Patent Application
20240372738
The present disclosure relates to a method of generating an output of a blockchain transaction that can be unlocked by any individual or threshold of individuals, and a method of generating a blockchain transaction for unlocking such an output.
A blockchain refers to a form of distributed data structure, wherein a duplicate copy of the blockchain is maintained at each of a plurality of nodes in a distributed peer-to-peer (P2P) network (referred to below as a “blockchain network”) and widely publicised. The blockchain comprises a chain of blocks of data, wherein each block comprises one or more transactions. Each transaction, other than so-called “coinbase transactions”, points back to a preceding transaction in a sequence which may span one or more blocks going back to one or more coinbase transactions. Coinbase transactions are discussed further below.
Transactions that are submitted to the blockchain network are included in new blocks. New blocks are created by a process often referred to as “mining”, which involves each of a plurality of the nodes competing to perform “proof-of-work”, i.e. solving a cryptographic puzzle based on a representation of a defined set of ordered and validated pending transactions waiting to be included in a new block of the blockchain. It should be noted that the blockchain may be pruned at some nodes, and the publication of blocks can be achieved through the publication of mere block headers.
The transactions in the blockchain may be used for one or more of the following purposes: to convey a digital asset (i.e. a number of digital tokens), to order a set of entries in a virtualised ledger or registry, to receive and process timestamp entries, and/or to time-order index pointers. A blockchain can also be exploited in order to layer additional functionality on top of the blockchain. For example blockchain protocols may allow for storage of additional user data or indexes to data in a transaction. There is no pre-specified limit to the maximum data capacity that can be stored within a single transaction, and therefore increasingly more complex data can be incorporated. For instance this may be used to store an electronic document in the blockchain, or audio or video data.
Nodes of the blockchain network (which are often referred to as “miners”) perform a distributed transaction registration and verification process, which will be described in more detail later. In summary, during this process a node validates transactions and inserts them into a block template for which they attempt to identify a valid proof-of-work solution. Once a valid solution is found, a new block is propagated to other nodes of the network, thus enabling each node to record the new block on the blockchain. In order to have a transaction recorded in the blockchain, a user (e.g. a blockchain client application) sends the transaction to one of the nodes of the network to be propagated. Nodes which receive the transaction may race to find a proof-of-work solution incorporating the validated transaction into a new block. Each node is configured to enforce the same node protocol, which will include one or more conditions for a transaction to be valid. Invalid transactions will not be propagated nor incorporated into blocks. Assuming the transaction is validated and thereby accepted onto the blockchain, then the transaction (including any user data) will thus remain registered and indexed at each of the nodes in the blockchain network as an immutable public record.
The node who successfully solved the proof-of-work puzzle to create the latest block is typically rewarded with a new transaction called the “coinbase transaction” which distributes an amount of the digital asset, i.e. a number of tokens. The detection and rejection of invalid transactions is enforced by the actions of competing nodes who act as agents of the network and are incentivised to report and block malfeasance. The widespread publication of information allows users to continuously audit the performance of nodes. The publication of the mere block headers allows participants to ensure the ongoing integrity of the blockchain.
In an “output-based” model (sometimes referred to as a UTXO-based model), the data structure of a given transaction comprises one or more inputs and one or more outputs. Any spendable output comprises an element specifying an amount of the digital asset that is derivable from the proceeding sequence of transactions. The spendable output is sometimes referred to as a UTXO (“unspent transaction output”). The output may further comprise a locking script specifying a condition for the future redemption of the output. A locking script is a predicate defining the conditions necessary to validate and transfer digital tokens or assets. Each input of a transaction (other than a coinbase transaction) comprises a pointer (i.e. a reference) to such an output in a preceding transaction, and may further comprise an unlocking script for unlocking the locking script of the pointed-to output. So consider a pair of transactions, call them a first and a second transaction (or “target” transaction). The first transaction comprises at least one output specifying an amount of the digital asset, and comprising a locking script defining one or more conditions of unlocking the output. The second, target transaction comprises at least one input, comprising a pointer to the output of the first transaction, and an unlocking script for unlocking the output of the first transaction.
In such a model, when the second, target transaction is sent to the blockchain network to be propagated and recorded in the blockchain, one of the criteria for validity applied at each node will be that the unlocking script meets all of the one or more conditions defined in the locking script of the first transaction. Another will be that the output of the first transaction has not already been redeemed by another, earlier valid transaction. Any node that finds the target transaction invalid according to any of these conditions will not propagate it (as a valid transaction, but possibly to register an invalid transaction) nor include it in a new block to be recorded in the blockchain.
An alternative type of transaction model is an account-based model. In this case each transaction does not define the amount to be transferred by referring back to the UTXO of a preceding transaction in a sequence of past transactions, but rather by reference to an absolute account balance. The current state of all accounts is stored by the nodes separate to the blockchain and is updated constantly.
In a blockchain transaction, typically a UTXO is locked to a pre-defined user or users, by including in the locking script the public key of the recipient. However, there may be situations in which the recipient of the UTXO is unknown at the time of generating the locking script.
According to one aspect disclosed herein, there is provided a computer-implemented method of generating a blockchain transaction, wherein each party of a plurality of second parties has a respective secret number k from which a respective r-portion is derivable, wherein a Merkle tree is defined, at least one leaf node of the Merkle tree being based on at least one of the respective r-portions, wherein the method is performed by a first party and comprises: generating a first blockchain transaction, wherein the first blockchain transaction comprises a first locking script comprising a target Merkle tree portion corresponding to the Merkle tree, the first locking script configured, when executed together with a first unlocking script of a second blockchain transaction, to: extract an r-portion from the first unlocking script, generate a hash value derived from the extracted r-portion, calculate a candidate Merkle tree portion based on the generated hash and a Merkle proof portion of the first unlocking script, verify that the calculated candidate Merkle tree portion matches the target Merkle tree portion, verify a signature portion of the first unlocking script based on a public key of the first unlocking script, and making the first blockchain transaction available to one or more nodes of a blockchain network.
The r-portion corresponds to an “R-puzzle”, which is a form of knowledge proof. This allows a user to unlock the UTXO using any valid public-private keypair, which are independent of the locking script, provided the user knows the secret number k which is used to derive r. Therefore, the UTXO can be locked knowing the r-puzzle, and subsequently the secret value k can be provided to the recipient without any need to alter the previous transaction in which the UTXO is locked. That is, the transaction locking the UTXO can be generated without knowledge of the recipient.
Merkle trees are another form of knowledge proof. The unlocker must know both the data from which the leaf node is derived and a Merkle proof, which provides the information required to calculate the Merkle root of a Merkle tree. Therefore, by combining r-puzzles with the Merkle tree, the unlocker must know the secret value k, from which the leaf node is derived, and the Merkle proof corresponding to that particular leaf node.
Thus, the disclosed method improves both flexibility of the blockchain transaction and confidentiality for external parties. Flexibility is improved by allowing any one or any set of a group of users to unlock the transaction, whilst confidentiality is improved by requiring the user(s) to have knowledge of a range of data items (namely, the Merkle proof portion, the signature portion, and the public key).
The use of the Merkle tree in script also provides a more efficient and scalable method to known threshold unlocking schemes. This is because the locking script does not comprise all of the keys which may be used to unlock the UTXO, or all of the possible sets of keys in the case of an M-of-N scheme.
To assist understanding of embodiments of the present disclosure and to show how such embodiments may be put into effect, reference is made, by way of example only, to the accompanying drawings in which:
Each blockchain node 104 comprises computer equipment of a peer, with different ones of the nodes 104 belonging to different peers. Each blockchain node 104 comprises processing apparatus comprising one or more processors, e.g. one or more central processing units (CPUs), accelerator processors, application specific processors and/or field programmable gate arrays (FPGAs), and other equipment such as application specific integrated circuits (ASICs). Each node also comprises memory, i.e. computer-readable storage in the form of a non-transitory computer-readable medium or media. The memory may comprise one or more memory units employing one or more memory media, e.g. a magnetic medium such as a hard disk; an electronic medium such as a solid-state drive (SSD), flash memory or EEPROM; and/or an optical medium such as an optical disk drive.
The blockchain 150 comprises a chain of blocks of data 151, wherein a respective copy of the blockchain 150 is maintained at each of a plurality of blockchain nodes 104 in the distributed or blockchain network 106. As mentioned above, maintaining a copy of the blockchain 150 does not necessarily mean storing the blockchain 150 in full. Instead, the blockchain 150 may be pruned of data so long as each blockchain node 150 stores the block header (discussed below) of each block 151. Each block 151 in the chain comprises one or more transactions 152, wherein a transaction in this context refers to a kind of data structure. The nature of the data structure will depend on the type of transaction protocol used as part of a transaction model or scheme. A given blockchain will use one particular transaction protocol throughout. In one common type of transaction protocol, the data structure of each transaction 152 comprises at least one input and at least one output. Each output specifies an amount representing a quantity of a digital asset as property, an example of which is a user 103 to whom the output is cryptographically locked (requiring a signature or other solution of that user in order to be unlocked and thereby redeemed or spent). Each input points back to the output of a preceding transaction 152, thereby linking the transactions.
Each block 151 also comprises a block pointer 155 pointing back to the previously created block 151 in the chain so as to define a sequential order to the blocks 151. Each transaction 152 (other than a coinbase transaction) comprises a pointer back to a previous transaction so as to define an order to sequences of transactions (N.B. sequences of transactions 152 are allowed to branch). The chain of blocks 151 goes all the way back to a genesis block (Gb) 153 which was the first block in the chain. One or more original transactions 152 early on in the chain 150 pointed to the genesis block 153 rather than a preceding transaction.
