The final six months or so have seen a number of proposals for enhancements to Bitcoin Script: CAT, 64-bit arithmetic, in addition to some older concepts (CTV) and far-future concepts (Chialisp and Simplicity). This exercise has largely overshadowed some revolutionary modifications in our understanding of the prevailing Bitcoin Script, modifications which type the idea of BitVM however which can additionally type the idea of different, equally-exciting enhancements.
This text tries to summarize and manage analysis into Script by different individuals. I make no declare to originality or authorship of something described right here.
Bitcoin Script
As many readers are conscious, Bitcoin Script is an easy programming language embedded within the Bitcoin blockchain, which is used to regulate underneath what situations cash might transfer. By far the commonest use of Script is to easily examine a signature with a single signature verification key. Although Bitcoin addresses have modified all through the years, each type of tackle has supported this use of script in a first-class manner: signing keys may be encoded straight into Bitcoin addresses, and wallets know find out how to develop these keys into full applications that examine signatures on these keys.
Script can do many extra issues: it may examine hash preimages, examine relative and absolute timelocks, and it may do some easy reasoning to mix these checks in numerous methods. That is the premise behind Miniscript: we are able to generalize the notion of increasing a key right into a Script to the notion of increasing an arbitrarily-large set of signing situations right into a Script.
Script can technically do much more than this: it may add and subtract 32-bit numbers, it may hash information and examine the hash values for equality, and it may rearrange and manipulate a “stack” of values in numerous attention-grabbing methods. Nonetheless, Script has many limitations: it lacks opcodes to do easy arithmetic resembling multiplication, it’s (practically) incapable of reasoning about objects bigger than 32 bits, and it has (practically) no capacity to introspect transaction information. The latter limitation is why covenant help seems to require a softfork, and the previous limitations are why Script, till lately, was by no means used to compute any “attention-grabbing” features.
For instance, to multiply two 16-bit numbers in Script, utilizing solely the addition and subtraction opcodes that Script supplies, you’ll want to break them into bits (by requiring the bits be supplied as witness information, then doubling and including them to reconstruct the unique quantity) after which implementing multiplication when it comes to additions of those bits. The ensuing code would contain a number of dozen opcodes for a single multiplication.
Previous to Taproot, Script had a synthetic restrict of 201 opcodes per program, and with particular person multiplications taking greater than 1 / 4 of this funds, it was unimaginable to do a lot of something. After Taproot, the 201-opcode restrict was eliminated, however each opcode nonetheless takes up a witness byte, which means that multi-kilobyte applications could be prohibitively costly for abnormal wallets to placed on the blockchain.
And with out transaction introspection, it is not even clear what giant computations could be good for.
In any case, if you are able to do arbitrary computations on arbitrary values, however these values aren’t tied to transaction information on the blockchain, how can these computations add helpful semantics to Bitcoin?
Lamport Signatures
Lamport signatures have been invented in 1979 by Leslie Lamport — although they’re insecure with out fashionable cryptographic hash features, which didn’t exist till the Nineteen Nineties — and are one of many few cryptographic objects from that period which endure to this present day. Their lasting reputation comes from their simplicity and the truth that their safety in opposition to quantum computer systems relies upon solely on sufficiently-random-looking hash features, not like extra fashionable and environment friendly proposals for quantum-secure signature schemes.
Nonetheless, Lamport signatures include two giant drawbacks: (1) they’re horribly inefficient, taking a number of kilobytes of knowledge for each keys and signatures, and (2) they’re single-use. Which means if a consumer indicators multiple message, it turns into attainable for a third celebration to forge extra messages, making all signatures successfully nugatory. This may be mitigated, for instance by having your “public key” be a Merkle tree of thousands and thousands of single-use keys, however this stretches the bounds of practicality..
These limitations have made Lamport signatures widespread as a “backup signature scheme” for Bitcoin in case of a quantum computing breakthrough, however have prevented their use as major signatures in any extensively deployed system.
The way in which they work is easy: assume that the message to be signed is 256 bits huge. This may be assured by first operating an arbitrary-length message by means of the SHA256 hash perform. The consumer’s public key consists of 256 pairs of hashes – 512 in whole. To signal a message, they reveal a preimage for one hash in every pair, selecting the preimage to disclose primarily based on a little bit of the message.
