Taproot (and graftroot) complexity

      /\
     /  \
    /    \
   /      \
  /\      /\
 /  \    /  \
/\  /\  /\  /\
a b c d e f g h
       /\
      /\ {<pk> schnorr_checksig}
     /  \
    /    \
   /      \
  /\      /\
 /  \    /  \
/\  /\  /\  /\
a b c d e f g h
      /\
     /  \
    /    \
   /      \
  /\      /\
 /  \    /  \
/\  /\  /\  /\
a b c d e f/\ {<pk> schnorr_checksig}
          g  h
1) A separate soft-fork for Merkle Branch Witnesses based on Taproot;
2) A separate soft-fork for Schnorr Signatures
3) A separate follow up soft-fork which enables Taproot and Graftroot
1) A separate soft-fork for Merkle Branch Witnesses based on Taproot;
2) A separate soft-fork for Schnorr Signatures
3) A separate follow up soft-fork which enables Taproot and Graftroot
> Is Taproot just a probability assumption about the frequency and
> likelihood of
> the signature case over the script case? Is this a good assumption?  The BIP
> only goes as far as to claim that the advantage is apparent if the outputs
> *could be spent* as an N of N, but doesn't make representations about
> how likely
> that N of N case would be in practice compared to the script paths. Perhaps
> among use cases, more than half of the ones we expect people to be doing
> could be
> spent as an N of N. But how frequently would that path get used?
> Further, while
> the *use cases* might skew toward things with N of N opt-out, we might
> end up in
> a power law case where it's the one case that doesn't use an N of N opt
> out at
> all (or at a de minimis level) that becomes very popular, thereby making
> Taproot
> more costly then beneficial.
      /\
     /  \
    /    \
   /      \
  /\      /\
 /  \    /  \
/\  /\  /\  /\
a b c d e f g h
       /\
      /\ {<pk> schnorr_checksig}
     /  \
    /    \
   /      \
  /\      /\
 /  \    /  \
/\  /\  /\  /\
a b c d e f g h
      /\
     /  \
    /    \
   /      \
  /\      /\
 /  \    /  \
/\  /\  /\  /\
a b c d e f/\ {<pk> schnorr_checksig}
          g  h
1) A separate soft-fork for Merkle Branch Witnesses based on Taproot;
2) A separate soft-fork for Schnorr Signatures
3) A separate follow up soft-fork which enables Taproot and Graftroot
> In particular, you care more about privacy when you are contesting a
> close of a channel or other script path because then the miners could be
> likely to extract a rent from you as "ransom" for properly closing your
> (or in other words, in a contested close the value of the closing
> larger than usual).
> Responding purely to one point as this may be sufficient to clear up
> lots of discussion:
>
> On 2/9/20 8:19 PM, Bryan Bishop via bitcoin-dev wrote:
> > Is Taproot just a probability assumption about the frequency and
> > likelihood of
> > the signature case over the script case? Is this a good assumption?  The
> BIP
> > only goes as far as to claim that the advantage is apparent if the
> outputs
> > *could be spent* as an N of N, but doesn't make representations about
> > how likely
> > that N of N case would be in practice compared to the script paths.
> Perhaps
> > among use cases, more than half of the ones we expect people to be doing
> > could be
> > spent as an N of N. But how frequently would that path get used?
> > Further, while
> > the *use cases* might skew toward things with N of N opt-out, we might
> > end up in
> > a power law case where it's the one case that doesn't use an N of N opt
> > out at
> > all (or at a de minimis level) that becomes very popular, thereby making
> > Taproot
> > more costly then beneficial.
> Its not just about the frequency and likelihood, no. If there is a
> clearly-provided optimization for this common case in the protocol, then
> it becomes further more likely that developers put in the additional
> effort required to make this possibility a reality. This has a very
> significant positive impact on user privacy, especially those who wish
> to utilize more advanced functionality in Bitcoin. Further, yes, it is
> anticipated that the N of N case is possible to take in the vast
> majority of deployed use-cases for advanced scripting systems, ensuring
> that it is maximally efficient to do so (and thereby encouraging
> developers to do so) is a key goal in this work.