Each of the blockchain nodes 104 is configured to forward transactions 152 to other blockchain nodes 104, and thereby cause transactions 152 to be propagated throughout the network 106. Each blockchain node 104 is configured to create blocks 151 and to store a respective copy of the same blockchain 150 in their respective memory. Each blockchain node 104 also maintains an ordered set (or “pool”) 154 of transactions 152 waiting to be incorporated into blocks 151. The ordered pool 154 is often referred to as a “mempool”. This term herein is not intended to limit to any particular blockchain, protocol or model. It refers to the ordered set of transactions which a node 104 has accepted as valid and for which the node 104 is obliged not to accept any other transactions attempting to spend the same output.
In a given present transaction 152j, the (or each) input comprises a pointer referencing the output of a preceding transaction 152i in the sequence of transactions, specifying that this output is to be redeemed or “spent” in the present transaction 152j. In general, the preceding transaction could be any transaction in the ordered set 154 or any block 151. The preceding transaction 152i need not necessarily exist at the time the present transaction 152j is created or even sent to the network 106, though the preceding transaction 152i will need to exist and be validated in order for the present transaction to be valid. Hence “preceding” herein refers to a predecessor in a logical sequence linked by pointers, not necessarily the time of creation or sending in a temporal sequence, and hence it does not necessarily exclude that the transactions 152i, 152j be created or sent out-of-order (see discussion below on orphan transactions). The preceding transaction 152i could equally be called the antecedent or predecessor transaction.
The input of the present transaction 152j also comprises the input authorisation, for example the signature of the user 103a to whom the output of the preceding transaction 152i is locked. In turn, the output of the present transaction 152j can be cryptographically locked to a new user or entity 103b. The present transaction 152j can thus transfer the amount defined in the input of the preceding transaction 152i to the new user or entity 103b as defined in the output of the present transaction 152j. In some cases a transaction 152 may have multiple outputs to split the input amount between multiple users or entities (one of whom could be the original user or entity 103a in order to give change). In some cases a transaction can also have multiple inputs to gather together the amounts from multiple outputs of one or more preceding transactions, and redistribute to one or more outputs of the current transaction.
According to an output-based transaction protocol such as bitcoin, when a party 103, such as an individual user or an organization, wishes to enact a new transaction 152j (either manually or by an automated process employed by the party), then the enacting party sends the new transaction from its computer terminal 102 to a recipient. The enacting party or the recipient will eventually send this transaction to one or more of the blockchain nodes 104 of the network 106 (which nowadays are typically servers or data centres, but could in principle be other user terminals). It is also not excluded that the party 103 enacting the new transaction 152j could send the transaction directly to one or more of the blockchain nodes 104 and, in some examples, not to the recipient. A blockchain node 104 that receives a transaction checks whether the transaction is valid according to a blockchain node protocol which is applied at each of the blockchain nodes 104. The blockchain node protocol typically requires the blockchain node 104 to check that a cryptographic signature in the new transaction 152j matches the expected signature, which depends on the previous transaction 152i in an ordered sequence of transactions 152. In such an output-based transaction protocol, this may comprise checking that the cryptographic signature or other authorisation of the party 103 included in the input of the new transaction 152j matches a condition defined in the output of the preceding transaction 152i which the new transaction assigns, wherein this condition typically comprises at least checking that the cryptographic signature or other authorisation in the input of the new transaction 152j unlocks the output of the previous transaction 152i to which the input of the new transaction is linked to. The condition may be at least partially defined by a script included in the output of the preceding transaction 152i. Alternatively it could simply be fixed by the blockchain node protocol alone, or it could be due to a combination of these. Either way, if the new transaction 152j is valid, the blockchain node 104 forwards it to one or more other blockchain nodes 104 in the blockchain network 106. These other blockchain nodes 104 apply the same test according to the same blockchain node protocol, and so forward the new transaction 152j on to one or more further nodes 104, and so forth. In this way the new transaction is propagated throughout the network of blockchain nodes 104.
In an output-based model, the definition of whether a given output (e.g. UTXO) is assigned (e.g. spent) is whether it has yet been validly redeemed by the input of another, onward transaction 152j according to the blockchain node protocol. Another condition for a transaction to be valid is that the output of the preceding transaction 152i which it attempts to redeem has not already been redeemed by another transaction. Again if not valid, the transaction 152j will not be propagated (unless flagged as invalid and propagated for alerting) or recorded in the blockchain 150. This guards against double-spending whereby the transactor tries to assign the output of the same transaction more than once. An account-based model on the other hand guards against double-spending by maintaining an account balance. Because again there is a defined order of transactions, the account balance has a single defined state at any one time.
In addition to validating transactions, blockchain nodes 104 also race to be the first to create blocks of transactions in a process commonly referred to as mining, which is supported by “proof-of-work”. At a blockchain node 104, new transactions are added to an ordered pool 154 of valid transactions that have not yet appeared in a block 151 recorded on the blockchain 150. The blockchain nodes then race to assemble a new valid block 151 of transactions 152 from the ordered set of transactions 154 by attempting to solve a cryptographic puzzle. Typically this comprises searching for a “nonce” value such that when the nonce is concatenated with a representation of the ordered pool of pending transactions 154 and hashed, then the output of the hash meets a predetermined condition. E.g. the predetermined condition may be that the output of the hash has a certain predefined number of leading zeros. Note that this is just one particular type of proof-of-work puzzle, and other types are not excluded. A property of a hash function is that it has an unpredictable output with respect to its input. Therefore this search can only be performed by brute force, thus consuming a substantive amount of processing resource at each blockchain node 104 that is trying to solve the puzzle.
The first blockchain node 104 to solve the puzzle announces this to the network 106, providing the solution as proof which can then be easily checked by the other blockchain nodes 104 in the network (once given the solution to a hash it is straightforward to check that it causes the output of the hash to meet the condition). The first blockchain node 104 propagates a block to a threshold consensus of other nodes that accept the block and thus enforce the protocol rules. The ordered set of transactions 154 then becomes recorded as a new block 151 in the blockchain 150 by each of the blockchain nodes 104. A block pointer 155 is also assigned to the new block 151n pointing back to the previously created block 151n−1 in the chain. The significant amount of effort, for example in the form of hash, required to create a proof-of-work solution signals the intent of the first node 104 to follow the rules of the blockchain protocol. Such rules include not accepting a transaction as valid if it assigns the same output as a previously validated transaction, otherwise known as double-spending. Once created, the block 151 cannot be modified since it is recognized and maintained at each of the blockchain nodes 104 in the blockchain network 106. The block pointer 155 also imposes a sequential order to the blocks 151. Since the transactions 152 are recorded in the ordered blocks at each blockchain node 104 in a network 106, this therefore provides an immutable public ledger of the transactions.
Note that different blockchain nodes 104 racing to solve the puzzle at any given time may be doing so based on different snapshots of the pool of yet-to-be published transactions 154 at any given time, depending on when they started searching for a solution or the order in which the transactions were received. Whoever solves their respective puzzle first defines which transactions 152 are included in the next new block 151n and in which order, and the current pool 154 of unpublished transactions is updated. The blockchain nodes 104 then continue to race to create a block from the newly-defined ordered pool of unpublished transactions 154, and so forth. A protocol also exists for resolving any “fork” that may arise, which is where two blockchain nodes 104 solve their puzzle within a very short time of one another such that a conflicting view of the blockchain gets propagated between nodes 104. In short, whichever prong of the fork grows the longest becomes the definitive blockchain 150. Note this should not affect the users or agents of the network as the same transactions will appear in both forks.
According to the bitcoin blockchain (and most other blockchains) a node that successfully constructs a new block 104 is granted the ability to newly assign an additional, accepted amount of the digital asset in a new special kind of transaction which distributes an additional defined quantity of the digital asset (as opposed to an inter-agent, or inter-user transaction which transfers an amount of the digital asset from one agent or user to another). This special type of transaction is usually referred to as a “coinbase transaction”, but may also be termed an “initiation transaction” or “generation transaction”. It typically forms the first transaction of the new block 151n. The proof-of-work signals the intent of the node that constructs the new block to follow the protocol rules allowing this special transaction to be redeemed later. The blockchain protocol rules may require a maturity period, for example 100 blocks, before this special transaction may be redeemed. Often a regular (non-generation) transaction 152 will also specify an additional transaction fee in one of its outputs, to further reward the blockchain node 104 that created the block 151n in which that transaction was published. This fee is normally referred to as the “transaction fee”, and is discussed blow.
Due to the resources involved in transaction validation and publication, typically at least each of the blockchain nodes 104 takes the form of a server comprising one or more physical server units, or even whole a data centre. However in principle any given blockchain node 104 could take the form of a user terminal or a group of user terminals networked together.
The memory of each blockchain node 104 stores software configured to run on the processing apparatus of the blockchain node 104 in order to perform its respective role or roles and handle transactions 152 in accordance with the blockchain node protocol. It will be understood that any action attributed herein to a blockchain node 104 may be performed by the software run on the processing apparatus of the respective computer equipment. The node software may be implemented in one or more applications at the application layer, or a lower layer such as the operating system layer or a protocol layer, or any combination of these.
Also connected to the network 101 is the computer equipment 102 of each of a plurality of parties 103 in the role of consuming users. These users may interact with the blockchain network 106 but do not participate in validating transactions or constructing blocks. Some of these users or agents 103 may act as senders and recipients in transactions. Other users may interact with the blockchain 150 without necessarily acting as senders or recipients. For instance, some parties may act as storage entities that store a copy of the blockchain 150 (e.g. having obtained a copy of the blockchain from a blockchain node 104).
