Distributed storage of blocks in blockchains

A method comprises: receiving, by a second node, a new transaction to be recorded to a block in a blockchain (510), determining, by the second node, based on an identity of the block that the second node is one of a subset of block managers in the blockchain to store the block (520), adding the block to the blockchain responsive to the determination (530) and transmitting a block list, wherein the block list comprises information about the block (540).

RELATED APPLICATION

This application was originally filed as PCT Application No. PCT/CN2019/083452, filed on Apr. 19, 2019, of which is incorporated herein by reference in its entirety.

FIELD

The present invention relates to blockchains, and more specifically to distributed storage of blocks in blockchains.

BACKGROUND

Blockchains are exploited in a diverse range of applications already and the usage of blockchains is expected to even increase in the near future. A blockchain may be seen as a public ledger with a chain of blocks, each of which is made up of several committed transactions, and it may grow continuously in chronological order when new blocks are appended to it. The core advantages of blockchains are decentralization, transparency, traceability, and tamper-resistance. Thus, blockchains may be applied in many fields, such as financial services, medicine, Internet of Things, IoT, software engineering, e-government and public services.

Blockchains generally provide a completely auditable log that includes every single transaction ever done in a blockchain network, which is very useful in a number of use cases. However, due to this nature, scalability of blockchains is affected by their ever-growing size. Improved methods, apparatuses and computer programs are therefore needed for exploiting blockchains.

SUMMARY

According to some aspects, there is provided the subject-matter of the independent claims. Some embodiments are defined in the dependent claims.

According to a first aspect, there is provided a first method comprising receiving, by a second node, a new transaction to be recorded to a block in a blockchain, determining, by the second node, based on an identity of the block that the second node is one of a subset of block managers in the blockchain to store the block, adding the block to the blockchain responsive to the determination and transmitting a block list, wherein the block list comprises information about the block.

According to the first aspect, the first method may further comprise receiving, from the first node, a request for a copy of the block and transmitting the copy of the block responsive to the request.

According to a second aspect, there is provided a second method comprising receiving, by a first node, a block list comprising information about a block in a blockchain, determining, by the first node, based on an identity of the block that a second node is one of a subset of block managers in the blockchain storing the block and transmitting, to the second node, a request for a copy of the block.

According to the first or the second aspect, the information about the block may comprise an identity of the block.

According to the first or the second aspect, the information about the block may comprise a total number of nodes at a time when the block was created and/or a number of block managers storing the block.

According to the first or the second aspect, said determining that the second node is one of the subset of block managers may be further based on a total number of nodes at a time when the block was created.

According to the first or the second aspect, said determining that the second node is one of the subset of block managers may be further based on a number of block managers storing the block.

According to the first or the second aspect, said determining that the second node is one of the subset of block managers may be done using a mapping function. In some embodiments, the mapping function may be N_IDi=B_IDk(mod nc−nm)+h, 1≤h≤nm, wherein N_IDimay be an identity of an i-th node, B_IDkmay be the identity of the block, ncmay be a total number of nodes at a time when the block was created and nmmay be a number of block managers storing the block.

According to a third aspect of the present invention, there is provided an apparatus comprising at least one processing core, at least one memory including computer program code, the at least one memory and the computer program code being configured to, with the at least one processing core, cause the apparatus at least to perform the first method.

According to a fourth aspect of the present invention, there is provided an apparatus comprising at least one processing core, at least one memory including computer program code, the at least one memory and the computer program code being configured to, with the at least one processing core, cause the apparatus at least to perform the second method.

According to a fifth aspect of the present invention, there is provided an apparatus comprising means for performing the first method. According to a sixth aspect of the present invention, there is provided an apparatus comprising means for performing the second method.

According to a seventh aspect of the present invention, there is provided non-transitory computer readable medium having stored thereon a set of computer readable instructions that, when executed by at least one processor, cause an apparatus to at least perform the first method.

According to an eighth aspect of the present invention, there is provided non-transitory computer readable medium having stored thereon a set of computer readable instructions that, when executed by at least one processor, cause an apparatus to at least perform the second method.

According to a ninth aspect of the present invention, there is provided a computer program configured to perform the first method. According to a tenth aspect of the present invention, there is provided a computer program configured to perform the second method.

EMBODIMENTS

Usage of blockchains may be improved by the procedures described herein. More specifically, embodiments of the present invention provide procedures for distributed storage of blocks in blockchains. For instance, a subset of block managers, but not all, may store one block. The subset of block managers for the block in question may be identified, or located, based on an identity of the block. Moreover, a total number of nodes at a time when the block was created and/or a number of block managers storing the block may be exploited for identifying, or locating, the subset of block managers as well. In some embodiments of the present invention, a mapping function may be used for this purpose.