>
> Matt
> _______________________________________________
> bitcoin-dev mailing list
> bitcoin-dev@lists•linuxfoundation•org
> https://lists.linuxfoundation.org/mailman/listinfo/bitcoin-dev
>
> 1) Is Taproot actually more private than bare MAST and Schnorr separately?
> What are the actual anonymity set benefits compared to doing the separately?
    - single-sig spends (similar to current use of P2PKH and P2WPKH)
    - n-of-n spends with musig or equivalent (similar to current use of
      P2SH and P2WSH 2-of-2 multisig without special features as used by
      Blockstream Green and LN mutual closes)
    - k-of-n (for low values of n) using the most common k signers
      (similar to BitGo-style 2-of-3 where the keys involved are
      alice_hot, alice_cold, and bob_hot and almost all transactions are
      expected to be signed by {alice_hot, bob_hot}; that common case
      can be the key-path spend and the alternatives {alice_hot,
      alice_cold} and {alice_cold, bob_hot} can be script-path spends)
    - contract protocols that can sometimes result in all parties
      agreeing on an outcome (similar to LN mutual closes, cross-chain
      atomic swaps, and same-chain coinswaps)
> 2) Is Taproot actually cheaper than bare MAST and Schnorr separately?
- (36 vbytes) outpoint
- (1) scriptSig compactSize uint
- (4) nSequence
- (16.25) schnorr signature (includes size byte)
> What evidence do we have that the assumption it will be more common to
> use Taproot with a key will outweigh Script cases?
> However, after
> our review, we're left perplexed about the development of Taproot (and
> Graftroot, to a lesser extent).
> In essence, Taproot is fundamentally the same as doing
> https://github.com/bitcoin/bips/blob/master/bip-0114.mediawiki and Schnorr
> signatures separately.
>
> Suppose a MAST for {a,b,c,d,e,f,g,h} spending conditions it looks something
> like this:
>
>       /\
>      /  \
>     /    \
>    /      \
>   /\      /\
>  /  \    /  \
> /\  /\  /\  /\
> a b c d e f g h
>
> If we want this to be functionally equivalent to Taproot, we add a new path:
>
>        /\
>       /\ {<pk> schnorr_checksig}
>      /  \
>     /    \
>    /      \
>   /\      /\
>  /  \    /  \
> /\  /\  /\  /\
> a b c d e f g h
    /\
   a /\
    b  c
  60%: a [#(b,c)]
  20%: b [#a, #c]
  20%: c [#a, #b]
  60%: a-tweaked
  20%: b [a, #c]
  20%: c [a, #b]
> However, if we do the same script via taproot, we now need to provide the base
> public key (33 bytes) as well as the root hash (32 bytes) and path and then the
> actual scripts.
>       /\
>      /  \
>     /    \
>    /      \
>   /\      /\
>  /  \    /  \
> /\  /\  /\  /\
> a b c d e f/\ {<pk> schnorr_checksig}
>           g  h
>
> We could argue that this is more private than Taproot, because we don't
> distinguish between the Schnorr key case and other cases by default, so chain
> analyzers can't tell if the signature came from the Taproot case or from one of
> the Script paths.
> This
> allows you to not completely fracture the anonymity set between people who want
> plain Schnorr and people who want MAST
> (at least until they go to spend).
> Overall, we are left with concerns both about the merit of doing Taproot
> versus alternatives, as well as the process through which we got to be here.
> 1) Is Taproot actually more private than bare MAST and Schnorr separately? What
> are the actual anonymity set benefits compared to doing the separately?
> 2) Is Taproot actually cheaper than bare MAST and Schnorr separately? What
> evidence do we have that the assumption it will be more common to use Taproot
> with a key will outweigh Script cases?
 - spent via key path
 - spent via script path, internal key not known
 - spent via script path, internal key known NUMS point
     738  0.3% "pubkey"
    2091  0.8% "witness_v0_scripthash"
   42749 16.8% "witness_v0_keyhash"
  102962 40.4% "pubkeyhash"
  106441 41.7% "scripthash"
> 3) Is Taproot riskier than bare MAST and Schnorr separately given the new
> crypto?
> How well reviewed is the actual crypto parts?
> None of us personally feel
> comfortable reviewing the crypto in Schnorr -- what's the set of people who
> have
> thoroughly reviewed the crypto and aren't just ACKing because they trust other
> developers to have looked at it close enough?