Some or all of the parties 103 may be connected as part of a different network, e.g. a network overlaid on top of the blockchain network 106. Users of the blockchain network (often referred to as “clients”) may be said to be part of a system that includes the blockchain network 106; however, these users are not blockchain nodes 104 as they do not perform the roles required of the blockchain nodes. Instead, each party 103 may interact with the blockchain network 106 and thereby utilize the blockchain 150 by connecting to (i.e. communicating with) a blockchain node 106. Two parties 103 and their respective equipment 102 are shown for illustrative purposes: a first party 103a and his/her respective computer equipment 102a, and a second party 103b and his/her respective computer equipment 102b. It will be understood that many more such parties 103 and their respective computer equipment 102 may be present and participating in the system 100, but for convenience they are not illustrated. Each party 103 may be an individual or an organization. Purely by way of illustration the first party 103a is referred to herein as Alice and the second party 103b is referred to as Bob, but it will be appreciated that this is not limiting and any reference herein to Alice or Bob may be replaced with “first party” and “second “party” respectively.
The computer equipment 102 of each party 103 comprises respective processing apparatus comprising one or more processors, e.g. one or more CPUs, GPUs, other accelerator processors, application specific processors, and/or FPGAs. The computer equipment 102 of each party 103 further comprises memory, i.e. computer-readable storage in the form of a non-transitory computer-readable medium or media. This memory may comprise one or more memory units employing one or more memory media, e.g. a magnetic medium such as hard disk; an electronic medium such as an SSD, flash memory or EEPROM; and/or an optical medium such as an optical disc drive. The memory on the computer equipment 102 of each party 103 stores software comprising a respective instance of at least one client application 105 arranged to run on the processing apparatus. It will be understood that any action attributed herein to a given party 103 may be performed using the software run on the processing apparatus of the respective computer equipment 102. The computer equipment 102 of each party 103 comprises at least one user terminal, e.g. a desktop or laptop computer, a tablet, a smartphone, or a wearable device such as a smartwatch. The computer equipment 102 of a given party 103 may also comprise one or more other networked resources, such as cloud computing resources accessed via the user terminal.
The client application 105 may be initially provided to the computer equipment 102 of any given party 103 on suitable computer-readable storage medium or media, e.g. downloaded from a server, or provided on a removable storage device such as a removable SSD, flash memory key, removable EEPROM, removable magnetic disk drive, magnetic floppy disk or tape, optical disk such as a CD or DVD ROM, or a removable optical drive, etc.
The client application 105 comprises at least a “wallet” function. This has two main functionalities. One of these is to enable the respective party 103 to create, authorise (for example sign) and send transactions 152 to one or more bitcoin nodes 104 to then be propagated throughout the network of blockchain nodes 104 and thereby included in the blockchain 150. The other is to report back to the respective party the amount of the digital asset that he or she currently owns. In an output-based system, this second functionality comprises collating the amounts defined in the outputs of the various 152 transactions scattered throughout the blockchain 150 that belong to the party in question.
Note: whilst the various client functionality may be described as being integrated into a given client application 105, this is not necessarily limiting and instead any client functionality described herein may instead be implemented in a suite of two or more distinct applications, e.g. interfacing via an API, or one being a plug-in to the other. More generally the client functionality could be implemented at the application layer or a lower layer such as the operating system, or any combination of these. The following will be described in terms of a client application 105 but it will be appreciated that this is not limiting.
The instance of the client application or software 105 on each computer equipment 102 is operatively coupled to at least one of the blockchain nodes 104 of the network 106. This enables the wallet function of the client 105 to send transactions 152 to the network 106. The client 105 is also able to contact blockchain nodes 104 in order to query the blockchain 150 for any transactions of which the respective party 103 is the recipient (or indeed inspect other parties' transactions in the blockchain 150, since in embodiments the blockchain 150 is a public facility which provides trust in transactions in part through its public visibility). The wallet function on each computer equipment 102 is configured to formulate and send transactions 152 according to a transaction protocol. As set out above, each blockchain node 104 runs software configured to validate transactions 152 according to the blockchain node protocol, and to forward transactions 152 in order to propagate them throughout the blockchain network 106. The transaction protocol and the node protocol correspond to one another, and a given transaction protocol goes with a given node protocol, together implementing a given transaction model. The same transaction protocol is used for all transactions 152 in the blockchain 150. The same node protocol is used by all the nodes 104 in the network 106.
When a given party 103, say Alice, wishes to send a new transaction 152j to be included in the blockchain 150, then she formulates the new transaction in accordance with the relevant transaction protocol (using the wallet function in her client application 105). She then sends the transaction 152 from the client application 105 to one or more blockchain nodes 104 to which she is connected. E.g. this could be the blockchain node 104 that is best connected to Alice's computer 102. When any given blockchain node 104 receives a new transaction 152j, it handles it in accordance with the blockchain node protocol and its respective role. This comprises first checking whether the newly received transaction 152j meets a certain condition for being “valid”, examples of which will be discussed in more detail shortly. In some transaction protocols, the condition for validation may be configurable on a per-transaction basis by scripts included in the transactions 152. Alternatively the condition could simply be a built-in feature of the node protocol, or be defined by a combination of the script and the node protocol.
On condition that the newly received transaction 152j passes the test for being deemed valid (i.e. on condition that it is “validated”), any blockchain node 104 that receives the transaction 152j will add the new validated transaction 152 to the ordered set of transactions 154 maintained at that blockchain node 104. Further, any blockchain node 104 that receives the transaction 152j will propagate the validated transaction 152 onward to one or more other blockchain nodes 104 in the network 106. Since each blockchain node 104 applies the same protocol, then assuming the transaction 152j is valid, this means it will soon be propagated throughout the whole network 106.
Once admitted to the ordered pool of pending transactions 154 maintained at a given blockchain node 104, that blockchain node 104 will start competing to solve the proof-of-work puzzle on the latest version of their respective pool of 154 including the new transaction 152 (recall that other blockchain nodes 104 may be trying to solve the puzzle based on a different pool of transactions 154, but whoever gets there first will define the set of transactions that are included in the latest block 151. Eventually a blockchain node 104 will solve the puzzle for a part of the ordered pool 154 which includes Alice's transaction 152j). Once the proof-of-work has been done for the pool 154 including the new transaction 152j, it immutably becomes part of one of the blocks 151 in the blockchain 150. Each transaction 152 comprises a pointer back to an earlier transaction, so the order of the transactions is also immutably recorded.
Different blockchain nodes 104 may receive different instances of a given transaction first and therefore have conflicting views of which instance is ‘valid’ before one instance is published in a new block 151, at which point all blockchain nodes 104 agree that the published instance is the only valid instance. If a blockchain node 104 accepts one instance as valid, and then discovers that a second instance has been recorded in the blockchain 150 then that blockchain node 104 must accept this and will discard (i.e. treat as invalid) the instance which it had initially accepted (i.e. the one that has not been published in a block 151).
An alternative type of transaction protocol operated by some blockchain networks may be referred to as an “account-based” protocol, as part of an account-based transaction model. In the account-based case, each transaction does not define the amount to be transferred by referring back to the UTXO of a preceding transaction in a sequence of past transactions, but rather by reference to an absolute account balance. The current state of all accounts is stored, by the nodes of that network, separate to the blockchain and is updated constantly. In such a system, transactions are ordered using a running transaction tally of the account (also called the “position”). This value is signed by the sender as part of their cryptographic signature and is hashed as part of the transaction reference calculation. In addition, an optional data field may also be signed the transaction. This data field may point back to a previous transaction, for example if the previous transaction ID is included in the data field.
In a UTXO-based model, each transaction (“Tx”) 152 comprises a data structure comprising one or more inputs 202, and one or more outputs 203. Each output 203 may comprise an unspent transaction output (UTXO), which can be used as the source for the input 202 of another new transaction (if the UTXO has not already been redeemed). The UTXO includes a value specifying an amount of a digital asset. This represents a set number of tokens on the distributed ledger. The UTXO may also contain the transaction ID of the transaction from which it came, amongst other information. The transaction data structure may also comprise a header 201, which may comprise an indicator of the size of the input field(s) 202 and output field(s) 203. The header 201 may also include an ID of the transaction. In embodiments the transaction ID is the hash of the transaction data (excluding the transaction ID itself) and stored in the header 201 of the raw transaction 152 submitted to the nodes 104.
Say Alice 103a wishes to create a transaction 152j transferring an amount of the digital asset in question to Bob 103b. In
The preceding transaction Tx0 may already have been validated and included in a block 151 of the blockchain 150 at the time when Alice creates her new transaction Tx1, or at least by the time she sends it to the network 106. It may already have been included in one of the blocks 151 at that time, or it may be still waiting in the ordered set 154 in which case it will soon be included in a new block 151. Alternatively Tx0 and Tx1 could be created and sent to the network 106 together, or Tx0 could even be sent after Tx1 if the node protocol allows for buffering “orphan” transactions. The terms “preceding” and “subsequent” as used herein in the context of the sequence of transactions refer to the order of the transactions in the sequence as defined by the transaction pointers specified in the transactions (which transaction points back to which other transaction, and so forth). They could equally be replaced with “predecessor” and “successor”, or “antecedent” and “descendant”, “parent” and “child”, or such like. It does not necessarily imply an order in which they are created, sent to the network 106, or arrive at any given blockchain node 104. Nevertheless, a subsequent transaction (the descendent transaction or “child”) which points to a preceding transaction (the antecedent transaction or “parent”) will not be validated until and unless the parent transaction is validated. A child that arrives at a blockchain node 104 before its parent is considered an orphan. It may be discarded or buffered for a certain time to wait for the parent, depending on the node protocol and/or node behaviour.
