CAT and Schnorr Tricks II

Part 1


This is the second in a series of posts about about covenants in Bitcoin using Taproot and a (hypothetical) CAT opcode. Historically, and as has been implemented in Elements, CAT has been considered to be a covenant opcode only in conjunction with CHECKSIGFROMSTACK.

In our last post we introduced the idea of using CAT and fixed Schnorr signatures to create covenants, by allowing a Script writer to reconstruct a transaction hash, fixing some elements of the transaction but leaving others free. In this post we'll talk about recursive covenants, ones which use covenant logic to require that coins being spent from a covenant only go to the same (or similar) covenant.

Fears About Recursive Covenants

Before we get into the technical details, let me provide some historical context for recursive covenants and the controversy surrounding them.

The idea for covenants was first introduced by Greg Maxwell in 2013 in a BitcoinTalk post that invited users to come up with outlandish malicious applications for the tech. Users observed that they could be used to "permanently taint" coins, e.g. by constructing one-way covenants which users could send coins to, but the covenant logic prevented moving coins out of. They hinted, without giving specifics, that such covenants could include data that would enable surveillance or transaction censorship.

Meanwhile, in 2015 Blockstream launched Elements Alpha with support for covevants to encourage experimentation in a no-value setting. In early 2016, Möser, Eyal and Sirer described a "vault" construction which used covenants to prevent fast theft of coins.

In May 2019, Jeremy Rubin proposed a CHECKOUTPUTSHASHVERIFY opcode for Bitcoin that would enable "a rudimentary, limited form of covenant which does not bear the same technical and social risks of prior covenant designs". This opcode was later replaced by SECURETHEBAG and then by CHECKTEMPLATEVERIFY, which became BIP 0119 in January of 2020. In parallel, Russell O'Connor proposed simply adding CHECKSIGFROMSTACK and CAT to Bitcoin, to enable covenants without the limits of Rubin's proposal. This received some opposition and ultimately conversation died off, largely because of the inefficiency of CAT+CHECKSIG style covenants, but also because of the lingering stigma of fully-general covenants.

For several years, I shared these general fears, and while I was frustrated that Bitcoin lacked any means to support vaults, velocity limits, options contracts or timelocks greater than 65000 blocks, I wasn't comfortable supporting general covenant functionality.

However, in the Fall of 2019, Ethan Heilman pointed out to me in private conversation that the sorts of "permanently-tainted-coin covenants" that people were worried about could be constructed today using CHECKMULTISIG. For example, Coinbase could hypothetically refuse to allow withdrawals to any scripts that didn't require their signature (or that of a regulator), saying that they would countersign future transactions but only if the signature requirement were preserved. He pointed out that the massive social and technical barriers preventing this from happening would also prevent anybody using covenant opcodes to the same effect.

Essentially, you could create a poisonous covenant if you wanted, but wallets wouldn't recognize them, they wouldn't be able to spend them, and users wouldn't want them.

He later issued a challenge on Twitter for anyone to come up with a technically feasible dark covenant, using any opcode that had ever been proposed for Bitcoin, in exchange for a free coffee. On Telegram, I and others offered to extend Ethan's offer with various alcoholic beverages. Incentivized by the free beer I'd promised to buy myself, I spent several hours trying to construct such a thing, but to no avail. As of this writing, nobody has successfully taken Ethan up on this challenge.

This more-or-less convinced me that fear of covenants is overblown, and as this blog series will illustrate, even if it weren't, it's impossible to avoid covenants when extending Bitcoin anyway. It seems that the rest of the ecosystem is coming around to this viewpoint, and I hope that we see some form of general covenant seriously proposed for Bitcoin in the future.

Meanwhile, let's continue exploring what kind of covenants we can create with the not-for-covenants opcode CAT.


In the next section we are going to describe a Vault construction, which consists of two covenant scripts, which restrict how their coins can be spent.

First, there is a vault output, which requires a signature with a fixed "hot key" and only allows coins to be sent to a staging output.

Second, the staging output allows coins to be spent in one of two ways:

  1. After a timelock expires, the coins can move to a target destination, which is set when coins are first moved to the staging output; or
  2. At any time, by signing with a "cold key", the coins can be returned to the same script, setting a new target destination and resetting the timelock.

The idea here is that the coins can lay dormant, and when the user wants to move them, they first need to send them to a "waiting area" where they must sit until a timelock has expired. Therefore, as long as the original user has possession of the cold key, it is impossible for an attacker to get ahold of the funds. At worst, an attacker who gains access to the cold key can start an indefinite battle with the rightful owner, where they take turns resetting the timelock, but neither actually gains access to the funds.

Value Switching

As we observed in the previous post, there is no way (that I can find) to compute a taproot commitment in Script+CAT, at least not without knowing the discrete logarithm of the commitment. And unless we can construct a commitment using a nothing-up-my-sleeve point, for which nobody knows the discrete logarithm, we can't commit to a script.