> 4) Design wise, couldn't we forego the NUMS point requirement and be able to
> check if it's a hash root directly?
> 5) Is the development model of trying to jam a bunch of features into Bitcoin
> all at once good for Bitcoin development? Would we be better off if we embraced
> incremental improvements that can work together (e.g., MAST and then Schnorr)?
 - you can have either a simple public-key and signature to authorise
   a spend, or you can have a script, and decide which to use when
   you spend
 - using the key path comes at no cost compared to not using taproot
 - adding a script path comes at no cost if you don't end up using it
 - if you can interactively verify the script conditions off-chain,
   you can always use the key path
> SUBJECT: An Alternative Deployment Path for Taproot Technologies
> 1) A separate soft-fork for Merkle Branch Witnesses based on Taproot;
  https://notes.ethereum.org/@vbuterin/rkhCgQteN
  https://www.btc-way.com/?p=8153
> 2) A separate soft-fork for Schnorr Signatures
> 3) A separate follow up soft-fork which enables Taproot and Graftroot
> users only have to use Schnorr signing if they want to, and can otherwise
> continue to use ECDSA.
>
>       /\
>      /  \
>     /    \
>    /      \
>   /\      /\
>  /  \    /  \
> /\  /\  /\  /\
> a b c d e f g h
* If you can induce all participants to sign a transaction using the keypath spend, then you gain privacy (no part of the MAST is ever published, not even its root or the presence of the MAST!) *and* reduced onchain fees (because the MAST is not published and does not take up space on the blockchain).
  * You can incentivize cooperation (beyond just the incentive of improved privacy) by letting participants recover some of the saved onchain fees.
    Lightning does this, for example: the funder of the channel is the one paying for the closing fees, and the closing fee of the mutual close is almost always lower than the unilateral close case (or else is equal: the closing ritual has the unilateral close fee as the upper bound on whatever fee can be proposed at the mutual close ritual).
* Even if a participant does not cooperate (for example, it might have been hit by a rogue asteroid in the meantime) we still have the fallback of revealing the entire MAST.
> Apologies for my previous attempt at relaying the message- it looks like
> the emails got mangled on the archive. I am re-sending them in this
> combined email with what I hope will be better formatting. Again this is
> from some nym that had trouble posting to this mailing list; I didn't see
> any emails in the queue so I couldn't help to publish this sooner.
>
> SUBJECT: Taproot (and Graftroot) Complexity
>
> This email is the first of a collection of sentiments from a group of
> developers who in aggregate prefer to remain anonymous. These emails have
> been sent under a pseudonym so as to keep the focus of discussion on the
> merits of the technical issues, rather than miring the discussion in
> personal politics.  Our goal isn't to cause a schism, but rather to help
> figure out what the path forward is with Taproot. To that end, we:
>
> 1) Discuss the merits of Taproot's design versus simpler alternatives (see
> thread subject, "Taproot (and Graftroot) Complexity").
>
> 2) Propose an alternative path to deploying the technologies described in
> BIP-340, BIP-341, and BIP-342 (see thread subject, "An Alternative
> Deployment Path for Taproot Technologies").
>
> 3) Suggest a modification to Taproot to reduce some of the overhead (see
> thread subject, "Taproot Public NUMS Optimization").
>
> Now that the BIP has moved to draft we felt that now was the time to
> prioritize review to make sure it was an acceptable change for our
> activities. As a group, we're excited about the totality of what Taproot
> has to offer. However, after our review, we're left perplexed about the
> development of Taproot (and Graftroot, to a lesser extent).
>
> We also want to convey that we have nothing but respect for the developers
> and community who have poured their heart and soul into preparing Taproot.
> Self evidently, it is an impressive synthesis of ideas. We believe that the
> highest form of respect to pay such a synthesis of ideas is a detailed and
> critical review, as it's pertinent to closely consider changes to Bitcoin.
>
>
> In essence, Taproot is fundamentally the same as doing
> https://github.com/bitcoin/bips/blob/master/bip-0114.mediawiki and Schnorr
> signatures separately.
