Hi,

> A Taproot output is a SegWit output [...]  with
> version number 1, and a 33-byte witness program whose first byte is 0 or 1.

Given a secret key k and public key P=(x,y), a signer with the knowledge of k
can sign for -P=(x,p-y) since -k is the secret key for that point. Encoding the
y value of the public key therefore adds no security. As an alternative to
providing the y value of the taproot output key Q when constructing the taproot
output, the signer can provide it when signing. We can also restrict the y value
of the internal key P to be even (or high, or a quadratic residue). That gives
us 4 options for how to set the y signs for P and Q.

1. Q sign is explictly set in the witness program, P sign is explicitly set in the control block
    => witness program is 33 bytes, 32 possible leaf versions (one for each pair of 0xc0..0xff)
2. Q sign is explictly set in the witness program, P sign is implicitly even
    => witness program is 33 bytes, 64 possible leaf versions (one for each 0xc0..0xff)
3. Q sign is explictly set in the control block, P sign is explicitly set in the control block
    => witness program is 32 bytes, 16 possible leaf versions (one for each 4-tuple of 0xc0..0xff)
4. Q sign is explictly set in the control block, P sign is implicitly even
    => witness program is 32 bytes, 32 possible leaf versions (one for pair of 0xc0..0xff)

The current proposal uses (1). Using (3) or (4) would reduce the size of a
taproot output by one byte to be the same size as a P2WSH output. That means
that it's not more expensive for senders compared to sending to P2WSH.
 
(Credit to James Chiang for suggesting omitting the y sign from the public key and
to sipa for pointing out the 4 options above)

> (native or P2SH-nested, see BIP141)

I'd prefer to not support P2SH-nested TR. P2SH wrapping was useful for segwit
v0 for compatibility reasons. Most wallets/exchanges/services now support sending
to native segwit addresses (https://en.bitcoin.it/wiki/Bech32_adoption) and that
will be even more true if Schnorr/Taproot activate in 12+ months time.

On Mon, May 6, 2019 at 2:36 PM Pieter Wuille via bitcoin-dev <bitcoin-dev@lists.linuxfoundation.org> wrote:

Hello everyone,

Here are two BIP drafts that specify a proposal for a Taproot
softfork. A number of ideas are included:

* Taproot to make all outputs and cooperative spends indistinguishable
from eachother.
* Merkle branches to hide the unexecuted branches in scripts.
* Schnorr signatures enable wallet software to use key
aggregation/thresholds within one input.
* Improvements to the signature hashing algorithm (including signing
all input amounts).
* Replacing OP_CHECKMULTISIG(VERIFY) with OP_CHECKSIGADD, to support
batch validation.
* Tagged hashing for domain separation (avoiding issues like
CVE-2012-2459 in Merkle trees).
* Extensibility through leaf versions, OP_SUCCESS opcodes, and
upgradable pubkey types.

The BIP drafts can be found here:
* https://github.com/sipa/bips/blob/bip-schnorr/bip-taproot.mediawiki
specifies the transaction input spending rules.
* https://github.com/sipa/bips/blob/bip-schnorr/bip-tapscript.mediawiki
specifies the changes to Script inside such spends.
* https://github.com/sipa/bips/blob/bip-schnorr/bip-schnorr.mediawiki
is the Schnorr signature proposal that was discussed earlier on this
list (See https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2018-July/016203.html)

An initial reference implementation of the consensus changes, plus
preliminary construction/signing tests in the Python framework can be
found on https://github.com/sipa/bitcoin/commits/taproot. All
together, excluding the Schnorr signature module in libsecp256k1, the
consensus changes are around 520 LoC.

While many other ideas exist, not everything is incorporated. This
includes several ideas that can be implemented separately without loss
of effectiveness. One such idea is a way to integrate SIGHASH_NOINPUT,
which we're working on as an independent proposal.

The document explains basic wallet operations, such as constructing
outputs and signing. However, a wide variety of more complex
constructions exist. Standardizing these is useful, but out of scope
for now. It is likely also desirable to define extensions to PSBT
(BIP174) for interacting with Taproot. That too is not included here.

Cheers,

--
Pieter
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