One of the one or more outputs 203 of the preceding transaction Tx0 comprises a particular UTXO, labelled here UTXO0. Each UTXO comprises a value specifying an amount of the digital asset represented by the UTXO, and a locking script which defines a condition which must be met by an unlocking script in the input 202 of a subsequent transaction in order for the subsequent transaction to be validated, and therefore for the UTXO to be successfully redeemed. Typically the locking script locks the amount to a particular party (the beneficiary of the transaction in which it is included). I.e. the locking script defines an unlocking condition, typically comprising a condition that the unlocking script in the input of the subsequent transaction comprises the cryptographic signature of the party to whom the preceding transaction is locked.
The locking script (aka scriptPubKey) is a piece of code written in the domain specific language recognized by the node protocol. A particular example of such a language is called “Script” (capital S) which is used by the blockchain network. The locking script specifies what information is required to spend a transaction output 203, for example the requirement of Alice's signature. Unlocking scripts appear in the outputs of transactions. The unlocking script (aka scriptSig) is a piece of code written the domain specific language that provides the information required to satisfy the locking script criteria. For example, it may contain Bob's signature. Unlocking scripts appear in the input 202 of transactions.
So in the example illustrated, UTXO0 in the output 203 of Tx0 comprises a locking script [Checksig PA] which requires a signature Sig PA of Alice in order for UTXO0 to be redeemed (strictly, in order for a subsequent transaction attempting to redeem UTXO0 to be valid). [Checksig PA] contains a representation (i.e. a hash) of the public key PA from a public-private key pair of Alice. The input 202 of Tx1 comprises a pointer pointing back to Tx1 (e.g. by means of its transaction ID, TxID0, which in embodiments is the hash of the whole transaction Tx0). The input 202 of Tx1 comprises an index identifying UTXO0 within Tx0, to identify it amongst any other possible outputs of Tx0. The input 202 of Tx1 further comprises an unlocking script <Sig PA> which comprises a cryptographic signature of Alice, created by Alice applying her private key from the key pair to a predefined portion of data (sometimes called the “message” in cryptography). The data (or “message”) that needs to be signed by Alice to provide a valid signature may be defined by the locking script, or by the node protocol, or by a combination of these.
When the new transaction Tx1 arrives at a blockchain node 104, the node applies the node protocol. This comprises running the locking script and unlocking script together to check whether the unlocking script meets the condition defined in the locking script (where this condition may comprise one or more criteria). In embodiments this involves concatenating the two scripts:
<Sig PA><PA>∥[Checksig PA]
where “∥” represents a concatenation and “< . . . >” means place the data on the stack, and “[ . . . ]” is a function comprised by the locking script (in this example a stack-based language). Equivalently the scripts may be run one after the other, with a common stack, rather than concatenating the scripts. Either way, when run together, the scripts use the public key PA of Alice, as included in the locking script in the output of Tx0, to authenticate that the unlocking script in the input of Tx1 contains the signature of Alice signing the expected portion of data. The expected portion of data itself (the “message”) also needs to be included in order to perform this authentication. In embodiments the signed data comprises the whole of Tx1 (so a separate element does not need to be included specifying the signed portion of data in the clear, as it is already inherently present).
The details of authentication by public-private cryptography will be familiar to a person skilled in the art. Basically, if Alice has signed a message using her private key, then given Alice's public key and the message in the clear, another entity such as a node 104 is able to authenticate that the message must have been signed by Alice. Signing typically comprises hashing the message, signing the hash, and tagging this onto the message as a signature, thus enabling any holder of the public key to authenticate the signature. Note therefore that any reference herein to signing a particular piece of data or part of a transaction, or such like, can in embodiments mean signing a hash of that piece of data or part of the transaction.
If the unlocking script in Tx1 meets the one or more conditions specified in the locking script of Tx0 (so in the example shown, if Alice's signature is provided in Tx1 and authenticated), then the blockchain node 104 deems Tx1 valid. This means that the blockchain node 104 will add Tx1 to the ordered pool of pending transactions 154. The blockchain node 104 will also forward the transaction Tx1 to one or more other blockchain nodes 104 in the network 106, so that it will be propagated throughout the network 106. Once Tx1 has been validated and included in the blockchain 150, this defines UTXO0 from Tx0 as spent. Note that Tx1 can only be valid if it spends an unspent transaction output 203. If it attempts to spend an output that has already been spent by another transaction 152, then Tx1 will be invalid even if all the other conditions are met. Hence the blockchain node 104 also needs to check whether the referenced UTXO in the preceding transaction Tx0 is already spent (i.e. whether it has already formed a valid input to another valid transaction). This is one reason why it is important for the blockchain 150 to impose a defined order on the transactions 152. In practice a given blockchain node 104 may maintain a separate database marking which UTXOs 203 in which transactions 152 have been spent, but ultimately what defines whether a UTXO has been spent is whether it has already formed a valid input to another valid transaction in the blockchain 150.
If the total amount specified in all the outputs 203 of a given transaction 152 is greater than the total amount pointed to by all its inputs 202, this is another basis for invalidity in most transaction models. Therefore such transactions will not be propagated nor included in a block 151.
Note that in UTXO-based transaction models, a given UTXO needs to be spent as a whole. It cannot “leave behind” a fraction of the amount defined in the UTXO as spent while another fraction is spent. However the amount from the UTXO can be split between multiple outputs of the next transaction. E.g. the amount defined in UTXO0 in Tx0 can be split between multiple UTXOs in Tx1. Hence if Alice does not want to give Bob all of the amount defined in UTXO0, she can use the remainder to give herself change in a second output of Tx1, or pay another party.
In practice Alice will also usually need to include a fee for the bitcoin node 104 that successfully includes her transaction 104 in a block 151. If Alice does not include such a fee, Tx0 may be rejected by the blockchain nodes 104, and hence although technically valid, may not be propagated and included in the blockchain 150 (the node protocol does not force blockchain nodes 104 to accept transactions 152 if they don't want). In some protocols, the transaction fee does not require its own separate output 203 (i.e. does not need a separate UTXO). Instead any difference between the total amount pointed to by the input(s) 202 and the total amount of specified in the output(s) 203 of a given transaction 152 is automatically given to the blockchain node 104 publishing the transaction. E.g. say a pointer to UTXO0 is the only input to Tx1, and Tx1 has only one output UTXO1. If the amount of the digital asset specified in UTXO0 is greater than the amount specified in UTXO1, then the difference may be assigned by the node 104 that wins the proof-of-work race to create the block containing UTXO1. Alternatively or additionally however, it is not necessarily excluded that a transaction fee could be specified explicitly in its own one of the UTXOs 203 of the transaction 152.
Alice and Bob's digital assets consist of the UTXOs locked to them in any transactions 152 anywhere in the blockchain 150. Hence typically, the assets of a given party 103 are scattered throughout the UTXOs of various transactions 152 throughout the blockchain 150. There is no one number stored anywhere in the blockchain 150 that defines the total balance of a given party 103. It is the role of the wallet function in the client application 105 to collate together the values of all the various UTXOs which are locked to the respective party and have not yet been spent in another onward transaction. It can do this by querying the copy of the blockchain 150 as stored at any of the bitcoin nodes 104.
Note that the script code is often represented schematically (i.e. not using the exact language). For example, one may use operation codes (opcodes) to represent a particular function. “OP_. . . ” refers to a particular opcode of the Script language. As an example, OP_RETURN is an opcode of the Script language that when preceded by OP_FALSE at the beginning of a locking script creates an unspendable output of a transaction that can store data within the transaction, and thereby record the data immutably in the blockchain 150. E.g. the data could comprise a document which it is desired to store in the blockchain.
Typically an input of a transaction contains a digital signature corresponding to a public key PA. In embodiments this is based on the ECDSA using the elliptic curve secp256k1. A digital signature signs a particular piece of data. In some embodiments, for a given transaction the signature will sign part of the transaction input, and some or all of the transaction outputs. The particular parts of the outputs it signs depends on the SIGHASH flag. The SIGHASH flag is usually a 4-byte code included at the end of a signature to select which outputs are signed (and thus fixed at the time of signing).
The locking script is sometimes called “scriptPubKey” referring to the fact that it typically comprises the public key of the party to whom the respective transaction is locked. The unlocking script is sometimes called “scriptSig” referring to the fact that it typically supplies the corresponding signature. However, more generally it is not essential in all applications of a blockchain 150 that the condition for a UTXO to be redeemed comprises authenticating a signature. More generally the scripting language could be used to define any one or more conditions. Hence the more general terms “locking script” and “unlocking script” may be preferred.
As shown in
The side channel 107 may be established via the same packet-switched network 101 as the blockchain network 106. Alternatively or additionally, the side channel 301 may be established via a different network such as a mobile cellular network, or a local area network such as a local wireless network, or even a direct wired or wireless link between Alice and Bob's devices 102a, 102b. Generally, the side channel 107 as referred to anywhere herein may comprise any one or more links via one or more networking technologies or communication media for exchanging data “off-chain”, i.e. separately from the blockchain network 106. Where more than one link is used, then the bundle or collection of off-chain links as a whole may be referred to as the side channel 107. Note therefore that if it is said that Alice and Bob exchange certain pieces of information or data, or such like, over the side channel 107, then this does not necessarily imply all these pieces of data have to be send over exactly the same link or even the same type of network.
The UI layer 402 is configured to render a user interface via a user input/output (I/O) means of the respective user's computer equipment 102, including outputting information to the respective user 103 via a user output means of the equipment 102, and receiving inputs back from the respective user 103 via a user input means of the equipment 102. For example the user output means could comprise one or more display screens (touch or non-touch screen) for providing a visual output, one or more speakers for providing an audio output, and/or one or more haptic output devices for providing a tactile output, etc. The user input means could comprise for example the input array of one or more touch screens (the same or different as that/those used for the output means); one or more cursor-based devices such as mouse, trackpad or trackball; one or more microphones and speech or voice recognition algorithms for receiving a speech or vocal input; one or more gesture-based input devices for receiving the input in the form of manual or bodily gestures; or one or more mechanical buttons, switches or joysticks, etc.