A signature verifier re-hashes the message and preimages to make sure they’re all constant.
In 2021, Jeremy Rubin posted a weblog submit claiming that Bitcoin Script can straight confirm Lamport signatures on 33-bit values. His mechanism was very intelligent. Bitcoin Script doesn’t have an opcode to learn particular person bits from a quantity, nor can it do the bitwise operations wanted to assemble a quantity from bits. However Script does have an opcode so as to add two numbers, and by including totally different numbers the place every has solely a single bit set, it’s attainable to bitwise-construct or bitwise-deconstruct a quantity.
Utilizing this perception, Rubin checks a Lamport signature, encoded as a collection of hash preimages, as follows:
- For every preimage, compute its hash and examine it in opposition to a pair of goal values (comprising the general public key) embedded within the Script.
- If the hash matches the primary member of the pair, this can be a 0-bit; the script does nothing on this case.
- If it matches the second member, this can be a 1-bit. On this case, add a selected energy of two to an accumulator.
- (If it matches neither member, the signature is invalid and the script ought to abort.)
The ultimate worth of the accumulator will then equal the signed quantity, constructed by including powers of two corresponding to every 1 bit in its binary enlargement.
Already that is attention-grabbing: it implies that for sure sorts of “oracle signature” purposes, you possibly can straight examine signatures in Bitcoin Script, assuming you’ve got an oracle that’s prepared to supply one-time Lamport signatures on particular occasions and that you understand a Lamport public key prematurely for every occasion. For instance, a particular sports activities match end result may be encoded as a single bit. The actual rating may be encoded utilizing a number of bits. A specific timestamp can (in all probability) be encoded in 33 bits. And so forth. And naturally, by checking a number of Lamport signatures, you possibly can successfully get signatures on as many bits as you need.
With out the flexibility to signal giant messages, you possibly can’t get a signature on transaction information and due to this fact can’t get covenants. (Or can we?)
BitVM and Equivocation
This weblog submit by Jeremy Rubin was largely thought-about to be a curiosity on the time and was misplaced amongst bigger discussions round his OP_CTV proposal and covenants. In December of 2023, it was not directly cited within the OP_CAT BIP by Ethan Heilman and Armin Sabouri, which gave it a contemporary viewers amongst individuals who have been considering in a different way about Bitcoin Script.
Folks have been considering in a different way as a result of in October 2023, simply two months prior, Robin Linus had introduced on the mailing record his venture BitVM—an bold venture to do arbitrary computations in Bitcoin Script by splitting applications throughout a number of transactions. The person transactions every do a single easy operation, with outputs from one operation hooked to inputs of one other by way of a hash-preimage-revealing development that appears suspiciously just like a Lamport signature.
The trick right here is that if a consumer Lamport-signs a number of messages with the identical key, the outcome might be two hashes in the identical hash-pair whose preimages are each recognized. That is simple to examine for in Script, which can be utilized to assemble a “slashing transaction” that can take cash from a consumer in the event that they do that. Such a slashing transaction would then grow to be legitimate precisely within the case {that a} consumer publicly signed two messages with the identical key. Slashing transactions are used inside multi-transaction protocols to punish customers who misbehave, usually by forfeiting a bond that they have to submit forward of time.
So these Lamport signatures, somewhat than merely dropping safety when they’re used greater than as soon as, may be configured to actively punish a consumer who indicators a number of occasions. This has apparent purposes for an oracle signature the place a signer is meant to attest to precisely one end result of a real-life occasion; we need to disincentivize such a signer from claiming that e.g. each groups gained a selected sports activities match. However that is an much more highly effective concept than it appears.
Within the cryptographic literature, when a celebration reveals two values for one thing that’s purported to be distinctive, that is known as equivocation. We will consider a slashing transaction as an anti-equivocation measure, as a result of it punishes any signer who produces signatures on the identical key with the identical message.