>
> The main reason for putting them together -- as mentioned in the BIP -- is
> a gain in efficiency. But this efficiency pre-supposes a specific use case
> and probability distribution of use cases.
>
> Compare:
>
> Suppose a MAST for {a,b,c,d,e,f,g,h} spending conditions it looks something
> like this:
>
>
>       /\
>      /  \
>     /    \
>    /      \
>   /\      /\
>  /  \    /  \
> /\  /\  /\  /\
> a b c d e f g h
>
> If we want this to be functionally equivalent to Taproot, we add a new path:
>
>        /\
>       /\ {<pk> schnorr_checksig}
>      /  \
>     /    \
>    /      \
>   /\      /\
>  /  \    /  \
> /\  /\  /\  /\
> a b c d e f g h
>
> Now, to spend from this MBV you have to reveal 32 bytes on the stack for
> the not taken branch, and 35 bytes for the <pk> schnorr_checksig (1 byte
> push, 33 bytes PK, 1 byte checksig).
>
> This is 67 bytes more than Taproot would require for the same spending
> condition.
>
> However, suppose we wanted to use one of the script paths instead. We still
> need to have one extra hash for the {<pk> schnorr_checksig} (depending on
> if we put the key in this position or not--see below). But now we can spend
> with just a logarithmic control program path.
>
> However, if we do the same script via taproot, we now need to provide the
> base public key (33 bytes) as well as the root hash (32 bytes) and path and
> then the actual scripts. With the need for 2 push bytes, this ends up being
> back at 67 bytes extra.
>
> Is Taproot just a probability assumption about the frequency and likelihood
> of the signature case over the script case? Is this a good assumption?  The
> BIP only goes as far as to claim that the advantage is apparent if the
> outputs *could be spent* as an N of N, but doesn't make representations
> about how likely that N of N case would be in practice compared to the
> script paths. Perhaps among use cases, more than half of the ones we expect
> people to be doing could be spent as an N of N. But how frequently would
> that path get used? Further, while the *use cases* might skew toward things
> with N of N opt-out, we might end up in a power law case where it's the one
> case that doesn't use an N of N opt out at all (or at a de minimis level)
> that becomes very popular, thereby making Taproot more costly then
> beneficial.
>
> Further, if you don't want to use a Taproot top-level key (e.g., you need
> to be able to audit that no one can spend outside of one of the script
> conditions), then you need to use a NUMS (nothing up my sleeve) point. This
> forces users who don't want Taproot to pay the expense, when if they just
> had a MAST based witness type they would be cheaper. So if this use case is
> at all common, Taproot leaves them worse off in terms of fees. Given that
> script paths are usually done in the case where there is some contested
> close, it's actually in the interest of protocol developers that the
> contested script path be as efficient as possible so that the fees paid
> maximally increase the feerate. We think this can be fixed simply in
> Taproot though, as noted below.
>
>
>
> On privacy, we're also a bit confused as to the goal of Taproot over MAST
> and Schnorr. Earlier, we presented a design with MAST which is very close
> to Taproot.  However, it'd also be possible to just add {<pk>
> schnorr_checksig} to the set {a,b,c,d,e,f,g,h}, shuffle them, and compute
> some MAST structure (perhaps probability encoded) on them. This has the
> effect of not having much additional fees for adding the extra Schnorr path
> at redeem time (only 1 extra branch on 2/8 script paths), e.g.
>
>
>       /\
>      /  \
>     /    \
>    /      \
>   /\      /\
>  /  \    /  \
> /\  /\  /\  /\
> a b c d e f/\ {<pk> schnorr_checksig}
>           g  h
>
> We could argue that this is more private than Taproot, because we don't
> distinguish between the Schnorr key case and other cases by default, so
> chain analyzers can't tell if the signature came from the Taproot case or
> from one of the Script paths. There's also no NUMS point required, which
> means chain analyzers can't tell when you spend that there was no top level
> key if the NUMS point is not per-output indistinguishable. By using a
> semi-randomized MAST structure, chain analyzers also can't tell exactly how
> big your spend condition MAST was. In particular, you care more about
> privacy when you are contesting a close of a channel or other script path
> because then the miners could be more likely to extract a rent from you as
> "ransom" for properly closing your channel (or in other words, in a
> contested close the value of the closing transaction is larger than usual).