Note: whilst the various functionality herein may be described as being integrated into the same client application 105, this is not necessarily limiting and instead they could be implemented in a suite of two or more distinct applications, e.g. one being a plug-in to the other or interfacing via an API (application programming interface). For instance, the functionality of the transaction engine 401 may be implemented in a separate application than the UI layer 402, or the functionality of a given module such as the transaction engine 401 could be split between more than one application. Nor is it excluded that some or all of the described functionality could be implemented at, say, the operating system layer. Where reference is made anywhere herein to a single or given application 105, or such like, it will be appreciated that this is just by way of example, and more generally the described functionality could be implemented in any form of software.
By way of illustration
For example, the UI elements may comprise one or more user-selectable elements 501 which may be, such as different on-screen buttons, or different options in a menu, or such like. The user input means is arranged to enable the user 103 (in this case Alice 103a) to select or otherwise operate one of the options, such as by clicking or touching the UI element on-screen, or speaking a name of the desired option (N.B. the term “manual” as used herein is meant only to contrast against automatic, and does not necessarily limit to the use of the hand or hands). The options enable the user (Alice) to select one or more of: a secret values k from which an r-portion is derived, and a Merkle tree or identifier thereof.
Alternatively or additionally, the UI elements may comprise one or more data entry fields 502, through which the user can input select one or more of: a secret values k from which an r-portion is derived, and a Merkle tree or identifier thereof. These data entry fields are rendered via the user output means, e.g. on-screen, and the data can be entered into the fields through the user input means, e.g. a keyboard or touchscreen. Alternatively the data could be received orally for example based on speech recognition.
Alternatively or additionally, the UI elements may comprise one or more information elements 503 output to output information to the user. E.g. this/these could be rendered on screen or audibly.
It will be appreciated that the particular means of rendering the various UI elements, selecting the options and entering data is not material. The functionality of these UI elements will be discussed in more detail shortly. It will also be appreciated that the UI 500 shown in
The script engine 452 thus has the locking script of Txi and the unlocking script from the corresponding input of Txj. For example, transactions labelled Tx0 and Tx1 are illustrated in
By running the scripts together, the script engine 452 determines whether or not the unlocking script meets the one or more criteria defined in the locking script—i.e. does it “unlock” the output in which the locking script is included? The script engine 452 returns a result of this determination to the protocol engine 451. If the script engine 452 determines that the unlocking script does meet the one or more criteria specified in the corresponding locking script, then it returns the result “true”. Otherwise it returns the result “false”.
In an output-based model, the result “true” from the script engine 452 is one of the conditions for validity of the transaction. Typically there are also one or more further, protocol-level conditions evaluated by the protocol engine 451 that must be met as well; such as that the total amount of digital asset specified in the output(s) of Txj does not exceed the total amount pointed to by its inputs, and that the pointed-to output of Txi has not already been spent by another valid transaction. The protocol engine 451 evaluates the result from the script engine 452 together with the one or more protocol-level conditions, and only if they are all true does it validate the transaction Txj. The protocol engine 451 outputs an indication of whether the transaction is valid to the application-level decision engine 454. Only on condition that Txj is indeed validated, the decision engine 454 may select to control both of the consensus module 455C and the propagation module 455P to perform their respective blockchain-related function in respect of Txj. This comprises the consensus module 455C adding Txj to the node's respective ordered set of transactions 154 for incorporating in a block 151, and the propagation module 455P forwarding Txj to another blockchain node 104 in the network 106. Optionally, in embodiments the application-level decision engine 454 may apply one or more additional conditions before triggering either or both of these functions. E.g. the decision engine may only select to publish the transaction on condition that the transaction is both valid and leaves enough of a transaction fee.
Note also that the terms “true” and “false” herein do not necessarily limit to returning a result represented in the form of only a single binary digit (bit), though that is certainly one possible implementation. More generally, “true” can refer to any state indicative of a successful or affirmative outcome, and “false” can refer to any state indicative of an unsuccessful or non-affirmative outcome. For instance in an account-based model, a result of “true” could be indicated by a combination of an implicit, protocol-level validation of a signature and an additional affirmative output of a smart contract (the overall result being deemed to signal true if both individual outcomes are true).
Public key cryptography is used as a basis for securing transactions in a number of different blockchain architectures. Uses of public key cryptography include public key encryption and digital signature schemes. Public key cryptography is founded on the principle that certain functions are easy to compute but hard to reverse without some special knowledge. Such a function is called a trapdoor function and the special knowledge needed to reverse it is referred to as a trapdoor of that function. Easy to compute means it is computationally feasible to compute the trapdoor function for a given input (or set of inputs) in a reasonable time frame, and hard to reverse that it is computationally infeasible to infer that input (or those inputs) from the result without knowledge of the trapdoor.
In the context of public key cryptography, a key pair means a public key (which can be made freely available to anyone) and a corresponding private key (which is assumed to be secret in the sense that it is only known to a specific entity or group). The public key defines a trapdoor function and the corresponding private key is the trapdoor needed to reverse that function.
In a public key encryption context, encryption is based on the trapdoor function (i.e. encryption is performed in the “forward direction”), whereas decryption is based on the reversal the trapdoor function (i.e. decryption is performed in the “reverse direction”) which is only feasible when the trapdoor is known.
In a digital signature context, signature verification is performed in the forward direction, using the public key, and signature generation is performed in the reverse direction and can only feasibly be performed using the private key.
In a blockchain context, digital signatures based on public key cryptography are used as a basis for cryptographically signing transactions and verifying transaction signatures.
ECC is a form of public key cryptography which harnesses the mathematical properties of elliptical curves, and has various benefits over other cryptographic schemes such as DSA (Digital Secure Algorithm).
The “Elliptic Curve Digital Signature Algorithm” (ECDSA) refers to a class of digital signature schemes which use ECC as a basis for digital signature generation and verification. Certain principles of the ECDSA are outlined below.
In mathematical terminology, ECC exploits the algebraic structure of elliptic curves over finite fields of prime order. A finite field means a finite set of elements and a set of associated operations of multiplication, addition, subtraction and division which satisfy the normal rules of arithmetic (associativity, commutativity etc.) when applied to the elements in the set. That is to say, operations which need not be addition, multiplication etc. in the “normal” sense, but which do behave in essentially the same way.
In the context of ECC, the addition, subtraction and multiplication operations are, respectively, elliptic curve point addition, denoted “+” herein, elliptic curve point subtraction, denoted “−” herein, and elliptic curve scalar multiplication, denoted “·” herein. The addition and subtraction operations are each applied to two points on an elliptic curve and return a third point on the elliptic curve; however, the multiplication operation is applied to a scalar and a single point on an elliptic curve, and returns a second point on the elliptic curve. Division is, by contrast, defined on scalars.
For the purposes of illustration, 2,
2 being the set of all real-valued two-dimensional coordinates and (x, y)∈
2 denoting an element of
2. The elliptical curve ε is the set of points which satisfy the following equation:
Hence, in
The point at infinity ∞ is more formally referred to as an “identity element” (note both the parallel with and the deviation from normal arithmetic: in normal arithmetic, the sum of any number a with its additive inverse −a is 0, with 0 being the identity element for normal arithmetic). Another property of the identity element, ∞ which mirrors normal arithmetic, is that A+∞=A for any point A on ε including ∞ itself (analogous to the statement a+0=0 for any real number a)
That is, as ν elliptic curve point additions of A with itself.
Note: elliptic curve scalar multiplication is also referred to in the art as elliptic curve point multiplication. Those two terms have the same meaning in the present disclosure.
.
where p is a prime number (the prime modulus) and mod denotes the modulo operation. The set of points which satisfy the above equation is finite, and all but one of those points are represented in
The subscript “n”, by contrast, is referred to herein as the order of the group formed by the elliptic curve points under the point addition defined above (as shorthand, this may be called the order of the elliptic curve εn)—see below.
In other words, n is the order of the group, and p is the order of the field. There will be n elliptic curve points in total. Each point on the elliptic curve is represented by two numbers/coordinates (x, y), where x and y are all in the range −(p−1), . . . 0, . . . , (p−1).
It can be seen that εn in
The “line” lA,B intersecting two points A and B on εn becomes a finite set of points, represented by smaller black circles, satisfying analogous geometric requirements, and the definition of elliptic curve scalar multiplication still holds. Analogous with
The elliptic curve addition A+B=−C of any two points on εn can be defined algebraically by the following equations:
For the purposes of the above, the definition of the multiplicate inverse ν−1 of an integer ν is modified as:
That is, the multiplicate inverse of the integer ν is the modular inverse of ν mod p.
The case of B=−A is special, and resolved by the introduction of the identity element ∞—as noted, in that case A+B=A+ (−A)=∞. The case of B=∞ is also a special case, resolved as noted above as A+∞=A.
The definition of elliptic curve scalar multiplication adopts this definition of elliptic curve addition and otherwise remains the same.
In other contexts, the definition of the multiplicative inverse ν−1 of a scalar ν with respect is:
It will be clear in context whether a multiplicative inverse is defined with respect to mod n or mod p.
In practice, to identify whether a number should be treated as mod n or mod p, the following checks may be applied:
Note that, there are occasions where both checks give positive answer, in which case that the number has to be mod p and mod n.
Elliptic curve arithmetic provides unique capabilities in obscuring a secret value and forms the basis of many contemporary cryptographic systems. In particular, reversing scalar multiplication of elliptic curve points over finite fields is an intractable problem (it is computationally infeasible to perform).
