Taproot is where Bitcoin script, elliptic-curve math, and cryptographic commitments finally come together cleanly.

I’ll build this from first principles, then assemble the full Taproot construction. No hand-waving, but I’ll keep the math conceptual, not abstract-symbol heavy.

0. Prerequisites (what I’ll assume you know)

I’ll assume you’re comfortable with:

Elliptic curves conceptually: scalar × generator → point
secp256k1 group arithmetic
Hash functions as random oracles
Bitcoin scripts at a high level

I won’t assume Schnorr familiarity beyond basics.

1. The core Taproot idea (one sentence)

Taproot commits to multiple spending conditions using a single public key, while revealing only the condition you actually use.

Mathematically:

All spend paths are cryptographically committed
Only one path is revealed
Observers cannot tell whether alternatives existed

2. Why Taproot needed new math

Before Taproot:

P2SH / P2WSH:
- Reveal the entire script
- Reveal all branches
Multisig vs single-sig are distinguishable
Privacy and efficiency suffer

Taproot solves this by:

Replacing script hashes with key tweaks
Replacing ECDSA with Schnorr
Using Merkle trees of scripts

3. Schnorr signatures (minimal but critical)

Taproot relies on BIP340 Schnorr signatures, which give us:

Key property we need

Public keys add linearly

If:

P = x·G
Q = y·G

Then:

P + Q = (x + y)·G

ECDSA does not support this cleanly. Schnorr does.

This single fact enables Taproot.

4. X-only public keys (why 32 bytes works)

Traditional pubkey:

(x, y)

Taproot uses:

x-only pubkey

Why safe?

For every x, there are only two possible y values
Schnorr signatures define a canonical choice
The parity of y is encoded elsewhere (control block)

So:

Public key = 32 bytes
Parity is handled implicitly

5. Internal key: the starting point

Taproot begins with an internal public key:

P = x·G

Chosen by wallet
Can represent:
- Single key
- Aggregated MuSig key
By itself, it already allows a key-path spend

But Taproot wants optional scripts too.

6. Scripts become Merkle leaves

Suppose you want multiple spending conditions:

Timelock recovery
Multisig backup
Emergency clause

Each script becomes a TapLeaf:

TapLeaf = H_tapleaf( leaf_version || script )

Where:

leaf_version currently = 0xC0
script = raw Bitcoin script bytes
H_tapleaf is a tagged hash:

H_tapleaf(x) = SHA256(SHA256("TapLeaf") || SHA256("TapLeaf") || x)

(Tagged hashes prevent cross-protocol collisions.)

7. Build the Taproot Merkle tree

All TapLeaf hashes are combined pairwise:

TapBranch(a, b) = H_tapbranch( min(a,b) || max(a,b) )

Important:

Ordering is lexicographic
Tree shape doesn’t matter
Only the final root matters

Final result:

merkle_root

If you have no scripts:

merkle_root = ∅

8. The Taproot tweak (THIS is the key math)

Now comes the heart of Taproot.

Compute tweak scalar

t = H_taptweak( P || merkle_root )

If no scripts:

t = H_taptweak( P )

Where:

H_taptweak(x) = SHA256(SHA256("TapTweak") || SHA256("TapTweak") || x)

Interpret t as a scalar mod n.

Compute output key

This is the Taproot output key:

Q = P + t·G

This is what goes on-chain.

Critical insight

The blockchain only sees Q
No one can tell:
- whether scripts exist
- how many scripts exist
Unless a script path is actually used

This is cryptographic commitment via elliptic-curve addition.

9. Address creation (P2TR)

The output script is:

OP_1 <x(Q)>

x(Q) = 32-byte x-coordinate of Q
Address encodes this as bc1p...

10. Spending via key path (most common)

If you control the private key x:

Private key tweak

q = x + t   (mod n)

Now:

Q = q·G

So you can produce a normal Schnorr signature using q.

Witness:

<64-byte schnorr signature>

That’s it.

Observers learn:

Only that a valid Schnorr signature exists
No scripts
No branches
No policy

This is Taproot’s privacy jackpot.

11. Spending via script path (when needed)

Now suppose you use a script.

You must reveal:

The script
The Merkle proof
The internal key
The parity bit

This is encoded in the control block.

12. Control block: exact structure

Control block bytes:

[1 byte]   control
[32 bytes] internal public key P
[32 bytes] merkle sibling #1
[32 bytes] merkle sibling #2
...

Control byte (very important)

Bits:

bit 0: parity of output key Q (even/odd y)
bits 1–7: leaf version (currently 0xC0)

So:

control = leaf_version | parity_bit

13. How script-path verification works

The verifier does:

Extract P from control block
Hash the revealed script → TapLeaf
Recompute Merkle root using provided siblings
Compute:

t = H_taptweak( P || merkle_root )
Q' = P + t·G

Check:

x(Q') == x(Q_from_scriptPubKey)

If equal → the script was committed at output creation time.

Then:

Script is executed
Witness stack items are applied
Script must evaluate to true

14. Why control blocks don’t leak privacy

Only one leaf is revealed
Other branches are hidden by hashes
Tree size and structure are unknown
Even existence of alternative scripts is unknowable

15. Why Taproot is strictly better than P2WSH

Feature	P2WSH	Taproot
Reveal all scripts	❌	✅ no
Hide unused paths	❌	✅
Single-sig indistinguishable	❌	✅
Key aggregation	❌	✅
Signature size	Large	Fixed 64B

16. Mental model (this helps a lot)

Think of Taproot as:

“A public key with optional encrypted clauses attached.”

Key-path spend = don’t open the envelope
Script-path spend = open exactly one envelope
Control block = proof the envelope existed

17. One extremely important consequence (often missed)

Because:

Q = P + H(P || scripts)·G

👉 Scripts influence the public key

This means:

You cannot change scripts later
You cannot lie about policy
Wallets must commit carefully

Taproot is binding, not just hiding.

18. Where this matters in real systems (your world)

Given your interest in hardware wallets / signers:

Key-path spends:
- Signer only needs tweaked private key
- No script parsing
- Very MCU-friendly
Script-path spends:
- Host provides script + control block
- Signer verifies tweak math
- Excellent for air-gapped PSBT flows

This is why Taproot is ideal for:

Cold wallets
Multisig with recovery paths
Policy wallets (miniscript → taproot)

If you want, next I can:

Derive a Taproot output numerically step-by-step
- actual hex
- actual hashes
Explain MuSig2 inside Taproot
Explain Taproot sighash math (BIP341)
Design a Taproot-based hardware wallet policy
Show Rust / embedded-friendly code for tweak verification

If you want maximum payoff, say “1” or “4”.