This means that if we hope to create recursive covenants, we are limited to sending coins only to the same destination from which they came from. (For non-recursive covenants, we can send to any of a number of pre-computed output scripts, but we cannot create them dynamically.) How, then, can we construct a vault, which seems to require a dynamically-generated output in order that the user can change the target destination?

In an early version of this post, I was going to describe a solution called "value switching", where there would be a large number of predefined destinations, and the user would choose which one by setting the low byte of the output value. The script would then check this value (which is directly committed in the sighash) when deciding what spends were allowable. The idea was that each tapleaf would be "gated" on the input value's low byte being a specific number.

This seems generally useful, in that you can use it to build multi-script state machines that have loops in them, so I'm mentioning it here, but actually I don't think I need it for vaults, so I leave the details as an exercise for the reader.

Transaction Preimaging

Since the recursive structure of vaults is very simple (you can send coins from the vault script to itself but the only other paths lead out of the covenant) we do not need value switching. However, we do have this "target destination" parameter that we need to set. Our strategy for handling this is as follows:

The vault script, which simply prevents the coins from moving anywhere except to a staging output, requires that it be spent with exactly two outputs. The covenant will restrict the first output such that its scriptPubKey will be the staging script and its value equal to that of the input.

The second output has no restrictions on it, but serves three purposes: * Most importantly, our target destination will be defined to have its scriptPubKey * It is a change output for the transaction * It is an output we can spend to use child-pays-for-parent to feebump the transaction

Since our restrictions are entirely on the output side of the equation, it is easy to control this by using a CAT and CHECKSIG as in our last post, requiring that the signature hash consist of the following data:

  • 80: SIGHASH_ANYONECANPAY | ALL (sign all outputs but only the current input)
  • 20000000000000: version 2, locktime 0
  • <32-byte hash of all outputs>
  • 01: no annex
  • <36-byte outpoint being spent>: irrelevant to us
  • <8-byte amount being spent>: must match the first 8 bytes of the output hash
  • <35-byte spk of output being spent>: irrelevant to us
  • <4-byte sequence number of output being spent>: irrelevant to us

where the third item, the hash of all outputs, covers the following data

  • <8-byte amount>: must equal input amount
  • <35-byte staging scriptpubkey>: fixed
  • <43-byte second output>: arbitrary but must be length-checked and checked that its 9th byte is decimal 34

It is fairly straightforward, but tedious, to implement these restrictions using the techniques discussed in the previous blog post. In the interest of keeping this series moving forward, I will elide details for now, and later edit the post when these are actually implemented, perhaps on Elements. Remember that in addition to the covenant restrictions, there should be a signature requirement from the user's hot key.

This second output may seem eyebrow-raising. I am ascribing it meaning but so far our covenant has made no effort to restrict it or use it in any way. To see where the interesting step comes in, let's look at the staging output's script.

Recall that there are two ways to spend the staging output: by waiting for the timelock and then sending to the target destination, or by signing with the cold key and sending back to the staging output, resetting the clock.

Let's cover the second first, since it's simpler: it is essentially identical to the vault script, except for two differences!

  • The cold key is used in place of the hot key
  • We cannot fix the output scriptPubKey without attempting to store the script's own hash inside the script. So rather than enforcing that the coins go to a fixed destination, we ensure that the input value+scriptPubKey (included in the BIP-0341SigMsg) matches the output scriptPubKey (by checking that sha_outputs included in SigMsg is the hash of some data starting with the required value+scriptPubKey).

Next, we consider the "unvaulting" spending path, in which the user waits for the timelock to expire and then moves the coins to the target destination. The covenant script needs to check that the coins actually do move to the target destination.

Again, we elide the exact script details, but the premise is simple: we check that the sha_outputs of SigMsg field begins with the input value followed by the target scriptPubKey, where the "target scriptPubKey" is determined by hashing the entire funding transaction, checking its txid against the input outpoint in SigMsg, and copying the second output's scriptPubKey.

We observe that there is no need to enforce, at funding time, that the funding transaction is well-formed (has a second output, whose scriptPubKey has the right size, etc). If any of these conditions fail, the coins will be stuck in the staging output until the user uses the coin key to re-fund with a well-formed transaction.


Although we elided many details, in this post we covered two ways to produce recursive covenants using CAT: value switching and transaction preimaging. We used these techniques to implement a vault.

In our next post, we'll talk about how to use these techniques to emulate the "rebinding" functionality that Lightning Network developers have proposed to get from SIGHASH_ANYPREVOUT, showing that CAT is sufficient to get constant-sized payment channel backups.

In our concluding post, we will look at using the Miniscript model of Script to make these complex and difficult-to-analyze Script constructions tractable.