Subject: Re: [bitcoin-dev] Soft-forks and schnorr signature aggregation

From: ZmnSCPxj <[email-masked]>
Date: Wed, 21 Mar 2018 03:53:59 -0400

Good morning aj,

I am probably wrong, but could solution 2 be simplified by using the below opcodes for aggregated signatures?

OP_ADD_AGG_PUBKEY - Adds a public key for verification of an aggregated signature.

OP_CHECK_AGG_SIG[VERIFY] - Check that the gathered public keys matches the aggregated signature.

Then:

 pubkey1 OP_ADD_AGG_PUBKEY
 OP_IF
   pubkey2 OP_ADD_AGG_PUBKEY
 OP_ELSE
   cond OP_CHECKCOVENANT
 OP_ENDIF
 OP_CHECK_AGG_SIG

(omitting the existence of buckets)

I imagine that aggregated signatures, being linear, would allow pubkey to be aggregated also by adding the pubkey points (but note that I am not a mathematician, I only parrot what better mathematicians say) so OP_ADD_AGG_PUBKEY would not require storing all public keys, just adding them linearly.

The effect is that in the OP_CHECKCOVENANT case, pre-softfork nodes will not actually do any checking.

OP_CHECK_AGG_SIG might accept the signature on the stack (combined signature of pubkey1 and pubkey2 and from other inputs), or the bucket the signature is stored in.

We might even consider using the altstack: no more OP_ADD_AGG_PUBKEY (one less opcode to reserve!), just push pubkeys on the altstack, and OP_CHECK_AGG_SIG would take the entire altstack as all the public keys to be used in aggregated signature checking.

This way, rather than gathering signatures, we gather public keys for aggregate signature checking.  OP_RETURN_TRUE interacts with that by not performing aggregate signature checking at all if we encounter OP_RETURN_TRUE first (which makes sense: old nodes have no idea what OP_RETURN_TRUE is really doing, and would fail to understand all its details).

I am very probably wrong but am willing to learn how to break the above, though.  I am probably making a mistake somewhere.

Regards,
ZmnSCPxj

​Sent with ProtonMail Secure Email.​

‐‐‐‐‐‐‐ Original Message ‐‐‐‐‐‐‐

On March 21, 2018 12:06 PM, Anthony Towns via bitcoin-dev <[email-masked]> wrote:

> Hello world,
> 
> There was a lot of discussion on Schnorr sigs and key and signature
> 
> aggregation at the recent core-dev-tech meeting (one relevant conversation
> 
> is transcribed at \[0\]).
> 
> Quick summary, with more background detail in the corresponding footnotes:
> 
> signature aggregation is awesome \[1\], and the possibility of soft-forking
> 
> in new opcodes via OP\_RETURN\_VALID opcodes (instead of OP_NOP) is also
> 
> awesome \[2\].
> 
> Unfortunately doing both of these together may turn out to be awful.
> 
> RETURN_VALID and Signature Aggregation
> 
> 
> -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
> 
> Bumping segwit script versions and redefining OP_NOP opcodes are
> 
> fairly straightforward to deal with even with signature aggregation,
> 
> the straightforward implementation of both combined is still a soft-fork.
> 
> RETURN_VALID, unfortunately, has a serious potential pitfall: any
> 
> aggregatable signature operations that occur after it have to go into
> 
> separate buckets.
> 
> As an example of why this is the case, imagine introducing a covenant
> 
> opcode that pulls a potentially complicated condition from the stack
> 
> (perhaps, "an output pays at least 50000 satoshi to address xyzzy"),
> 
> checks the condition against the transaction, and then pushes 1 (or 0)
> 
> back onto the stack indicating compliance with the covenant (or not).
> 
> You might then write a script allowing a single person to spend the coins
> 
> if they comply with the covenant, and allow breaking the covenant with
> 
> someone else's sign-off in addition. You could write this as:
> 
> pubkey1 CHECKSIGVERIFY
> 
> cond CHECKCOVENANT IFDUP NOTIF pubkey2 CHECKSIG ENDIF
> 
> If you pass the covenant, you supply "SIGHASHALL|BUCKET_1" and aggregate
> 
> the signature for pubkey1 into bucket1 and you're set; otherwise you supply
> 
> "SIGHASHALL|BUCKET\_1 SIGHASHALL|BUCKET\_1" and aggregate signatures for both
> 
> pubkey1 and pubkey2 into bucket1 and you're set. Great!
> 
> But this isn't a soft-fork: old nodes would see this script as:
> 
> pubkey1 CHECKSIGVERIFY
> 
> cond RETURN_VALID IFDUP NOTIF pubkey2 CHECKSIG ENDIF
> 
> which it would just interpret as:
> 
> pubkey1 CHECKSIGVERIFY cond RETURN_VALID
> 
> which is fine if the covenant was passing; but no good if the covenant
> 
> didn't pass -- they'd be expecting the aggregted sig to just be for
> 
> pubkey1 when it's actually pubkey1+pubkey2, so old nodes would fail the
> 
> tx and new nodes would accept it, making it a hard fork.
> 
> Solution 0a / 0b
> 
> 
> ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
> 
> There are two obvious solutions here:
> 
> 0a) Just be very careful to ensure any aggregated signatures that
> 
> are conditional on an redefined RETURN_VALID opcode go into later
> 
> buckets, but be careful about having separate sets of buckets every
> 
> time a soft-fork introduces a new redefined opcode. Probably very
> 
> complicated to implement correctly, and essentially doubles the
> 
> number of buckets you have to potentially deal with every time you
> 
> soft fork in a new opcode.
> 
> 0b) Alternatively, forget about the hope that RETURN_VALID
> 
> opcodes could be converted to anything, and just reserve OP_NOP
> 
> opcodes and convert them to CHECK\_foo\_VERIFY opcodes just as we
> 
> have been doing, and when we can't do that bump the segwit witness
> 
> version for a whole new version of script. Or in twitter speak:
> 
> "non-verify upgrades should be done with new script versions" \[3\]
> 
> I think with a little care we can actually salvage RETURN_VALID though!
> 
> Solution 1
> 
> 
> ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
> 
> You don't actually have to write your scripts in ways that can cause
> 
> this problem, as long as you're careful. In particular, the problem only
> 
> occurs if you do aggregatable CHECKSIG operations after "RETURN_VALID"
> 
> \-\- if you do all the CHECKSIGs first, then all nodes will be checking
> 
> for the same signatures, and there's no problem. So you could rewrite
> 
> the script above as:
> 
> pubkey1 CHECKSIGVERIFY
> 
> IF pubkey2 CHECKSIG ENDIF
> 
> cond CHECKCOVENANT OR
> 
> which is redeemable either by:
> 
> sig1 0 \[and covenant is met\]
> 
> sig1 1 sig2 \[covenant is not checked\]
> 
