Summary

OP_CHECKMULTISIG and OP_CHECKMULTISIGVERIFY will be upgraded to accept Schnorr signatures in a way that increases verification efficiency and is compatible with batch verification.

note: this document assumes knowledge of the prior Schnorr signature upgrade.

Motivation

In the last upgrade, we added Schnorr support to OP_CHECKSIG and OP_CHECKDATASIG, but not OP_CHECKMULTISIG.

Although we could have added support to OP_CHECKMULTISIG as well (which would have been overall simpler), this would conflict with the desire to do batch verification in future: Currently with OP_CHECKMULTISIG validation, it is needed to check a signature against multiple public keys in order to find a possible match. In Schnorr batch verification however, it is required to know ahead of time, which signatures are supposed to match with which public keys. Without a clear path forward on how to resolve this, we postponed the issue and simply prevented Schnorr signatures from being used in OP_CHECKMULTISIG.

Schnorr aggregated signatures (with OP_CHECKSIG) are one way to do multisignatures, but they have different technical properties than the familiar Bitcoin multisig, and thus are far from being a drop-in replacement for it. Besides that, it is also desirable that any existing coin can be spent using Schnorr signatures, and there are numerous OP_CHECKMULTISIG-based wallets and coins in existence that we want to be able to take advantage of Schnorr signatures.

Specification

OP_CHECKMULTISIG and OP_CHECKMULTISIGVERIFY will be upgraded to allow two execution modes, based on the value of the dummy element.

Mode 1 (legacy ECDSA support, M-of-N; consumes N+M+3 items from stack):

<dummy> <sig0> ... <sigM> M <pub0> ... <pubN> N OP_CHECKMULTISIG

The precise validation mechanics of this are complex and full of corner cases; the source code is the best reference. Most notably, for 2-of-3 (M=2, N=3), sig0 may be a valid ECDSA transaction signature from pub0 or from pub1; sig1 may be from pub1 (if sig0 is from pub0) or pub2. Historical transactions (prior to FORKID, STRICTENC and NULLFAIL rules) had even more freedoms and weirdness). Upon activation, the dummy element must be null, i.e., an empty byte array.

Mode 2 (new Schnorr support, M-of-N; consumes N+M+3 items from stack):

<checkbits> <sig0> ... <sigM> M <pub0> ... <pubN> N OP_CHECKMULTISIG

• The dummy element has now been repurposed as a bitfield that we call checkbits, and indicates which public keys should have a signature checked against them.
• This mode activates when dummy (checkbits) is non-null, i.e., not an empty byte array.
• Crucially, each of the signature checks requested by checkbits must be valid, or else the script fails.
• In mode 2, ECDSA signatures are not allowed.

Triggering and execution mechanism

Whether to execute in mode 1 or mode 2 is determined by the size of the dummy / checkbits element.

• If the checkbits element is NULL (length 0), then Mode 1 is executed
• If the checkbits element is non-NULL (length > 0), then Mode 2 is executed.

The new mode operates similar to legacy mode but only checks signatures as requested, according to the checkbits field. If the least significant bit of checkbits is set, then the bottom (first-pushed) signature should be checked against the bottom public key, and so on. For a successful verification in the new mode, checkbits must have exactly M bits set, and the signatures must be correctly ordered. On stack, checkbits is encoded as a byte array of length floor((N + 7)/8), i.e., the shortest byte array that can hold N bits. It is encoded in little-endian order, i.e., the least significant bit occurs in the first byte.

In pseudocode, the full OP_CHECKMULTISIG code is:

Get N (number of pubkeys) from stack ; check bounds 0 <= N <= 20.
Add N to nOpCount; if nOpCount exceeds 201 limit, fail script.
Get M (number of signatures) from stack ; check bounds 0 <= M <= N.
Calculate scriptCode.
If activated, and the dummy element is not null, then:
    # New mode (2)
    Set a cursor on the bottom signature (first signature pushed on stack).
    Set another cursor on the bottom public key (first key pushed on stack).
    Fail if the dummy element does not have length in bytes = floor((N+7)/8)
    Set checkbits := 0, then iterate over the bytes in the dummy element in reverse order:
        For each byte X,  checkbits := (checkbits << 8) | X
    Loop while the signature and key cursors are not depleted:
        If the least significant bit of checkbits is 1, then:
            Check public key encoding.
            Check signature encoding; exclude non-Schnorr signatures.
            Validate the current signature against the current public key; if invalid, fail script.
            Move the signature cursor up by one position.
        Bitshift checkbits down by one bit. (checkbits := checkbits >> 1)
        Move the public key cursor up by one position.
    If the final checkbits value is nonzero, fail script.
    If the signature cursor has not been depleted, fail script.
Else:
    # Legacy mode (1)
    Set a cursor on the top signature (last signature pushed on stack).
    Set another cursor on the top public key (last key pushed on stack).
    If pre-BCH-fork, then run findAndDelete on scriptCode.
    Loop while the signature cursor is not depleted:
        Check public key encoding.
        Check signature encoding; exclude Schnorr signatures (64+1 bytes).
        Validate the current signature against the current public key.
        If valid, then move signature cursor deeper by one position.
        Move the public key cursor deeper by one position.
        If more signatures remain than public keys, set success=False and abort loop early.
    If loop was not aborted, set success=True.
    [non-consensus] Check NULLDUMMY rule.