>
> It would also be possible to do something really simple which is to allow
> the witness type to be either a MAST hash OR a schnorr key (but not a
> Taproot). This allows you to not completely fracture the anonymity set
> between people who want plain Schnorr and people who want MAST (at least
> until they go to spend). This fix can also be used in Taproot in place of a
> NUMS point, to decrease extra fees. It's unclear if this plays negatively
> with any future batch validation mechanism though, but the contextual
> checks to exclude a witness program from the batch are relatively simple.
> See thread subject, "Taproot Public NUMS Optimization".
>
> The considerations around Graftroot, a proposed delegation mechanism, is a
> bit similar. Delegation is a mechanism by which a UTXO with script S can
> sign a script R which can then be executed in addition to S without
> requiring a transaction. This allows an output to monotonically and
> dynamically increase the number of conditions under which it can be spent.
> As noted by Pieter Wiulle here:
> https://github.com/kanzure/diyhpluswiki/commit/a03f6567d714f8733b578de263a4b149441cd058
> delegation was originally possible in Bitcoin, but got broken during an
> emergency fork to split the scriptSig and scriptpubkey separation. Rather
> than adding some fancy delegation mechanism in Bitcoin, why not just have a
> P2SH-like semantic which allows a delegated script to be evaluated? See
> BIP-117 https://github.com/bitcoin/bips/blob/master/bip-0117.mediawiki.
> This way we aren't special casing where delegation can occur, and we can
> allow taproot nested spending conditions (i.e., with timelocks) to generate
> their own delegations. As I've seen Graftroot discussed thus far, it is as
> a top-level witness program version like Taproot and non-recursive. Similar
> to the above discussion, top-level is more efficient if you suspect that
> delegation will be most likely occurring at the top level, but it's not
> clear that's a good assumption as it may be common to want to allow
> different scripts to delegate.
>
>
> Overall, we are left with concerns both about the merit of doing Taproot
> versus alternatives, as well as the process through which we got to be here.
>
> 1) Is Taproot actually more private than bare MAST and Schnorr separately?
> What are the actual anonymity set benefits compared to doing the separately?
>
> 2) Is Taproot actually cheaper than bare MAST and Schnorr separately? What
> evidence do we have that the assumption it will be more common to use
> Taproot with a key will outweigh Script cases?
>
> 3) Is Taproot riskier than bare MAST and Schnorr separately given the new
> crypto? How well reviewed is the actual crypto parts? None of us personally
> feel comfortable reviewing the crypto in Schnorr -- what's the set of
> people who have thoroughly reviewed the crypto and aren't just ACKing
> because they trust other developers to have looked at it close enough?
>
> 4) Design wise, couldn't we forego the NUMS point requirement and be able
> to check if it's a hash root directly? This would encumber users who don't
> need the key path a cheaper spend path. See thread subject, "Taproot Public
> NUMS Optimization".
>
> 5) Is the development model of trying to jam a bunch of features into
> Bitcoin all at once good for Bitcoin development? Would we be better off if
> we embraced incremental improvements that can work together (e.g., MAST and
> then Schnorr)?  Although the BIP raises some points about anonymity sets
> being why to do them all at once, it's not clear to me this argument holds
> water (same goes for businesses not upgrading). If we can take things as
> smaller steps, we are not only more secure, but we also have more time to
> dedicate review to each change independently. We also end up co-mingling
> changes that people end up accepting only because they want one and they're
> bundled (e.g., MAST and Schnorr, MAST seems like a much less risky addition
> versus Schnorr). See thread subject, "An Alternative Deployment Path for
> Taproot Technologies".
>
>
>
>
> Our provocation with this email is primarily that we think we should more
> carefully consider the benefits of Taproot over simpler primitives that are
> not only easier to review, but could have been made available much sooner
> rather than waiting on putting everything all together for an unclear
> aggregate benefit.
>
> We do think that most of the developers have been honest about the benefits
> of Taproot, but that on closer look we feel the general ecosystem has
> oversold Taproot as being the key enabler for a collection of techniques
> that we could do with much simpler building blocks.
