Abstract: A computer-implemented method is disclosed. The method includes providing a blockchain transaction comprising a public key combination verification function. The blockchain transaction is configured to be redeemable to permit access to, or transfer control of, a resource by providing to the blockchain transaction an input comprising: a plurality of public keys; a gradient value (?) related to two of the plurality of public keys; and a group public key derived from a combination of the public keys and the gradient value (?). The blockchain transaction is configured to apply the public key verification function to the input to verify, upon successful redemption of the transaction, that the group public key is derived from the combination of the plurality of public keys.

Abstract: A computer-implemented method for adapting connections between nodes of a layered network. The layered network comprises a plurality of nodes arranged in an ordered set of layers. The ordered set of layers comprises, in order, a core layer comprising a set of core nodes, a second layer comprising a set of second nodes, and one or more outer layers each comprising a respective set of outer nodes. Each core node is connected to at least one other core node. The method is performed by an adapting node, the adapting node being a node of the layered network that is connected to one or more second nodes and multiple core nodes. The method comprises, based on one or more network properties of the layered network, disabling a respective connection between the adapting node and at least one but not all of the multiple core nodes.

Abstract: A method of validating transactions for recordal in a blockchain comprises receiving one or more transactions at a node of a blockchain network. For each received transaction a protocol for validating the transaction is applied. The protocol is configured to allow a termination opcode to be included in an output script of the transaction. The termination opcode is configured to, upon being executed by the node, a) terminate execution of the output script, and b) not to invalidate the transaction based only on the inclusion of the termination opcode in the output script. The protocol is also configured to disallow any instance of the termination opcode from being included in an input script of the transaction, said disallowing comprising the node at least invalidating the transaction if any instance of the termination opcode is included in the input script.

Abstract: A computer-implemented method for connecting to a layered network. The layered network comprises a plurality of nodes arranged in an ordered set of layers. The ordered set of layers comprises, in order, a core layer comprising a set of core nodes, a second layer comprising a set of second nodes, and one or more outer layers each comprising a respective set of outer nodes. Each core node is connected to at least one other core node. The method is performed by a connecting node and comprises connecting to the network according to a connection protocol. The connection protocol requires that each node must connect to at least one node of a preceding layer, and each outer node must also connect to at least one core node.

Abstract: The disclosure provides improved methods and systems for processing, storing, sharing, retrieving, writing, and accessing data (content) on a blockchain. In particular, improved efficiency and also enhanced access control permissions are provided. An embodiment of the present disclosure comprises the step of processing at least one blockchain transaction (Tx) comprising: a protocol flag; a discretionary public key (DPK); and a discretionary transaction ID (DTxID). These are discretionary in the sense that they are not required as part of the underlying blockchain protocol but in accordance with the present disclosure. This combination of features enables portions of data to be identified, retrieved, and shared on a blockchain, and also to be linked/associated with one another when provided in a plurality of transactions. It enables a graph or tree-like structure to be constructed, which reflects the hierarchical relationships between portions of data, facilitating their processing, searching, and sharing.

Abstract: A computer implemented method for generating a multisignature transaction for a blockchain comprising: receiving a first output and a non-signature portion of a first input, the first input comprising an outpoint to a first spendable transaction output encoding a multisignature requirement; electronically signing the first output and the non-signature portion of the first input, thereby generating a signature for satisfying the multisignature requirement when included with: (i) an associated sighash single flag, and (ii) at least one other signature; receiving a second output and a non-signature portion of a second input, the second input comprising an outpoint to a second spendable transaction output encoding a signature requirement requiring at least one signature; and electronically signing the second output and the non-signature portion of the second input, thereby generating a signature for at least partially satisfying the signature requirement when included in a signature portion with an associated sig

Abstract: A method of validating transactions for recordal in a blockchain comprises receiving one or more transactions at a node of a blockchain network. For each received transaction a protocol for validating the transaction is applied. The protocol is configured to allow a termination opcode to be included in an output script of the transaction. The termination opcode is configured to, upon being executed by the node, a) terminate execution of the output script, and b) not to invalidate the transaction based only on the inclusion of the termination opcode in the output script. The protocol is also configured to disallow any instance of the termination opcode from being included in an input script of the transaction, said disallowing comprising the node at least invalidating the transaction if any instance of the termination opcode is included in the input script.

Abstract: Terpolymers with more than one constituent cationic monomer, further copolymerized with acrylamide, are conditioning additives for hair, skin and nail care products which, with regard to hair, improve wet and dry hair combability, especially detangling and reduced static flyaway, sheen, and fixative properties, especially curl retention, and with regard to skin and nails, improve softening or lubricating (lubricity), moisture retention and attraction (moisturizing), feel, and reduction of irritation. The dual cationic polymers may have a weight average molecular weight of from about 10 thousand to 10 million, and comprise (a) from at least 30 to as much as 75 weight percent of acrylamide (AM), (b) from at least 25 to as much as 80 weight percent of the cationic monomer dimethyldiallylammonium chloride (DMDAAC), and (c) from at least 5 to as much as 35 weight percent of a second cationic monomer of the formula: ##STR1## where Y is --O--A-- or --NH--A--, where A is ethyl or propyl; R.sup.1 is H or CH.sub.3 ; R.