A private key V takes the form of an integer, and the corresponding public key P is a point P on the elliptic curve εn derived from a “generator point” G, which is also a point on the elliptic curve εn, as:
where ‘·’ denotes elliptic curve scalar multiplication on the elliptic curve εn defined by a, b and n (the elliptic curve parameters).
For a sufficiently large V, actually performing V elliptic curve additions to derive P is hard, i.e. computationally infeasible. However, if V is known, then P can be computed much more efficiently by exploiting the algebraic properties of the elliptic curve operations. An example of an efficient algorithm that can be used to compute P is the “double and add” algorithm—crucially, this can only be implemented if V is known.
Conversely, if V is not known, then there is no computationally feasible way of deriving V (i.e. reversing the scalar multiplication) even if both G and P are known (this is the so-called “discrete-logarithm problem”). An attacker could attempt to “brute force” P by starting from G and repeatedly performing elliptic curve point additions until he gets to P; at that point, he would know V to be the number of elliptic curve point additions he had to perform; but that turns out to be computationally infeasible. Hence, V satisfies the requirements of a trapdoor in the above sense.
In ECC, the public key P, generator key G and elliptic curve εn are public and assumed to be known, whereas the private key V is secret.
In a blockchain system, a user or other entity will typically hold a private key V that is used to prove their identity and the corresponding public key P would be calculated by:
The private key V can be used sign a piece of data m (“the message”) using the ECDSA.
Further details of the ECDSA may for example be found in the following, which is incorporated herein by reference in its entirety: “RFC 6979—Deterministic Usage of the Digital Signature Algorithm (DSA) and Elliptic Curve Digital Signature Algorithm (ECDSA)”, Tools.ietf.org, 2019.
The signature generation is based on the same elliptic curve εn and generator point G used to derived the public key P, hence the elliptic curve parameters a, b and n and the generator point G are shows as inputs to the signature generator 600.
An ephemeral key generator 602 of the signature generator 600 generates an “ephemeral” key k∈[1, n−1], i.e. in the range from 1 to n−1 inclusive.
An r-part generator 604 calculates a corresponding public ephemeral key from k as follows:
and then take the x-coordinate (with [ ]x denoting the process of taking the x coordinate of an elliptic curve point) of the calculated point:
r=[R]
xmod n
which is the r-part of the signature.
As s-part generator 606 calculates an s-part of signature(s) using the modular inverse k−1 of k mod n (i.e. such that k−1k≡1 (mod n)—see above) and a hash of the message m, denoted H(m) (truncated if necessary) as follows:
In the present example, the message m comprises data to be included in a transaction 608 (one or more transaction outputs in the present example). This may be referred to as the process of signing the message m, and the message m may be referred to as a signed part of the transaction.
The message m and the signature (r, s), in turn, form part of the transaction 608. In the present example, the signature (r, s) in included in an input of the transaction 608 as part of an unlocking script.
Whilst the signature requires the private key Vas input, that is, one requires knowledge of it in order to generate a valid signature, only the signature pair (r, s), the message m, and the public key P are needed to validate the signature (r, s). In order to verify the signature, the signature verifier 620 hashes the signed part of the transaction m (applying the same hash function H as used to generate the signature (r, s)). The verification process is then performed using the following calculation:
The signature is valid (i.e. the signature verification will succeed) if and only if [R′]x mod n=r, otherwise it is invalid (i.e. the signature verification fails). In the present example, r denotes the r-part of the signature included in the transaction 608.
The public key P used in the signature verification process could for example be specified in the locking script of a preceding transaction. The signature verification is performed, in that case, using the public key specified in the locking script of the preceding transaction, and the signed part m and the signature (r, s) of the (later) transaction 608—and will fail unless the signature (r, s) has been generated based the private key V corresponding to the public key P specified in the preceding transaction and the signed part m of the later transaction 608. Hence, only the person who holds the private key V can claim the output of the preceding transaction (typically by including their own public key in the output of the later transaction 608), and the signed part m of the later transaction 608 cannot be altered without invalidating the signature (r, s).
An “r-puzzle” is a type of transaction script that allows the spending party to sign the input UTXO using any valid public-private keypair. The key pair can be application-specific or randomly generated for one-off use.
In such a knowledge proof, the signer derives the r-part of the signature (r, s) using a secret number k.
An example locking script that uses an r-puzzle can have the following form:
It will be appreciated that any opcode that performs hashing would be used instead of OP_HASH160, for example, OP_RIPEMD160, OP_SHA1, OP_SHA256, or OP_Hash256.
For brevity, the locking script can be represented as:
The above script drops the first 3 bytes of the signature, then extracts the r part of the signature, then drops the rest of the signature.
The unlocking script has the following form:
The signer, who provides the unlocking script, should know r such that kG=(x, y) and r=x mod n, i.e., the signer knows k which when multiplied by G produces a point whose x-coordinates is r when taken modulus n.
<Hash (r)> is provided in the locking script and therefore is public. The secret is k. The script execution is in the following table.
The signer can use any private-public key pair to sign the message. This is because neither the private key nor the corresponding public key is used in creating the locking script. Therefore, the party providing the locking script does not need to know who will provide the unlocking script at the time of generating the transaction.
A pay-to-R-Puzzle can also be represented as P2RP HASH(r).
Merkle Trees are hierarchical data structures that enable secure verification of collections of data. In a Merkle tree, each node in the tree has been given an index pair (i, j) and is represented as N(i, j). The indices i, j are simply numerical labels that are related to a specific position in the tree.
An important feature of the Merkle tree is that the construction of each of its nodes is governed by the following equations
where and H is a cryptographic hash function.
A binary Merkle tree constructed according to these equations is shown in
For example, the node N(0,3) is constructed from the four data packets D0, . . . , D3 as
The tree depth M is defined as the lowest level of nodes in the tree, and the depth m of a node is the level at which the node exists. For example, mroot=0 and mleaf=M, where M=3 in
For Merkle trees in Bitcoin and some other blockchains, the hash function is double SHA256, which is to apply the standard hash function SHA-256 twice: H(x)=SHA256 (SHA256 (x)).
The primary function of a Merkle tree is to verify that some data packet Di is a member of a list or set of N data packets ∈{D0, . . . , DN-1}. The mechanism for verification is known as a Merkle proof and involves obtaining a set of hashes known as the Merkle path for a given data packet Di and Merkle root R. The Merkle proof for a data packet is simply the minimum list of hashes required to reconstruct the root R by way of repeated hashing and concatenation, often referred to as the ‘authentication proof’.
A proof of existence could be performed trivially if all packets D0, . . . , DN-1 and their order are known to the prover. This does however require a much larger storage overhead than the Merkle proof, as well as requiring that the entire data set is available to the prover. The comparison between using a Merkle proof and using the entire list is shown in the table below, where we have used a binary Merkle tree and assumed that the number of data blocks N is exactly equal to an integer power 2.
The following table shows the relationship between the number of leaf nodes in a Merkle tree and the number of hashes required for a Merkle proof (or Merkle proof).
In this simplified scenario—where the number of data packets is equal to the number of leaf nodes—we find that the number of hash values required to compute a Merkle proof scales logarithmically. It is clearly far more efficient and practical to compute a Merkle proof involving log2 N hashes than to store N data hashes and compute the explicit proof.
If, given a Merkle root R, we wish to prove that the data block D0 belongs to the ordered list £{D0, . . . , DN-1} represented by R we can perform a Merkle proof as follows
Γ={N(1,1),N(2,3),N(4,7)}.
N(0,0)=H(D0).
N(0,1)=H(N(0,0)|N(1,1)).
N(0,3)=H(N(0,1)|N(2,3)).
N(0,7)=H(N(0,3)∥N(4,7)),
R′=N(0,7).
This is an efficient mechanism for providing a proof of existence for some data as part of the data set represented by a Merkle tree and its root. For example, if the data D0 corresponded to a blockchain transaction and the root R is publicly available as part of a block header then we can quickly prove that the transaction was included in that block.
The process of authenticating the existence of D0 as part of our example Merkle tree is shown in
When constructing a Merkle proof of a single leaf, the minimal information required is
To explain how the index of the leaf works, consider the Merkle tree in
Assume that a Merkle tree has N=2M leaves. Given an index i at layer 0, let i0=i, b0=i0 mod 2, p0=i0+(−1)b
p0 is the index of the pair leaf node of the leaf node with index i0. These are referred to as pairs since they are concatenated and hashed to calculate their parent hash node in the Merkle Tree (see above). The node with index p0 is also referred to as “the provided hash” or “the required data” since it must be provided when calculating the Merkle root of the i0 leaf node.
Thus, we can define at layer m, we have
Then the index of the provided hash is
The above equations assume that the index starts at 0.
Signature trees are an efficient alternative for a threshold signature requirement, where a threshold number of parties are required to sign a transaction. In this method, a Merkle tree is created, where each of its leaves can represent one spending condition. The locking script includes the root of the Merkle tree and opcodes to verify the Merkle proof. The unlocking script includes the Merkle proof and one of the spending conditions (i.e. one of the leaf nodes or information from which the leaf node can be derived).
Referring to the Merkle tree of
1-of-N Multisig
In a 1-of-N multisignature equivalent Merkle tree, the locking script generation steps are:
Where the string (OP_SWAP OP_IF OP_SWAP OP_ENDIF OP_CAT OP_SHA256)*3 means that the opcodes between brackets are repeated [log 2 number of leaves]=3 number of times.
The unlocking script consists of: signature, public key and the Merkle proof.
The Merkle proof consists of <sibling_node><0 if sibling is to the left and 1 if it is to the right>
For example for User 5, with Public key Ps being data item D4 of the Merkle tree of
The script execution is as follows:
Herein, [Merkle Root Derivation] is used as an abbreviation for: (OP_SWAP OP_IF OP_SWAP OP_ENDIF OP_CAT OP_SHA256) repeated t times, where t is the length of the Merkel proof and given by t=┌log2 Number of leaves┐.