>
>
> At the end of the day, we do not strongly advocate not deploying Taproot at
> this point in the review cycle. We think the Taproot Public NUMS
> Optimization may be a good idea, worth considering if it's not insecure, as
> it cuts through the case where you would otherwise need a NUMS point.
> Things like TapScript and its MAST mechanisms are well designed and offer
> exciting new deployment paths, and would be something we would use even if
> we opted for MAST instead of Taproot. However, we also believe it is our
> duty to raise these concerns and suggestions, and we look forward to
> listening to the responses of the community.
>
> Great thanks,
>
> The Group
>
> ----
>
> SUBJECT: An Alternative Deployment Path for Taproot Technologies
>
> This email is the second of a collection of sentiments from a group of
> developers who in aggregate prefer to remain anonymous. These emails have
> been sent under a pseudonym so as to keep the focus of discussion on the
> merits of the technical issues, rather than miring the discussion in
> personal politics. Our goal isn't to cause a schism, but rather to help
> figure out what the path forward is with Taproot. To that end, we: [clip
> repeat]
>
> As a follow up to our prior message, we propose a different path forward
> for the Taproot family of changes:
>
> 1) A separate soft-fork for Merkle Branch Witnesses based on Taproot;
>
> 2) A separate soft-fork for Schnorr Signatures
>
> 3) A separate follow up soft-fork which enables Taproot and Graftroot
>
> We think that the first 2 forks can be offered at the same time or one at a
> time.
>
> Taproot, as a follow up to changes 1 and 2, can be enabled as a soft-fork
> on the existing semantics, but requiring a new witness version. With the
> Public NUMS Optimization, wallets could upgrade by just changing one
> version byte to be in the same anonymity set as Taproot.
>
> It's not clear to us that the time to prepare a BIP and implementation for
> 1 and 2 at this point would be any less than the time to do Taproot as
> currently proposed. However, we believe that such a deployment plan is a
> reasonable option as it is more conservative, as Merkle Branch witnesses
> are relatively simple and users only have to use Schnorr signing if they
> want to, and can otherwise continue to use ECDSA. A further benefit of
> waiting on 3 is that we get to collect real world protocol engineering
> experience to see how frequently the Taproot frequency of use assumption
> holds, and if it is worth doing or not.
>
>
> Great thanks,
>
> The Group
>
>
> ----
>
> SUBJECT: Taproot Public NUMS Optimization
>
> This email is the third of a collection of sentiments from a group of
> developers who in aggregate prefer to remain anonymous. These emails have
> been sent under a pseudonym so as to keep the focus of discussion on the
> merits of the technical issues, rather than miring the discussion in
> personal politics. Our goal isn't to cause a schism, but rather to help
> figure out what the path forward is with Taproot. To that end, we: [clipped
> again]
>
> We propose to modify Taproot's specification in BIP-341 by adding the rule:
>
> If there is one element on the witness stack:
>
> 1) Attempt hashing it to see if it's equal to  the witness program. The
> first byte is the control byte for leaf versioning.
>
> 2) If it's not the witness program, and it's 65 bytes, try signature
> validation
>
> If there is more than one element on the witness stack:
>
> If the control block is even, treat it as a non-Taproot MAST and get the
> leaf version as the last byte of the script (so you can pop it off before
> hashing).
>
>
> If greater anonymity is required, a NUMS point can still be used in
> Taproot, at the expense of the additional data. However, if NUMS points are
> just a couple well known constants this could actually decrease privacy as
> then the NUMS points could differ from application to application
> fingerprinting wallets.  Instead, the NUMS point should only be used when a
> single use nonce can be sent, so that NUMS cannot be distinguished from a
> normal Taproot to a third party who doesn't know the setup (e.g., that the
> NUMS is H(X) for known X).
