I agree with most of the comments so far, but the group brings up an often
overlooked point with respect to the privacy benefits of taproot. In the extreme
case, if there would be no policies that have both a key and a script spend
path, then taproot does not improve anonymity sets compared to the "Taproot
Public NUMS Optimization" proposal (which saves 8 vbytes in a script-spend). (*)

In fact, the cases where scripts would have to be used given usage of Bitcoin
today are be rare because threshold policies, their conjunctions and
disjunctions can be expressed with a single public key. Even if we disregard
speculation that timelocks, ANYPREVOUT/NOINPUT and other interesting scripts
will be used in the future (which can be added through the leaf or key versions
without affecting key-spend anonymity sets), not all of today's applications are
able to be represented single public keys because there are applications that
can not deal with interactive key setups or interactive signing. For
applications where this is possible it will be a gradual change because of the
engineering challenges involved. For example, k-of-n threshold policies could
have the most likely k-of-k in the taproot output key and other k-of-k in the
leaves, instead of going for a k-of-n taproot output key immediately.

Given that anonymity sets in Bitcoin are permanent and software tends to be
deployed longer than anyone would expect at the time of deployment,
realistically Taproot is superior to the "Public NUMS Optimization" and "An
Alternative Deployment Path".

(*) One could argue that the little plausible deniability gained by a very small
probability of the change of a script-spend being a key-spend and vice versa is
significantly better than no probability at all.

On 2/9/20 8:47 PM, Bryan Bishop via bitcoin-dev wrote:
> 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
>
>
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