Each time, the string causes two sibling tree nodes to be concatenated and the hash of the concatenation to be calculated, the hash being the parent node of the two sibling tree nodes.
To illustrate, the execution of this part of the string is as follows:
The above method can be used to allow a threshold of M signers out of N. In that case, each leaf corresponds to one of the M-of-N combinations (i.e. N choose M).
For example, if there are five users {U1, U2, U3, U4, U5} and there is a requirement for 4 of the 5 users to sign, i.e. 4-of-5 multisig, a tree is constructed with leaves Xi given by:
X
1
→U
1
,U
2
,U
3
,U
4
X
2
→U
1
,U
2
,U
3
,U
5
X
3
→U
1
,U
2
,U
4
,U
5
X
4
→U
1
,U
3
,U
4
,U
5
X
5
→U
2
,U
3
,U
4
,U
5
Since there are an odd number of leaves, a “dummy” leaf X6 is introduced so that there is an even number of leaves. The dummy leaf may have a value of 1, some other pre-defined constant, or the value of its sibling leaf (i.e. setting X6:=X5).
The locking script would include the root of the tree. The unlocking script would need to provide the Merkle proof that proves the link to the root.
In-script elliptic curve point addition checking (IECPAC) provides a method for checking that P1+P2=P3 in the script, where Pi=(xi, yi) is an elliptic curve (EC) point. It establishes a function:
<Point Add P1,P2,λ,P3>
that takes P1, P2, λ, P3 as its inputs and returns TRUE if λ is the gradient of the line between P1 and P2, and if P3=P1+P2. These two verifications are done in script.
The function comprises two steps
This method can be extended to an arbitrary number of points. For example, to verify the addition of three EC-points P=P1+P2+P3, the operations are:
<Point Add Multi P1,P2,P3,Δ′,Δ,P>:=<Point Add P1,P2,Δ′,P′><Point Add P′,P3,Δ,P>
Where P′=P1+P2, the value Δ′ is the gradient between P1 and P2, and the value λ is the gradient between P′ and P3.
P1, P2, P3 and Δ can be provide in the unlocking script and the check preformed in script.
According to the embodiments of the present disclosure, the concepts of R-puzzles and signature trees may be advantageously combined to provide a flexible and secure way of locking an output of a blockchain transaction. A first user (e.g. Alice 103a) constructs a first transaction that includes a locking script that locks an output of the first transaction. The locking script is configured to require an unlocking script of a second transaction, generated by a second user (e.g. Bob 103b) or group of users, to comprise a Merkle proof portion, a signature portion, and a public key. In order to successfully unlock the locking script, the signature portion must include an r-portion, which when hashed produces a hash value that forms part of a Merkle tree. The locking script performs a Merkle proof using the hash value and the Merkle proof, and verifies that it matches a node (or portion) of the Merkle tree that is included in the locking script (test 1). For instance, the node may be the Merkle root of the Merkle tree. In addition, the signature portion is validated using the public key (test 2). Only if each test is met will the locking script and unlocking script successfully execute.
Each of a set of users knows a secret value k. A Merkle tree is defined with the nodes being based on the r or R values (or “r-portions”) derived from the secret values of the users. As set out above:
r=[R]
x mod n
R=kG
where the ECDSA signature takes the form (r, s).
The term “r-portion” is used herein to refer to both R and r, with r being the r-part of the ECDSA signature.
Note that whilst reference herein is made to ECDSA signatures, it should be appreciated that ECDSA is just one type of digital signature scheme, and in general any equivalent scheme may be used.
1-of-N R-Values
In this case, each user Ui has a secret ki. Each leaf node of the Merkle tree is the hash of the r-part derived from the secret value ki of one of the users.
The following steps are implemented:
Since ri cannot be derived from Xi, sharing Xi with the other users does not affect security.
The Merkle root is also referred to as a “Merkle tree portion”. Similarly, a leaf node or intermediate node of the Merkle tree may also be referred to as a Merkle tree portion.
The locking script may take the following form:
Where t=┌log2(Number of leaves)┐.
The unlocking script may be in the form:
That is, the unlocking script comprise a signature of the form (ri, si), where ri is the r-portion of the locking script, a public key PK of the signer, and a Merkle proof portion.
The unlocking script 202A is provided in the second transaction Tx2, generated by a second user 103b. In the example of
When executed together with the unlocking script 202A, the unlocking script 203A causes the following steps to be implemented:
The unlocking script is valid for any <Sig><PK> provided as long as the following conditions are all true:
Again, each user Ui has a secret ki. However, in this case, each leaf node of the Merkle tree is the hash of a combination of the r-parts derived from the secret value ki of a group of users who together can unlock the locking script 202B.
In 2-of-4 system, each leaf Xi corresponds to one of 6 possible 4-choose-2 combinations
The corresponding Merkle tree has 6 leaves. That is, there is no X7 or X8. In order to balance the tree, Y4, which has no leaf nodes, is set to 1. Y4 is referred to as a “dummy” node. If instead there were an odd number of leaf nodes, the dummy node would be a dummy leaf node, also equal to 1.
While the rUi values above are summed, it will be appreciated that these values may be combined in any other way, for example by concatenating or multiplying the r-part of each user in the group.
The Locking script may include 2 r-extractors, a single Merkle proof, and 2 OP_CHECKSIG:
The unlocking script comprises two <Sig><PK> and one Merkle proof:
That is, the unlocking script 202B comprises a signature-key pair for each of the users, each signature comprising an r-part, and the Merkle proof portion.
The script execution for X6, where U3 uses k3 to calculate r3 and provides <SigU3><PKU3>, and U4 similarly uses k4 and provides <SigU4><PKU4> is as follows:
Referring to
When the locking script 203B is executed together with the unlocking script 202B, the following steps are implemented:
The unlocking script is valid for any <Sig><PK> provided as long as the following conditions are all true:
The above method can be extended to any M-of-N, such that the locking script may be given by:
Where t is the length of the Merkle proof,
is the threshold, i.e. number of required signers, and OP_CHECKSIG*M means repeating OP_CHECKSIG M-times.
The use of the Merkle tree set out above in which the leaf nodes are the hash of a combination of the M r values is a more efficient solution for M-of-N signature requirements. This is because the locking and unlocking script are smaller than for the alternative solution to the M-of-N system in which there are M distinct paths for M signers, with each leaf corresponds to a single user, such that the Merkle tree would have N leaves.
In the case of M distinct paths, the unlocking script contains M Merkle proofs, where a single Merkle proof is of length ┌log2 N┐ nodes.
However, if each leaf node is set to correspond to one of the N-choose-M possibilities, the tree has
leaves, and the unlocking script contains a single Merkle proof of
nodes.
There will always be
for all positive integers N and M where M≤N.
Therefore, the use of the M-of-N tree set out above is less expensive and more efficient since the size, and therefore the cost, of the tree and scripts are reduced.
The method proposed above requires repeating OP_CHECKSIG M times. To remove this requirement, each group of users (corresponding to a leaf in the Merkle tree) have a single signing key. This reduces the number of OP_CHECKSIG opcodes to one, and reduces the size of the unlocking script. This reduces cost and improves efficiency as both the unlocking and locking scripts are shorter, and their execution has fewer steps.
The signature tree comprising r-puzzle leaf nodes can be modified such that EC point addition is used to generate the leaves. IECPAC is then used to check that an EC-point, RX
As in the previous section, a Merkle tree is formed, where each of the leaves corresponds to each of the possible N-choose-M combinations.
The use of the IEPCA method provides a technique for proving that the user knows both the leaf data (i.e. the summation R-value) and the R-values (or r-portions) from which the leaf data is derived (i.e. the R-values of the M users).
Again, each user Ui has a secret ki. Each leaf node of the Merkle tree is the hash of an x-coordinate derived from a sum of the M r-values derived from the secret value ki of a group of users who together can unlock the locking script 202C. That is, for M=2,
where RX
A 2-of-4 multi-sig system is provided as an example. Each leaf Xi of the Merkle tree corresponds to one of 6 possible 4-choose-2 combinations of users, as in the example of
In an individual signature scheme, each group member provides their own signature.
The following steps are implemented:
Each of the N-choose-M groups share their Xi with either the other users or a coordinator in order to construct the Merkle tree.
While the Merkle tree itself is similar to that shown in
The locking script 203C contains:
The second transaction is provided by users 1 and 3, corresponding to leaf node X3. The unlocking script 202C comprises:
The gradient <λR
The Merkle tree may be constructed using either R or r for each of the users. In the example of
When the locking script 203C is executed together with the unlocking script 202C, the following steps occur:
The value xX
In the general case of M>2, the locking script contains M OP_CHECKSIG opcodes, M Extract_r opcode strings, (M−1) IECPAC strings, and a single Merkle Root derivation opcodes string.
The unlocking script contains M signatures and public keys, (M−1) gradient values <λ1> . . . <λM-1> and M−EC points Ri, M−2 intermediate EC-points and the EC-point RX
In order to reduce the length of both the locking and unlocking scripts, and subsequently the execution time, the method of
In this case each of the N-choose-M groups choose a single R-puzzle to represent the group. The IECPAC is used to prove to an outside observer that the provided signature is generated from multiple users. This embodiment assumes a high trust level between members of each group, since any group member can provide the signature and modify the transaction.
The Merkle tree has
leaves
and constructed as follows:
In the example of
The locking script 203D contains:
The unlocking script 202D is provided by a single one of the users of the group, either U1 or U3, and contains:
Generally for any M-of-N group, the unlocking script is:
To be able to generate a signature (rX
When the locking script 203D is executed together with the unlocking scrip 202D, the following step are implemented:
In the more general M-of-N embodiment, step 1 above uses all of the (M−1) gradients and M R-values provided in the unlocking script, the IEPAC method is run M−1 times, and M−2 intermediate EC-points are included.