> The witness in this case is essentially committing to the execution path
> 
> that would have been taken in the first script by a fully validating node,
> 
> then the new script checks all the signatures, and then validates that the
> 
> committed execution path was in fact the one that was meant to be taken.
> 
> If people are clever enough to write scripts this way, I believe you
> 
> can make RETURN_VALID soft-fork safe simply by having every soft-forked
> 
> RETURN_VALID operation set a state flag that makes every subsequent
> 
> CHECKSIG operation require a non-aggregated sig.
> 
> The drawback of this approach is that if the script is complicated
> 
> (eg it has multiple IF conditions, some of which are nested), it may be
> 
> difficult to write the script to ensure the signatures are checked in the
> 
> same combination as the later logic actually requires -- you might have
> 
> to store the flag indicating whether you checked particular signatures
> 
> on the altstack, or use DUP and PICK/ROLL to organise it on the stack.
> 
> Solution 2
> 
> 
> ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
> 
> We could make that simpler for script authors by making dedicated opcodes
> 
> to help with "do all the signatures first" and "check the committed
> 
> execution path against reality" steps. I think a reasonable approach
> 
> would be something like:
> 
> 0b01 pubkey2 pubkey1 2 CHECK\_AGGSIG\_VERIFY
> 
> cond CHECKCOVENANT 0b10 CHECK\_AGG\_SIGNERS OR
> 
> which is redeemed either by:
> 
> sighash1 0 \[and passing the covenant cond\]
> 
> sighash2 sighash1 0b10
> 
> (I'm using the notation 0b10110 to express numbers as binary bitfields;
> 
> 0b10110 = 22 eg)
> 
> That is, two new opcodes, namely:
> 
> CHECK\_AGGSIG\_VERIFY which takes from the stack:
> 
> \- N: a count of pubkeys
> 
> \- pubkey1..pubkeyN: N pubkeys
> 
> \- REQ: a bitmask of which pubkeys are required to sign
> 
> \- OPT: a bitmask of which optional pubkeys have signed
> 
> \- sighashes: M sighashes for the pubkeys corresponding to the set
> 
> bits of (REQ|OPT)
> 
> CHECK\_AGGSIG\_VERIFY fails if:
> 
> \- the stack doesn't have enough elements
> 
> \- the aggregated signature doesn't pass
> 
> \- a redefined RETURN_VALID opcode has already been seen
> 
> \- a previous CHECK\_AGGSIG\_VERIFY has already been seen in this script
> 
> REQ|OPT is stored as state
> 
> CHECK\_AGG\_SIGNERS takes from the stack:
> 
> \- B: a bitmask of which pubkeys are being queried
> 
> and it pushes to the stack 1 or 0 based on:
> 
> \- (REQ|OPT) & B == B ? 1 : 0
> 
> A possible way to make sure the "no agg sigs after an upgraded
> 
> RETURN\_VALID" behaviour works right might be to have "RETURN\_VALID"
> 
> fail if CHECK\_AGGSIG\_VERIFY hasn't already been seen. That way once you
> 
> redefine RETURN\_VALID in a soft-fork, if you have a CHECK\_AGGSIG_VERIFY
> 
> after a RETURN_VALID you've either already failed (because the
> 
> RETURN\_VALID wasn't after a CHECK\_AGGSIG_VERIFY), or you automatically
> 
> fail (because you've already seen a CHECK\_AGGSIG\_VERIFY).
> 
> There would be no need to make CHECKSIG, CHECKSIGVERIFY, CHECKMULTISIG
> 
> and CHECKMULTISIGVERIFY do signature aggregation in this case. They could
> 
> be left around to allow script authors to force non-aggregate signatures
> 
> or could be dropped entirely, I think.
> 
> This construct would let you do M-of-N aggregated multisig in a fairly
> 
> straightforward manner without needing an explicit opcode, eg:
> 
> 0 pubkey5 pubkey4 pubkey3 pubkey2 pubkey1 5 CHECK\_AGGSIG\_VERIFY
> 
> 0b10000 CHECK\_AGG\_SIGNERS
> 
> 0b01000 CHECK\_AGG\_SIGNERS ADD
> 
> 0b00100 CHECK\_AGG\_SIGNERS ADD
> 
> 0b00010 CHECK\_AGG\_SIGNERS ADD
> 
> 0b00001 CHECK\_AGG\_SIGNERS ADD
> 
> 3 NUMEQUAL
> 
> redeemable by, eg:
> 
> 0b10110 sighash5 sighash3 sighash2
> 
> and a single aggregate signature by the private keys corresponding to
> 
> pubkey{2,3,5}.
> 
> Of course, another way of getting M-of-N aggregated multisig is via MAST,
> 
> which brings us to another approach...
> 
> Solution 3
> 
> 
> ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
> 
> All we're doing above is committing to an execution path and validating
> 
> signatures for that path before checking the path was the right one. But
> 
> MAST is a great way of committing to an execution path, so another