Then our Lamport signature with anti-equivocation development has the impact of mapping public keys to particular and immutable values. In different phrases, we’ve a world key-value retailer accessible from Script, with the curious property that every entry within the world retailer may be set by a particular individual (the one that is aware of the preimages for that key), however can solely be set as soon as all the time. This key-value retailer can be accessible from any Bitcoin transaction (or a transaction on any blockchain, actually) no matter its connection to different transactions.
This key-value retailer has on the order of two^256 entries, most of which aren’t accessible since no one is aware of the preimages to their keys, so whereas it’s a “world key-value retailer” shared by each attainable program utilizing this Lamport signature development, it can not replenish and there’s no danger that information from one program may by accident clobber information from one other, or {that a} worth which ought to be set by one consumer is perhaps set by one other. Neither is the key-value retailer truly saved wherever in its entirety.
BitVM and its variants use this truth to tie the output of 1 operation to the enter of the subsequent: a given program may be cut up into an extended collection of primary operations, for instance opcodes within the RISC-V instruction set, and every of those primary operations may be carried out by a self-contained Script program which seems to be up the operation’s inputs and outputs within the key-value retailer, checks that they’re associated appropriately, and by some means punishes the consumer if not.
The whole BitVM system is far more sophisticated than this: for every program, it carves out an addressable reminiscence house from the key-value retailer; every operation must search for its inputs and outputs from this reminiscence house; every operation wants to trace a program counter and different state past its inputs and outputs; and the entire thing is tied along with interactive protocols and bushes of unconfirmed transactions to make sure than slashing penalties are appropriately enforced and that even a single incorrect step in a multi-billion-step program may be zeroed-in-on and punished. However this text just isn’t about BitVM and we won’t discover this.
Interlude: Small Script and Massive Script
We take a second to remind the reader that Script can solely do non-trivial computations on values which might be 32 bits huge or smaller. Such values are “scriptnums” and Script has many opcodes to control them by decoding them as signed integers or boolean values, generally as each.
Utilizing BitVM or an identical mechanism to separate Script applications throughout a number of transactions, you are able to do arbitrary computations in Small Script, from ZKP verification to proof-of-work checking to quantity factoring.
Values which might be bigger than 32 bits can solely be manipulated by a small set of narrow-purpose opcodes: they are often hashed, interpreted as public keys or signatures to examine a transaction signature, their measurement in bytes may be computed, and they are often moved across the stack as opaque blobs. The one “actual” general-purpose computation that may be achieved on them is a examine for equality, which by itself supplies little or no worth.
We describe the world of 32-bit values as “Small Script”, which is computationally expressive however can not entry transaction information in any manner. The world of bigger values we name “Massive Script”, and may entry transaction information by means of the CHECKSIG opcode. It is usually attainable to examine hash preimages in Massive Script, and that is important to implementing Lamport signatures, however that is just about the one factor you are able to do in Massive Script.
It’s unimaginable to usefully bridge the 2 worlds: you possibly can hash a Small worth to get a Massive worth, however you can’t then study something in regards to the Massive worth that you just did not already know. And you should utilize the SIZE opcode to study the dimensions of a Massive worth, but when this worth represents a hash, public key or signature, then its measurement is mounted so that you study nothing new. (Though previous to Taproot, signatures had a variable measurement, and it’s attainable which you can extract transaction info from a suitably constrained CHECKSIG-passing transaction.)
All this to remind the reader that, whereas this new Script performance is thrilling, it supplies a variety of computation expressivity with out the flexibility to examine transaction information, and due to this fact can’t be used for vaults or different covenant purposes.
The CAT opcode supplies a mechanism to bridge the 2 Scripts, which is why CAT is adequate to offer covenants. That is additionally why there are such a lot of methods wherein Script “virtually” helps covenants, or wherein non-covenant-related proposals like CAT prove to allow covenants: just about any opcode that takes Small values and outputs Massive ones, or vice-versa, can be utilized to feed Massive Script transaction information right into a Small Script common program. Even a sufficiently dangerous break of the SHA1 opcode might in all probability be used as a bridge, as a result of then you might do “computations” on Massive values by decoding them as SHA1 hashes and discovering Small preimages for them.