>
>
> Great thanks,
>
> The Group
>
>
> _______________________________________________
> bitcoin-dev mailing list
> bitcoin-dev@lists•linuxfoundation•org
> https://lists.linuxfoundation.org/mailman/listinfo/bitcoin-dev
>
*When schnorr and taproot are done together, all of the following
transaction types can be part of the same set:     - single-sig spends
(similar to current use of P2PKH and P2WPKH)     - n-of-n spends with musig
or equivalent (similar to current use of       P2SH and P2WSH 2-of-2
multisig without special features as used by       Blockstream Green and LN
mutual closes)     - k-of-n (for low values of n) using the most common k
signers       (similar to BitGo-style 2-of-3 where the keys involved are
    alice_hot, alice_cold, and bob_hot and almost all transactions are
  expected to be signed by {alice_hot, bob_hot}; that common case       can
be the key-path spend and the alternatives {alice_hot,       alice_cold}
and {alice_cold, bob_hot} can be script-path spends)     - contract
protocols that can sometimes result in all parties       agreeing on an
outcome (similar to LN mutual closes, cross-chain       atomic swaps, and
same-chain coinswaps) *
1) Public NUMS -- this is a constant, HashToCurve("I am a NUMS Point").
Anyone scanning the chain can see spends are using this constant. Hopefully
everyone uses the same constant (or everyone uses 2,3,4) so that "what type
of NUMS you are using" isn't a new fingerprint.
2) Moslty Public NUMS -- I take the hash of some public data (like maybe
the txid) on some well defined protocol, and use that. Anyone scanning the
chain and doing an EC operation per-txid can see I'm using a constant --
maybe my HashToCurve takes 10 seconds (perhaps through a VDF to make it
extra annoying for anyone who hasn't been sent the shortcut), but in
practice it's no better than 1.
3) Interactive NUMS -- I swap H(Rx), H(Ry) with the other participant and
then NUMS with H(Rx || Ry). This is essentially equivalent to using a MuSig
key setup where one person's key is a NUMS. Now no one passively scanning
can see that it's NUMS, but I can prove to an auditor later.
4) 1/2 RTT Async-Interactive NUMS -- I take some public salt -- say the
txid T, and hash it with a piece of random data R and then HashToCurve(T ||
R)... I think this is secure? Not clear the txid adds any security. Now I
can prove to you that the hash was based on the txid, but I've blinded it
with R to stop passive observers. But I also need ot send you data out of
band for R (but I already had to do this for Taproot maybe?)
> Is the same if Schnorr + Merkle Branch without Taproot optimization, unless
> I'm missing something in one of the cases?
- P2WPKH scriptpubkey (22.00 vbytes): `OP_0 PUSH20 <hash160>`
- P2WPKH witness data (26.75): `size(72) <sig>, size(33) <pubkey>`
- P2TR scriptpubkey (34.00): `OP_1 PUSH32 <pubkey>`
- P2TR witness data (16.25): `size(64) <sig>`
- BIP116 MBV P2WSH scriptpubkey (34.00): `OP_0 PUSH32 <sha256>`
- BIP116 MBV P2WSH witness data (42.00): `size(64) <signature>, size(32)
  <pubkey>, size(32) <inclusion_proof>, size(36) <PUSH1 <n> PUSH32
  <merkle_root> OP_MBV>`
P.S. I think this branch of the thread is just rehashing points that
     were originally covered over two years ago and which haven't really
     changed since then.  E.g.:
    https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2018-January/015629.html
>
> On Fri, Feb 14, 2020 at 12:07:15PM -0800, Jeremy via bitcoin-dev wrote:
> > Is the same if Schnorr + Merkle Branch without Taproot optimization, unless
> > I'm missing something in one of the cases?
>
> That's fair.  However, it's only true if everyone constructs their
> merkle tree in the same way, with a single `<schnorr_pk> OP_CHECKSIG` as
> one of the top leaves.   Taproot effectively standardizes the position
> of the all-parties-agree condition and so its anonymity set may contain
> spends from scripts whose creators buried or excluded the the all-agree
> option because they didn't think it was likely to be used.
>
> More importantly, there's no incentive for pure single-sig users to use a
> merkle tree, since that would make both the scriptPubKey and the witness
> data are larger for them than just continuing to use v0 segwit P2WPKH.
> Given that single-sig users represent a majority of transactions at
> present (see AJ Towns's previous email in this thread), I think we
> really want to make it as convenient as possible for them to participate
> in the anonymity set.
> At the time you create a utxo, provided you don't reuse keys, all taproot
> spends are indistinguishable. At the time you spend a taproot utxo,
> At the time you create a utxo, provided you don't reuse keys, all taproot
> spends are indistinguishable. At the time you spend a taproot utxo,