - coin tosses, dice rolls, pure randomness. We learned how that randomness is encoded into a string of words: easier to memorize and backup than a really big number.

But that’s not the whole story. After generating a mnemonic, how does a wallet conjure up addresses? Why do those addresses always appear in the same order, every time? How does your seed phrase turn into extended keys, derivation paths, and the whole branching architecture of your wallet?

In this guide, we’ll go on an interactive journey to explore the invention which underpins these questions: hierarchical deterministic (HD) wallets.

If you haven’t already, I recommend you start with my mnemonic article - the seed and passphrase you generate there will follow you into this guide.

Why HD wallets exist

Before HD wallets (BIP 32) and mnemonics (BIP 39), people relied on “JBOK” wallets, or “Just a Bunch of Keys”. Every private key was generated individually, lived alone, and had to be backed up separately.

HD wallets provide an elegant solution. They are:

• Hierarchical: keys live in a tree structure
• Deterministic: the same seed always regenerates the same tree

Modern wallets extract everything from a single 512-bit seed. That seed becomes a master extended private key, which becomes the root of your entire key tree:

The master key - color-coded above - is created by putting the seed through an HMAC function to generate another set of 64 bytes.

• What is HMAC? Hash-based Message Authentication Code is a hashing method. It takes input data + a secret key and outputs fixed-length pseudo-random bytes. We use it repeatedly to “walk” down our wallet tree.
• Why not use the seed directly? The seed and master key are both 64 bytes in length - why do we hash it again? Before BIP 39, seeds weren’t always 64 bytes. Passing it through HMAC is a (somewhat redundant) way to normalise everything.

The wallet as a tree

Your master extended private key sits at the trunk. It’s as sensitive as the seed itself: compromise it, and the entire wallet is gone. This is the key your hardware wallet hides from the world.

Each branch in the tree is created by feeding two things into HMAC:

1. The current extended private key
2. An index number

Each extended key can produce over 4 billion children, giving us huge, flexible space to build a hierarchy. A specific sequence of these branches is called a derivation path.

Let’s explore the many possible derivation paths of your wallet’s tree:

Breaking down these first 3 levels:

• Purpose: Defines the script type. For example, 84 refers to BIP84 (Native SegWit with P2WPKH addresses).
• Coin: A single wallet seed can actually hold multiple cryptocurrencies. The number zero refers to the OG: Bitcoin.
• Account: Does what it says on the tin, enabling segmentation for different purposes (personal, business, savings, whatever).

So for example, if you were setting up a Segwit (BIP 84) Bitcoin wallet, your derivation path might be: m/84'/0'/0'. If you decided to create an additional account, that new account would be found at m/84'/0'/1'.

From extended keys to addresses

These first layers yield your account-level extended keys: the xpub and xprv.

• The xpub (Extended Public Key) is frequently the first key to be surfaced to users in wallet software. It can derive children, but only public keys. My open-source donation widget uses an xpub to derive all its addresses, for example.
• The xprv (Extended Private Key) can derive everything — private keys, public keys, addresses. For obvious reasons, it is highly sensitive.

I’ve glossed over some complexities here. Curious readers can go on deeper technical tangents below, but these points aren’t essential for the rest of the guide.

Now let’s finish the last two levels - the ones your wallet uses every day:

At this depth:

• Level 4 (0 or 1) - shown in pink
  • 0 = receiving addresses, shared for receiving funds
  • 1 = change addresses, used to receive change from transactions
• Level 5 (0, 1, 2, 3…) - shown in yellow
  • The index increments each time you request a fresh address

At this point you’re working with simple keys, not extended ones. The address format (P2WPKH, P2TR, etc.) comes from the purpose field at level 1. A BIP86 purpose (i.e. Taproot, m/86') indicates that the wallet should generate P2TR addresses (starting with bc1p).

Address generation is another topic entirely - perhaps one for my next visual walkthrough.

Bringing it all together

Between this article and the previous one, you’ve gone from pure randomness, to a mnemonic, to a seed, to extended keys, to child keys, to real Bitcoin addresses capable of receiving funds.

And here’s the part which (to me) feels like science-fiction: you could do every step of this offline, and it would still work perfectly.

Paper and dice. No computer. No website or permission needed.

If you meticulously worked through these steps yourself to generate an address, it could receive millions of dollars’ worth of bitcoin in minutes, even if no computer on Earth has ever seen it before. It isn’t registered anywhere. Nobody “issues” it to you. There is no approval list, no form to fill in, no gatekeeper. The keys are yours, and yours alone.

Your address - which is ultimately just an expression of a really big number - could be carved into a stone tablet, or communicated to the outside world via shortwave radio, and anyone in the world would be able to transmit value to it. That is the magic of bitcoin. It’s just mathematics.

Take a step back and appreciate the journey you’ve taken:

Wrapping up

HD wallets are a quiet miracle of Bitcoin UX. They give us:

• A single seed phrase that regenerates an entire wallet structure
• Organised accounts without juggling dozens of keys
• Recoverability across different devices, manufacturers, and software
• Privacy without complexity

I’ve tried to do the topic justice here. While remarkably elegant, there’s a lot of technical depth to this subject that I wanted to do cover properly. For the next article in this series, I’d like to explore address formats: P2PKH vs P2SH vs P2WPKH vs P2TR, how they work and what they reveal.

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