Interlude: Wormholes
Really, there’s a manner which you can get covenants in Small Script, assuming you’ve got sufficient computational energy. By going “exterior” of Script, customers can validate the Bitcoin blockchain itself, in addition to the transaction that incorporates the Script (it must keep away from straight encoding its personal information to keep away from being infinitely-sized, however this may be achieved by oblique means; see the subsequent part for extra particulars), after which impose extra constraints on the transaction by imposing these constraints on this internally-validated “view” of itself.
This concept might enable the creation of some restricted covenant performance, however you will need to keep in mind that appropriate entry to the key-value retailer, which is critical to be able to cut up giant computations, just isn’t straight enforced. As a substitute, some extra mechanism must be imposed to implement slashing penalties on incorrect entry. This enormously complicates the implementation of vault-like covenants whose performance is determined by sure spending patterns truly being unimaginable, not simply disincentivized.
Tic-Tac-Toe
So far we’ve talked in regards to the anti-equivocation characteristic of Lamport signatures, and the way this can be utilized to impact a “world key-value retailer” in Bitcoin Script, which in flip can be utilized to move information between Script applications and to separate giant computations into many impartial components. However there’s one other attention-grabbing and maybe stunning side of Lamport signatures, which is that they permit committing to a singular worth in a script with out that worth affecting the TXID of its transaction.
This has two penalties: one is that we are able to commit information in a transaction with out affecting its TXID, which means that we are able to probably change parameters inside a Script program with out invalidating chains of unconfirmed transactions. The opposite is that we are able to commit information with out affecting the signature hash, which means that customers can “pre-sign” a transaction with out first understanding all of its information.
(By the best way, these properties apply to any signature scheme, supplied there’s a examine to punish the signing of a number of values. What’s attention-grabbing about Lamport signatures is that we are able to use them in Bitcoin immediately.)
The flexibility to place information right into a Script program with out affecting the TXID of the contained transaction opens the door to constructions wherein a program is ready to confer with its personal code (for instance, by injecting the TXID itself into this system, which is a hash of all transaction information together with this system). That is known as Quining, and can be utilized to allow delegation and to create recursive covenants. This capacity is the motivation behind the disconnect combinator in Simplicity. Nonetheless, since we are able to solely validate Lamport signatures in Small Script, which excludes objects as giant as txids, it seems that there’s presently nothing we are able to do in that path. Nonetheless, nothing is stopping us from emulating non-recursive covenants with related methods.
Let’s describe an instance attributable to supertestnet on Github.
Contemplate the sport tic-tac-toe, performed between two individuals who take turns marking a three-by-three grid. The principles are easy: no participant might mark an already-marked sq., and if both participant marks three squares in a row (horizontally, vertically, or diagonally) then they win. Think about that these gamers need to play this sport on-chain, representing every flip by a transaction.
In fact, in parallel to those transactions, they might have a single “completely happy path” transaction the place each events would simply signal cash over to the winner in order that in the event that they agreed on the occasions of the sport, they wouldn’t truly must publish the person turns! Nevertheless it’s necessary to additionally assemble the total sport transcript in order that within the case of disputes, the blockchain can mediate.
One strategy they may take is to mannequin the sport as a collection of pre-signed transactions, which every require a signature from each gamers. The primary participant has 9 attainable strikes. So the second participant would signal all 9, and the primary participant would signal whichever one they needed to take. Then for every of the 9 attainable strikes, the second participant has eight strikes; so the primary participant indicators all eight, and the second participant picks one to signal, and so forth.
It seems that this doesn’t fairly work – as a result of both participant may refuse to signal a selected transfer, there isn’t a approach to assign blame within the case that the sport stalls out, and due to this fact no incentive for the dropping participant to finish the sport. To forestall this example, every participant should signal all of his counterparty’s strikes earlier than the sport begins. Then a participant can solely refuse to signal his personal strikes, and this may be simply disincentivized by including timelocked forfeit situations to the transactions.