If success is False, then ensure all signatures were null. (NULLFAIL rule)
Clean up the used stack items.

Push success onto stack
If opcode is OP_CHECKMULTISIGVERIFY:
    Pop success from stack
    If not success:
        FAIL

Notes

The mechanics of CHECKMULTISIG are complicated due to the order of signature checking, the timing of when key/signature encodings are checked, the ability to either hard-fail (fail script & invalidate transaction) or soft-fail (return False on stack), and the interaction with previously activated consensus rules. Some features of the specification are worth emphasizing:

• The legacy mode has unaltered functionality, except being restricted to only use a null dummy element.
• Compatibility is good, as basically any reasonable smart contract using OP_CHECKMULTISIG can be spent using either legacy or new mode. (Of course, with effort a script could be deliberately crafted to only allow one mode.)
• In both modes, public keys only have their encoding checked just prior to performing a signature check. The unchecked public keys may be arbitrary data.
  • In legacy mode, the precise order of checking is critical to obtaining a correct implementation, due to the public key encoding rule. Signature and pubkey iteration always starts at the top public key and signature (the last pushed on stack).
  • Some multisig scripts were made unspendable on Aug 1 2017, due to the last-pushed public key having incorrect encoding. These will now be spendable, but only in the new mode.
• Note that the numbers N, M will require minimal encoding, upon activation of the minimal number encoding rule (see https://github.com/bitcoincashorg/bitcoincash.org/pull/376/files).
• In the new mode, checkbits must have exactly M of the lower N bits set, and all other bits must be clear:
  • Only the least significant N bits may be set in checkbits, i.e., if checkbits taken as an integer exceeds 2^N-1 then the script will fail.
  • If checkbits has more than M bits set, the script will fail.
  • If checkbits is nonzero but has fewer than M bits set, then the script will fail because too few signature verifications were performed.
• In normal circumstances the new mode cannot be third-party malleated, since the new mode design means that checkbits should have only one valid value for a given set of signatures
  • Third-party malleation can still occur in some very unusual cases. For example, if some public key points are repeated in the list of keys, then signatures can be reordered and/or the checkbits can be adjusted. Also, if M=0 then two possible values of the dummy element are permitted.
  • Likewise the design stops the malleation vector of the legacy mode, since the dummy element now must be null for it to execute. The non-consensus NULLDUMMY rule will thus be made redundant, after this rule activates.
• The legacy mode can require up to N signature checks in order to complete. In the new mode, exactly M signature checks occur for a sucessful operation.
• A soft-failing CHECKMULTISIG (that returns False on stack) can only occur with all null signatures, due to NULLFAIL. For simplicity and avoiding malleability, the new mode does not allow a failing case, and a soft-failing CHECKMULTISIG must execute in the legacy mode (which will require a NULL dummy element). Note that even such a soft-failing checkmultisig still requires the top public key to be correctly encoded due to the legacy mechanics.
• For M=0, the opcode returns True without checking any key encodings. This is true in both new and legacy mode.

And, some clarifications:

• As usual, checking public key encoding means permitting only 65-long byte arrays starting with 0x04, or 33-long byte arrays starting with 0x02 or 0x03.
• As usual, checking signature encoding for either ECDSA or Schnorr involves permitting only recognized hashtype bytes; Schnorr signatures must have a given length, while ECDSA signatures must follow DER encoding and Low-S rules, and must not have the length allocated to Schnorr signatures. Null signatures (empty stack elements) are also treated as 'correctly encoded'.
• The findAndDelete operation only applies to old transactions prior to August 2017, and does not impact current transactions, not even in legacy mode.

Wallet implementation guidelines

(Currently, the common multisig wallet uses P2SH-multisig, i.e., a redeemScript of the form M <pub0> ... <pubN> N OP_CHECKMULTISIG. We'll focus on this use case and assume M > 0.)

In the new Schnorr mode, all signatures must be Schnorr; no mixing with ECDSA is supported. Multisig wallets that wish to use the new Schnorr signatures will need to update their co-signing pools infrastructure to support a new type of signing. If some parties are unable to generate a Schnorr signature, then it will not be possible to generate a successful transaction except by restarting to make an ECDSA multisig. This creates some problems in particular when some of the parties are a hardware wallet, which may only support ECDSA for the forseeable future.

We suggest the following for wallet software producers that wish to make Schnorr multisig spends while remaining backwards compatible:

• Add an optional marker to the initial setup process, such as appending ?schnorr=true to the xpub.
• Add a new kind of non-backwards-compatible multisignature request that indicates schnorr signatures are needed.
• If it is not known that all parties can accept Schnorr requests, then only generate ECDSA multisignature requests.
• Have the ability to participate in either ECDSA or Schnorr multisignatures, as requested.