Blockchains are increasingly applied for recording transactions without relying to a centralized entity. Changes in resource ownership in a blockchain network take the form of blockchain transactions secured by strong cryptography. Information provided in a blockchain transaction may be stored as a new block in the blockchain in response to validation of the respective blockchain transaction. Blockchain state information shared by the nodes may store all transactions and history carried out in the blockchain network. Application of blockchain technology and a ledger make it possible to track the unique history of transactions by the individual nodes in the network.

The following steps may be performed for creating a new block in a blockchain network. First, a new transaction may be generated and broadcasted to all nodes in the blockchain network. Each node may then collect the new transaction into a block. A consensus mechanism may be exploited for selecting one node and when a node is selected for creating a block using the consensus mechanism, the selected node may create the block and broadcast the created block to all nodes in the blockchain network. Upon receiving the created block, all nodes may accept the created block only if all transactions in the created block are valid and not spent already. All nodes may express, or acknowledge, their acceptance of the created block by working on creating a next block in the blockchain. A hash of the created block that was accepted may be used for the next block.

In some embodiments of the present invention, a blockchain network may comprise lightweight nodes, such as wallets, and full nodes. Lightweight nodes may keep a header of a block only, but not the entire block. A header of a block may comprise for example a hash of a previous block, a time stamp, a nonce and Merkle root. In some embodiments of the present invention, full nodes may be referred to as block managers. That is to say, full nodes may be capable of storing a complete and up-to-date replica of blocks, i.e., a canonical blockchain may be stored locally by full nodes.

For lightweight nodes it may be difficult to verify transactions contained in blocks. For instance, in some blockchain applications, a blockchain may be used as a database for storing data generated by the blockchain network. However, it may be difficult, if not impossible, for lightweight nodes to maintain a distributed database. Full nodes on the other hand may be able to autonomously verify the transactions contained in blocks without external reference. Nevertheless, a hard disk capacity of a full node, i.e., a block manager, may be limited so once a size of all blocks of the blockchain is greater than the hard disk capacity, the full node may not be able to store new blocks. That is to say, in such a case the full node in question may not be able to add new blocks to the blockchain. Development of the blockchain may be therefore limited due to limited hard disk capacity of full nodes.

Some embodiments of the present invention therefore enable storing of blocks in a distributed manner to enable distribution of the load. Thus, embodiments of the present invention provide improvements for various applications to be used in different fields, such as financial services, medicine, Internet of Things, IoT, software engineering, e-government and public services, etc.

Moreover, some embodiments of the present invention provide a mechanism for allocating new blocks amongst the full nodes that store blocks, i.e., block managers. In addition, some embodiments of the present invention provide a mechanism for lightweight nodes, for determining which full nodes have stored a particular block of interest, so that a lightweight node can identify full nodes storing the block and request at least one of the identified full nodes to provide a copy of the block. Therefore, only a subset of block managers may store one block but it is not necessary that all block managers store every block.

So according to some embodiments of the present invention, a distributed block storage scheme may be employed in a blockchain. The distributed block storage scheme may refer to a scheme, wherein a subset of block managers stores a particular block, but not all blocks. The distributed block storage scheme may further employ a block list. The block list may be distributed to all nodes, i.e., each node in the blockchain may have the latest updated version of the block list. The block list may be useful for holding up-to-date information about the entire blockchain and each node may maintain a single-chain structure of the blockchain using the block list. Thus, the block list may be used for looking up blocks of the blockchain rapidly.

FIG.1illustrates an exemplary architecture of a blockchain in accordance with at least some embodiments of the present invention. InFIG.1, a first node of the blockchain is denoted by110, a second node of the blockchain is denoted by120, a third node of the blockchain is denoted by130and a fourth node of the blockchain is denoted by140. For instance, first node110and third node130may be lightweight nodes which do not, or cannot, store blocks of the blockchain while second node120and fourth node140may be full nodes, i.e., block managers. Thus, second node120and fourth node140may store blocks of the blockchain. However, it should be noted that in some embodiments of the present invention a fully distributed blockchain network may be achieved if all nodes110,120,130and140are full nodes, i.e., block managers.

In general, nodes110,120,130and140may comprise corporate, authority, and/or user devices, such as a server, a desktop/tablet/laptop computer, smartphone, set-top box or any other suitable electronic device. In addition, a system may comprise an administrator or management node, a hub, relay or other kind of intermediate device for connecting nodes110,120,130and140to further networks or services, such as another distributed or centralized computing system or a cloud service. Nodes110,120,130and140may be mutually addressable in a suitable way, for example, they may communicate using an Internet Protocol, IP. Messages released into an IP network with a recipient address may be routed by the network to the recipient node identified by the recipient address. Other protocols may be used for communication as well so IP is not the only suitable communication protocol that may be used. For example, other peer-to-peer networking models are also suitable.