> 
> approach would just be "don't have alternative execution paths, just have
> 
> MAST and CHECK/VERIFY codes". Taking the example I've been running with,
> 
> that would be:
> 
> branch1: 2 pubkey2 pubkey1 2 CHECKMULTISIG
> 
> branch2: pubkey1 CHECKSIGVERIFY cond CHECKCOVENANT
> 
> So long as MAST is already supported when signature aggregation becomes
> 
> possible, that works fine. The drawback is MAST can end up with lots of
> 
> branches, eg the 3-of-5 multisig check has 10 branches:
> 
> branch1: 3 pubkey3 pubkey2 pubkey1 3 CHECKMULTISIG
> 
> branch2: 3 pubkey4 pubkey2 pubkey1 3 CHECKMULTISIG
> 
> branch3: 3 pubkey5 pubkey2 pubkey1 3 CHECKMULTISIG
> 
> branch4: 3 pubkey4 pubkey3 pubkey1 3 CHECKMULTISIG
> 
> branch5: 3 pubkey5 pubkey3 pubkey1 3 CHECKMULTISIG
> 
> branch6: 3 pubkey5 pubkey4 pubkey1 3 CHECKMULTISIG
> 
> branch7: 3 pubkey4 pubkey3 pubkey2 3 CHECKMULTISIG
> 
> branch8: 3 pubkey5 pubkey3 pubkey2 3 CHECKMULTISIG
> 
> branch9: 3 pubkey5 pubkey4 pubkey2 3 CHECKMULTISIG
> 
> branch10: 3 pubkey5 pubkey4 pubkey3 3 CHECKMULTISIG
> 
> while if you want, say, 6-of-11 multisig you get 462 branches, versus
> 
> just:
> 
> 0 pubkey11 pubkey10 pubkey9 pubkey8 pubkey7 pubkey6
> 
> pubkey5 pubkey4 pubkey3 pubkey2 pubkey1 11 CHECK\_AGGSIG\_VERIFY
> 
> 0b10000000000 CHECK\_AGG\_SIGNERS
> 
> 0b01000000000 CHECK\_AGG\_SIGNERS ADD
> 
> 0b00100000000 CHECK\_AGG\_SIGNERS ADD
> 
> 0b00010000000 CHECK\_AGG\_SIGNERS ADD
> 
> 0b00001000000 CHECK\_AGG\_SIGNERS ADD
> 
> 0b00000100000 CHECK\_AGG\_SIGNERS ADD
> 
> 0b00000010000 CHECK\_AGG\_SIGNERS ADD
> 
> 0b00000001000 CHECK\_AGG\_SIGNERS ADD
> 
> 0b00000000100 CHECK\_AGG\_SIGNERS ADD
> 
> 0b00000000010 CHECK\_AGG\_SIGNERS ADD
> 
> 0b00000000001 CHECK\_AGG\_SIGNERS ADD
> 
> 6 NUMEQUAL
> 
> Provided doing lots of hashes to calculate merkle paths is cheaper than
> 
> publishing to the blockchain, MAST will likely still be better though:
> 
> you'd be doing 6 pubkeys and 9 steps in the merkle path for about 1532bytes in MAST, versus showing off all 11 pubkeys above for 11(32+4)
> 
> bytes, and the above is roughly the worst case for m-of-11 multisig
> 
> via MAST.
> 
> If everyone's happy to use MAST, then it could be the only solution:
> 
> drop OP_IF and friends, and require all the CHECKSIG ops to occur before
> 
> any RETURN_VALID ops: since there's no branching, that's just a matter of
> 
> reordering your script a bit and should be pretty easy for script authors.
> 
> I think there's a couple of drawbacks to this approach that it shouldn't
> 
> be the only solution:
> 
> a) we don't have a lot of experience with using MAST
> 
> b) MAST is a bit more complicated than just dealing with branches in
> 
> a script (probably solvable once (a) is no longer the case)
> 
> c) some useful scripts might be a bit cheaper expressed with
> 
> of branches and be better expressed without MAST
> 
> If other approaches than MAST are still desirable, then MAST works fine
> 
> in combination with either of the earlier solutions as far as I can see.
> 
> Summary
> 
> 
> ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
> 
> I think something along the lines of solution 2 makes the most sense,
> 
> so I think a good approach for aggregate signatures is:
> 
> -   introduce a new segwit witness version, which I'll call v2 (but which
>     
>     might actually be v1 or v3 etc, of course)
>     
> -   v2 must support Schnorr signature verification.
> -   v2 should have a "pay to public key (hash?)" witness format. direct
>     
>     signatures of the transaction via the corresponding private key should
>     
>     be aggregatable.
>     
> -   v2 should have a "pay to script hash" witness format: probably via
>     
>     taproot+MAST, possibly via graftroot as well
>     
> -   v2 should support MAST scripts: again, probably via taproot+MAST
> -   v2 taproot shouldn't have a separate script version (ie,
>     
>     the pubkey shouldn't be P+H(P,version,scriptroot)), as signatures
>     
>     for later-versioned scripts couldn't be aggregated, so there's no
>     
>     advantage over bumping the segwit witness version
>     
> -   v2 scripts should have a CHECK\_AGG\_SIG_VERIFY opcode roughly as
>     
>     described above for aggregating signatures, along with CHECK\_AGG\_SIGNERS
>     