It may also be helpful to include both an ECDSA and Schnorr signature in the partially signed transaction format, so that if one cosigner is unable to sign Schnorr, then an ECDSA fallback is possible without needing a retry. This introduces no additional malleability concerns since already any of the cosigners is able to malleate their own signature.

Calculating and pushing checkbits

In order to complete a multisignature, whether in the new mode or legacy mode, wallets need to keep track of which signatures go with which public keys. In the new mode, wallets must not just correctly order the signatures, but must also correctly include the checkbits parameter.

Once the checkbits parameter is determined, it needs to be encoded to bytes, and then minimally pushed in the scriptSig. While the encoding to bytes is straight forward, it is worth emphasizing that certain length-1 byte vectors must be pushed using special opcodes.

• For N <= 8, a length-1 byte array is to be pushed.
  • The byte arrays {0x01} through {0x10} must be pushed using OP_1 through OP_16, respectively.
  • The byte array {0x81} must be pushed using OP_1NEGATE. This can only occur for a 2-of-8 multisig, where the checkbits bit pattern is 10000001.
  • Other cases will be pushed using no special opcode, i.e., using 0x01 <checkbits>.
• For 9 <= N <= 16, a length-2 byte array is to be pushed.
  • The push will always be 0x02 LL HH, where LL is the least significant byte of checkbits, and HH is the remaining high bits.
• For 17 <= N <= 20, a length-3 byte array is to be pushed.
  • The push will always be 0x03 LL II HH, where where LL is the least significant byte of checkbits, II is the next-least significant byte, and HH is the remaining high bits.

ScriptSig size

Wallets need to know ahead of time the maximum transaction size, in order to set the transaction fee.

Let R be the length of the redeemScript and its push opcode, combined.

The legacy mode scriptSig <dummy> <sig0> ... <sigM> can be as large as 73M + 1 + R bytes, which is the upper limit assuming all max-sized ECDSA signatures.

In the new mode scriptSig <checkbits> <sig0> ... <sigN>, each Schnorr signature will contribute a fixed size of 66 bytes (including push opcode), however the length of checkbits will vary somewhat. Wallets should allocate for fees based on the largest possible encoding, which gives a scriptSig size of:

• N <= 4: checkbits will always be pushed using OP_1 through OP_15, so always 66M + R + 1 bytes.
• 5 <= N <= 8: checkbits may sometimes be pushed using a single-byte opcode, or may need to be pushed as 0x01 0xnn -- up to 66M + R + 2 bytes.
• 9 <= N <= 16: checkbits will be pushed as 0x02 0xnnnn -- always 66M + R + 3 bytes.
• 17 <= N <= 20: checkbits will be pushed as 0x03 0xnnnnnn -- always 66M + R + 4 bytes.

Pubkey Encoding

It is strongly recommended that wallets never create scripts with invalid pubkeys, even though this specification allows them to exist in the public key list as long as they are unused. It is possible that a future rule may stipulate that all pubkeys must be strictly encoded. If that were to happen, any outputs violating this rule would become unspendable.

Rationale and commentary on design decisions

Repurposing of dummy element

In an earlier edition it was proposed to require N signature items (either a signature or NULL) for new mode instead of M items and a dummy element. The following problems inspired a move away from that approach:

• Triggering mechanics for the new mode were somewhat of a kluge.
• Some scripts rely on a certain expected stack layout. This is particularly the case for recently introduced high-level smart contracting languages that compile down to script, which reach deep into the stack using OP_PICK and OP_ROLL.

That said, a scan of the blockchain only found about a hundred instances of scripts that would be impacted by stack layout changes. All were based on a template as seen in this spend, where OP_DEPTH was used to choose an OP_IF execution branch.

Use of a bitfield instead of a number

Another draft of this specification proposed decoding the dummy element as a number, using the standard number decoding rules. The change to using a custom bitfield representation was motivated by the fact that the bitwise operators (OP_AND, OP_OR, OP_XOR) do not cleanly operate on bitcoin's numbers, since numbers are encoded using variable lengths whereas the bitwise operators require the operands to have equal lengths.

The current specification guarantees that for a successful multisig in the new mode, the dummy element always has a specific length of either 1, 2, or 3, depending only on N. Smart contracts can use this property to perform a multisignature and then do bit inspection on which signatures were actually checked.

No mixing ECSDA / Schnorr

Allowing mixed signature types might help alleviate the issue of supporting mixed wallet versions that do support / don't support Schnorr signatures. However, this would mean that an all-ECDSA signature list could be easily converted to the new mode, unless extra complicated steps were taken to prevent that conversion. As this is an undesirable malleability mechanism, we opted to simply exclude ECDSA from the new mode, just as Schnorr are excluded from the legacy mode.

Implementation

https://reviews.bitcoinabc.org/D3474

Acknowledgements

Thanks to Tendo Pein, Rosco Kalis, Amaury Sechet, and Antony Zegers for valuable feedback.