The value Xi may be calculated using rxi as extracted from the group signature as set out above, or it may be calculated from using the R-values provided in the unlocking script corresponding to each individual users of the group (RUi), i.e. X3=hash ([RU1+RU3]x).
Whether a single group signature or multiple individual signatures embodiment is used, the IECPAC complexity grows linear with M.
In the examples set out above, the Merkle root is provided in the locking script and any user may provide the unlocking script, provided they know k and the Merkle proof, where k=k1+k2+ . . . +kM.
To restrict the users who can provide unlocking scripts, the locking script creator may, instead, provide a node of the Merkle tree in the locking script. In this way, only users whose r-values are used to generate leaf nodes located below said node in the Merkle tree can be unlocking script creators.
In each example, user 3 corresponding to X3 provides the unlocking script, which comprises r3 in the signature, as described above.
The following locking script can be unlocked by any of {U1, . . . , U8}:
If instead the locking script comprises Z1, as is shown in
In this case, the locking script is as follows:
Such that any of users {U1, . . . , U4} can provide the unlocking script.
The creator of the transaction can further limit the possible script unlockers to two by inserting, for example, Y2 in the locking script as shown in
In this case, the locking script may take the following form:
Note that [Merkle Root Derivation] in the first locking script derives the root from a 3-points Merkle proof, whereas [Merkle Root Derivation] in the 2nd and 3rd locking scripts require a 2-points and 1-point Merkle proof respectively. The number of points refers to the number of node values provided in the Merkle proof portion of the unlocking script, which is used to calculate the candidate Merkle proof.
The locking scripts of any of the above embodiments may be modified to provide a node of the Merkle tree rather than the Merkle root, and thus define a subset of users, or a subset of groups of users, who can create the unlocking script.
Other variants or use cases of the disclosed techniques may become apparent to the person skilled in the art once given the disclosure herein. The scope of the disclosure is not limited by the described embodiments but only by the accompanying claims.
For instance, some embodiments above have been described in terms of a bitcoin network 106, bitcoin blockchain 150 and bitcoin nodes 104. However it will be appreciated that the bitcoin blockchain is one particular example of a blockchain 150 and the above description may apply generally to any blockchain. That is, the present invention is in by no way limited to the bitcoin blockchain. More generally, any reference above to bitcoin network 106, bitcoin blockchain 150 and bitcoin nodes 104 may be replaced with reference to a blockchain network 106, blockchain 150 and blockchain node 104 respectively. The blockchain, blockchain network and/or blockchain nodes may share some or all of the described properties of the bitcoin blockchain 150, bitcoin network 106 and bitcoin nodes 104 as described above.
In preferred embodiments of the invention, the blockchain network 106 is the bitcoin network and bitcoin nodes 104 perform at least all of the described functions of creating, publishing, propagating and storing blocks 151 of the blockchain 150. It is not excluded that there may be other network entities (or network elements) that only perform one or some but not all of these functions. That is, a network entity may perform the function of propagating and/or storing blocks without creating and publishing blocks (recall that these entities are not considered nodes of the preferred bitcoin network 106).
In other embodiments of the invention, the blockchain network 106 may not be the bitcoin network. In these embodiments, it is not excluded that a node may perform at least one or some but not all of the functions of creating, publishing, propagating and storing blocks 151 of the blockchain 150. For instance, on those other blockchain networks a “node” may be used to refer to a network entity that is configured to create and publish blocks 151 but not store and/or propagate those blocks 151 to other nodes.
Even more generally, any reference to the term “bitcoin node” 104 above may be replaced with the term “network entity” or “network element”, wherein such an entity/element is configured to perform some or all of the roles of creating, publishing, propagating and storing blocks. The functions of such a network entity/element may be implemented in hardware in the same way described above with reference to a blockchain node 104.
It will be appreciated that the above embodiments have been described by way of example only. More generally there may be provided a method, apparatus or program in accordance with any one or more of the following Statements.
Statement 1. According to a first aspect, there is provided a computer-implemented method of generating a blockchain transaction, wherein each party of a plurality of second parties has a respective secret number k from which a respective r-portion is derivable, wherein a Merkle tree is defined, at least one leaf node of the Merkle tree being based on at least one of the respective r-portions, wherein the method is performed by a first party and comprises:
Statement 2. An embodiment of the method of statement 1, wherein the r-portion is extracted from an elliptical curve digital signature algorithm (ECDSA) signature of the signature portion of the first unlocking script, the r-portion being an r-part of the ECDSA signature.
Statement 3. An embodiment of the method of statement 2, wherein each leaf node of the Merkle tree is a hash of a combination of multiple r-portions, wherein the first locking script is configured to extract multiple r-portions from the first unlocking script and combine the multiple r-portions, wherein the hash value is a hash of the combined multiple r-portions.
Statement 4. An embodiment of the method of statement 3, wherein the r-portions are combined using one of addition and concatenation.
Statement 5. An embodiment of the method of statement 1, wherein the at least one leaf node of the Merkle tree is a hash of a sum of multiple r-portions, wherein the first locking script is configured to:
Statement 6. An embodiment of the method of statement 5, wherein the hash value is a hash of one of: the calculated sum of the multiple r-portions and the extracted summation r-portion.
Statement 7. An embodiment of the method of statement 5 or statement 6, wherein the first locking script is further configured to:
Statement 8. An embodiment of the method of statement 7, wherein the number of gradient values is one less than the number of extracted multiple r-portions corresponding to one of the secret numbers k.
Statement 9. An embodiment of the method of any of statements 3 to 8, wherein the first locking script is configured to verify each of multiple signature portions of the first unlocking script, each of the multiple signature portions provided with a corresponding public key, the verification of each of the multiple signature portions being based on the provided corresponding public key.
Statement 10. An embodiment of the method of statement 9, wherein the corresponding public keys are provided in the first unlocking script.
Statement 11. An embodiment of the method of any of statement 5 to 8 when dependent on claim 1, wherein the summation r-portion is extracted from a signature portion of the unlocking script.
Statement 12. An embodiment of the method of statement 1 or statement 2, wherein the at least one leaf node of the Merkle tree is a hash of a combination of multiple r-portions, wherein the extracted r-portion is equal to the combination of multiple r-portions of the leaf node corresponding to the Merkle tree portion of the first locking script.
Statement 13. An embodiment of the method of any preceding statement, wherein the Merkle tree portion is a Merkle root of the Merkle tree.
Statement 14. An embodiment of the method of any of statements 1 to 10, wherein the Merkle tree portion is a node of the Merkle tree other than the Merkle root.
Statement 15. An embodiment of the method of statement 1 or any of statements 5 to 14 when dependent on statement 1, wherein the r-portion is an elliptical curve point.
Statement 16. According to a second aspect, there is provided a computer-implemented method of generating a blockchain transaction, wherein each party of a plurality of second parties has a secret number k from which an r-portion is derivable, wherein a Merkle tree is defined, at least one leaf node of the Merkle tree being based on at least one of the respective r-portions, wherein a first blockchain transaction comprises a first locking script comprising a target Merkle tree portion corresponding to the Merkle tree, the first locking script configured, when executed together with a first unlocking script of a second blockchain transaction, to extract an r-portion from the first unlocking script, generate a hash value derived from the extracted r-portion, calculate a candidate Merkle tree portion based on the generated hash and a Merkle proof portion of the first unlocking script, verify that the calculated candidate Merkle tree portion matches the target Merkle tree portion of the first locking script, and verify a signature portion of the first unlocking script based on a public key of the first unlocking script, wherein the method is performed by at least one of said second parties and comprises:
Statement 17. An embodiment of the method of statement 16, wherein the signature portion comprises an elliptical curve digital signature algorithm (ECDSA) signature of the signature portion of the first unlocking script, the r-portion being an r-part of the ECDSA signature.
Statement 18. An embodiment of the method of statement 17, wherein the r-part is a combination of multiple r-portions derived from the secret number k of multiple of the plurality of second parties.
Statement 19. An embodiment of the method of statement 16, wherein the first unlocking script further comprises the r-portion.
Statement 20. An embodiment of the method of any of statements 15 to 18, wherein the first unlocking script further comprises one or more gradient values, the first locking script being configured to extract the one or more gradient values from the first unlocking script, and verify, based on the extracted one or more gradient values and multiple extracted r-portions extracted from the unlocking script, that each of the extracted one or more gradient values is equal to the gradient between two of the multiple extracted r-portions.
Statement 21. An embodiment of the method of statement 20, wherein the first unlocking script comprises a number of gradient values one less than a number of r-portions extracted from the unlocking script and corresponding to one of the multiple secret numbers k.
Statement 22. An embodiment of the method of any preceding statement, wherein the r-portion is derived from a public key corresponding to the secret number.
Statement 23. According to a third aspect, there is provided computer equipment comprising:
Statement 24. According to a fourth aspect, there is provided a computer program embodied on computer-readable storage and configured so as, when run on one or more processors, to perform the method of any of claims 1 to 21.
| Number | Date | Country | Kind |
|---|---|---|---|
| 2113092.7 | Sep 2021 | GB | national |
This application is the U.S. National Stage of International Application No. PCT/EP2022/072787 filed on Aug. 15, 2022, which claims the benefit of United Kingdom Patent Application No. 2113092.7, filed on Sep. 14, 2021, the contents of which are incorporated herein by reference in their entireties.
| Filing Document | Filing Date | Country | Kind |
|---|---|---|---|
| PCT/EP2022/072787 | 8/15/2022 | WO |