> -   CHECK{MULTI,}SIG{VERIFY,} in v2 scripts shouldn't support aggregated
>     
>     signatures, and possibly shouldn't be present at all?
>     
> -   v2 signers should be able to specify an aggregation bucket for each
>     
>     signature, perhaps in the range 0-7 or so?
>     
> -   v2 scripts should have a bunch of RETURN_VALID opcodes for future
>     
>     soft-forks, constrained so that CHECK\_AGG\_SIG_VERIFY doesn't appear
>     
>     after them. the currently disabled opcodes should be redefined as
>     
>     RETURN_VALID eg.
>     
>     For soft-fork upgrades from that point:
>     
> -   introducing new opcodes just means redefining an RETURN_VALID opcode
> -   introducing new sighash versions requires bumping the segwit witness
>     
>     version (to v3, etc)
>     
> -   if non-interactive half-signature aggregation isn't ready to go, it
>     
>     would likewise need a bump in the segwit witness version when
>     
>     introduced
>     
>     I think it's worth considering bundling a hard-fork upgrade something
>     
>     like:
>     
> -   ~5 years after v2 scripts are activated, existing p2pk/p2pkh UTXOs
>     
>     (either matching the pre-segwit templates or v0 segwit p2wpkh) can
>     
>     be spent via a v2-aggregated-signature (but not via taproot)
>     
>     \[4\]
>     
> -   core will maintain a config setting that allows users to prevent
>     
>     that hard fork from activating via UASF up until the next release
>     
>     after activation (probably with UASF-enforced miner-signalling that
>     
>     the hard-fork will not go ahead)
>     
>     This is already very complicated of course, but note that there's still
>     
>     more things that need to be considered for signature aggregation:
>     
> -   whether to use Bellare-Neven or muSig in the consensus-critical
>     
>     aggregation algorithm
>     
> -   whether to assign the aggregate sigs to inputs and plunk them in the
>     
>     witness data somewhere, or to add a new structure and commitment and
>     
>     worry about p2p impact
>     
> -   whether there are new sighash options that should go in at the same time
> -   whether non-interactive half-sig aggregation can go in at the same time
>     
>     That leads me to think that interactive signature aggregation is going to
>     
>     take a lot of time and work, and it would make sense to do a v1-upgrade
>     
>     that's "just" Schnorr (and taproot and MAST and re-enabling opcodes and
>     
>     ...) in the meantime. YMMV.
>     
>     Cheers,
>     
>     aj
>     
>     \[0\] http://diyhpl.us/wiki/transcripts/bitcoin-core-dev-tech/2018-03-06-taproot-graftroot-etc/
>     
>     \[1\] Signature aggregation:
>     
>     Signature aggregation is cool because it lets you post a transaction
>     
>     spending many inputs, but only providing a single 64 byte signature
>     
>     that proves authorisation by the holders of all the private keys
>     
>     for all the inputs. So the witnesses for your inputs might be:
>     
>     p2wpkh: pubkey1 SIGHASH_ALL
>     
>     p2wpkh: pubkey2 SIGHASH_ALL
>     
>     p2wsh: "3 pubkey1 pubkey3 pubkey4 3 CHECKMULTISIG" SIGHASH\_ALL SIGHASH\_ALL SIGHASH_ALL
>     
>     where instead of including full 65-byte signature for each CHECKSIG
>     
>     operation in each input witness, you just include the ~1-byte sighash,
>     
>     and provide a single 64-byte signature elsewhere, calculated either
>     
>     according to the Bellare-Neven algorithm, or the muSig algorithm.
>     
>     In the above case, that means going from about 500 witness bytes
>     
>     for 5 public keys and 5 signatures, to about 240 witness bytes for
>     
>     5 public keys and just 1 signature.
>     
>     A complication here is that because the signatures are aggregated,
>     
>     in order to validate any signature you have to be able to validate
>     
>     every signature.
>     
>     It's possible to limit that a bit, and have aggregation
>     
>     "buckets". This might be something you just choose when signing, eg:
>     
>     p2wpkh: pubkey1 SIGHASH\_ALL|BUCKET\_1
>     
>     p2wpkh: pubkey2 SIGHASH\_ALL|BUCKET\_2
>     
>     p2wsh: "3 pubkey1 pubkey3 pubkey4 3 CHECKMULTISIG" SIGHASH\_ALL|BUCKET\_1 SIGHASH\_ALL|BUCKET\_2 SIGHASH\_ALL|BUCKET\_2