As an alternative choice to having every participant signal the opposite gamers’ strikes, a trusted third celebration may very well be enlisted to pre-sign every transfer. However the outcome is similar: each attainable collection of transactions have to be signed. For the tic-tac-toe sport, there are 255168 paths for a complete of 549945 pre-signed transactions. That is pushing the bounds of practicality, and it’s clear that such a technique won’t generalize to nontrivial video games. For chess, for instance, these values are bounded under by the Shannon quantity, which is 10^120.
The rationale that we’ve such a big blow-up is that we’re distinguishing between strikes by distinct transactions which every have to be arrange earlier than the sport has began.
Nonetheless, utilizing Lamport signatures, we are able to do significantly better:
- Every sport of tic-tac-toe has (at most) 9 strikes,
- Every of which is a transition between two board states, which might be sufficiently small to be Lamport-signed,
- And every transition should obey guidelines that are easy sufficient to fairly encode inside Script.
We will due to this fact strategy the sport in a different way: every participant generates a Lamport public key with which to signal the sport state after every of their strikes (so the primary participant generates 5 keys, the second participant 4). They then generate a collection of 9 transactions whose output taptrees have three branches:
1. A “abnormal transfer” department, consisting of
- An abnormal signature from each gamers;
- A Lamport signature on the earlier sport state from the suitable participant,
- A Lamport signature on the subsequent sport state from the opposite participant,
- And a examine, carried out in Script, that the two-game states are appropriately associated (they differ by precisely one authorized transfer by the right participant).
2. A “win situation”, consisting of
- An abnormal signature from each gamers;
- A Lamport signature on the earlier ga)me state from the suitable participant,
- A examine, carried out in Script, that this participant has gained the sport.
3. A “timeout” situation, consisting of
- An abnormal signature from each gamers;
- A relative timelock that has expired.
The ultimate transaction, instead of an “abnormal transfer” department, has a “draw” department, since if all strikes have accomplished and not using a win, there isn’t a winner and presumably any cash at stake ought to return to their unique house owners.
As earlier than, every participant pre-signs all transactions, of which there are 27, together with “win situation” transactions (which ship all of the cash to the successful participant), “timeout situation” transactions (which ship all of the cash to the participant who didn’t day out) and “draw situation”.
And by the best way, whereas the foundations of Chess are a good bit extra sophisticated, and the board state might require a number of 32-bit values to symbolize, and there could also be greater than 9 strikes, it’s nonetheless possible to hold out precisely the identical development.
Transaction Timber
Within the earlier instance, we took nice benefit of the truth that the foundations of tic-tac-toe may be embedded in Script itself, which means {that a} consumer can not usefully signal an invalid sport state. (In the event that they signal an invalid transfer, the transaction representing the transfer might be invalid, and the transactions representing all future strikes can even be invalid as a result of they depend upon it. So all of the attacker can have completed is leaking a part of his Lamport signing key, permitting the opposite participant to probably forge strikes on his behalf.
We additionally took benefit of the truth that our full protocol was not very lengthy: at most 9 strikes. Which means if one participant refuses to finish the sport, or completes the sport however won’t acknowledge the outcome, it’s cheap to publish the whole sport transcript on-chain as a recourse. For a lot of video games that is adequate.
It’s out of scope of this weblog submit, however there are numerous methods we are able to play with this mannequin: checking single-party computations as a “sport” between a prover and verifier, outsourcing one or each roles, combining a number of steps into single transactions with giant taptrees, changing the linear transcript with a binary seek for invalid steps, and so forth. These methods type the idea for BitVM, BitVM 2, BitVMX, and so forth.
Utilizing such methods, we are able to cut back the price of current protocols that depend upon bushes of unsigned transactions. A basic 2017 Bitcoin paper by Bentov and Miller argues that stateful protocols within the UTXO mannequin all the time undergo an exponential blowup relative to analogous protocols within the account mannequin, e.g. on Ethereum. Utilizing Lamport signatures as a world key-value retailer, we are able to partially refute this paper. However we’re out of house and might want to discover this in our subsequent submit!
Acknowledgments
I want to thank Robin Linus and Ethan Heilman for reviewing an early draft of this submit.
It is a visitor submit by Andrew Poelstra. Opinions expressed are totally their very own and don’t essentially mirror these of BTC Inc or Bitcoin Journal.