>     
>     bucket1: 64 byte sig for (pubkey1, pubkey1)
>     
>     bucket2: 64 byte sig for (pubkey2, pubkey3, pubkey4)
>     
>     That way you get the choice to verify both of the pubkey1 signatures
>     
>     or all of the pubkey{2,3,4} signatures or all the signatures (or
>     
>     none of the signatures).
>     
>     This might be useful if the private key for pubkey1 is essentially
>     
>     offline, and can't easily participate in an interactive protocol
>     
>     \-\- with separate buckets the separate signatures can be generated
>     
>     independently at different times, while with only one bucket,
>     
>     everyone has to coordinate to produce the signature)
>     
>     (For clarity: each bucket corresponds to many CHECKSIG operations,
>     
>     but only contains a single 64-byte signature)
>     
>     Different buckets will also be necessary when dealing with new
>     
>     segwit script versions: if there are any aggregated signatures for
>     
>     v1 addresses that go into bucket X, then aggregate signatures for
>     
>     v2 addresses cannot go into bucket X, as that would prevent nodes
>     
>     that support v1 addresses but not v2 addresses from validating
>     
>     bucket X, which would prevent them from validating the v1 addresses
>     
>     corresponding to that bucket, which would make the v2 upgrade a hard
>     
>     fork rather than a soft fork. So each segwit version will need to
>     
>     introduce a new set of aggregation buckets, which in turn reduces
>     
>     the benefit you get from signature aggregation.
>     
>     Note that it's obviously fine to use an aggregated signature in
>     
>     response to CHECKSIGVERIFY or n-of-n CHECKMULTISIGVERIFY -- when
>     
>     processing the script you just assume it succeeds, relying on the
>     
>     fact that the aggregated signature will fail the entire transaction
>     
>     if there was a problem. However it's also fine to use an aggregated
>     
>     signature in response to CHECKSIG for most plausible scripts, since:
>     
>     sig key CHECKSIG
>     
>     can be treated as equivalent to
>     
>     sig DUP IF key CHECKSIGVERIFY OP_1 FI
>     
>     provided invalid signatures are supplied as a "false" value. So
>     
>     for the purpose of this email, I'll mostly be treating CHECKSIG and
>     
>     n-of-n CHECKMULTISIG as if they support aggregation.
>     
>     \[2\] Soft-forks and RETURN_VALID:
>     
>     There are two approaches for soft-forking in new opcodes that are
>     
>     reasonably well understood:
>     
>     1.  We can bump the segwit script version, introducing a new class of
>         
>         bc1 bech32 addresses, which behave however we like, but can't be
>         
>         validated at all by existing nodes. This has the downside that it
>         
>         effectively serialises upgrades.
>         
>     2.  We can redefine OP\_NOP opcodes as OP\_CHECK\_foo\_VERIFY
>         
>         opcodes, along the same lines as OP_CHECKLOCKTIMEVERIFY or
>         
>         OP_CHECKSEQUENCEVERIFY. This has the downside that it's pretty
>         
>         restrictive in what new opcodes you can introduce.
>         
>         A third approach seems possible as well though, which would combine
>         
>         the benefits of both approaches: allowing any new opcode to be
>         
>         introduced, and allowing different opcodes to be introduced in
>         
>         concurrent soft-forks. Namely:
>         
>     3.  If we introduce some RETURN_VALID opcodes (in script for a new
>         
>         segwit witness version), we can then redefine those as having any
>         
>         behaviour we might want, including ones that manipulate the stack,
>         
>         and have the change simply be a soft-fork. RETURN_VALID would
>         
>         force the script to immediately succeed, in contrast to OP_RETURN
>         
>         which forces the script to immediately fail.
>         
>         \[3\] https://twitter.com/bramcohen/status/972205820275388416
>         
>         \[4\] https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2018-January/015580.html
>         
> 
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