Second node120and fourth node140may comprise, or be connected to, block manager150. Block manager150may further comprise, or be connected to, local block database155. That is to say, for example second node120may comprise, or be connected to, local database155. Moreover, local database155may store a block when second node120is identified as a block manager for the block in question. In addition, block list is denoted by160. Block list160may be distributed to all nodes of the blockchain, e.g., after adding a new block to the blockchain. Thus, each node of the blockchain has up-to-date information about the entire blockchain. That is to say, all nodes110,120,130and140may have block list160, or at least have access to block list160.

FIG.2illustrates an example of a block list in accordance with at least some embodiments of the present invention. Block list260ofFIG.2may correspond to block list160of FIGURE. Block list260may comprise information, i.e., parameters, related to blocks but not the entire blocks. For instance, block list260may comprise first block header2201. First block header2201may be associated with first block2101, i.e., first block header2201may be a header of first block2101. Block list260may also comprise more block headers, e.g., k-th block header220K. Block header220Kmay be associated with k-th block210K, i.e., k-th block header220Kmay be a header of k-th block210K.

Block headers may comprise information about blocks of a blockchain. Thus, block list260may comprise information about the blocks of the blockchain as well. For example, k-th block header220Kmay comprise an identity of block K. The identity of block K may be numbered by creation order, i.e., an identity of new block that is added after block K may be K+1. In addition, or alternatively, k-th block header220Kmay comprise a total number of nodes at a time when block K was created and/or a number of block managers, such as second node120and fourth node140, storing block K.

When a new block is added to the blockchain, the information about the new block may be added to a header of the new block and block list260may be updated by adding the header of the new block to block list260. Updated block list may then be distributed to all nodes in the blockchain. Block list260may be hence helpful for maintaining a single-channel structure of the blockchain, thereby enabling decentralization, transparency, traceability, and tamper-resistance.

In some cases all nodes of the blockchain may not be able to store all blocks of the blockchain, e.g., due to limited hard disk capacity. Hence, in some embodiments of the present invention, a block manager, such as second node120, may become a block manager of certain blocks, but not all.

In some embodiments of the present invention, a subset of block managers for block K may be identified, or located, using a unique identity of block K. That is to say, the subset of block managers for block K may be identified, or located, based on the identity of block K. In addition, the subset of block managers for block K may be identified, or located, based on a total number of nodes at a time when block K is created and/or a number of block managers storing block K. The number of block managers storing block K may correspond to a size of the subset of block managers storing block K.

For instance, an exemplary mapping function may be exploited. In some embodiments of the present invention, the exemplary mapping function may be defined as follows. Taking block K as an example, the subset of block managers storing block K may be identified, or located, using the exemplary mapping function
N_IDi=B_IDk(modnc−nm)+h,1≤h≤nm(1)
, wherein N_IDiis an identity of an i-th node participating in the blockchain, B_IDkis the identity of block K, ncis a total number of nodes at a time when block K was created and nmis a number of block managers storing block K. The identity of the i-th node participating in the blockchain, N_IDi, may denote a running number of the i-th node among all nodes participating in the blockchain.

The exemplary mapping function provides various benefits. For example, regardless of how big B_IDk, the exemplary mapping function ensures that N_IDi≤nc−nm, i.e., the identity of located, or identified, block managers always falls within a range of nodes.

Moreover, the exemplary mapping function makes it possible for all block managers of block K to have different identities. As B_IDk(mod nc−nm) is a unique value, there are different N_IDivalues for the number of block managers storing block, nm, and the subset of block managers can be located, or identified, by adjusting h from 1 to nm. For instance, if we assume that B_IDk=1650, nc=60 and nm=8, then the exemplary mapping function gives a subset N_IDi={N39, N40, N41, N42, N43, N44, N45, N46} as a result for locations, or identities, of the block managers of block K.

In addition, the exemplary mapping function makes it possible for all nodes of the blockchain to quickly locate, or identify, the subset of block managers for a certain block, such as block K, thereby saving resources as the nodes do not need to store the identities of these block managers.

FIG.3illustrates an example apparatus capable of supporting at least some embodiments of the present invention. Illustrated is device300, which may comprise, for example, first node110or second node120ofFIG.1. Comprised in device300is processor310, which may comprise, for example, a single- or multi-core processor wherein a single-core processor comprises one processing core and a multi-core processor comprises more than one processing core. Processor310may comprise, in general, a control device. Processor310may comprise more than one processor. A processing core may comprise, for example, a Cortex-A8 processing core manufactured by ARM Holdings or a Steamroller processing core produced by Advanced Micro Devices Corporation. Processor310may comprise at least one Qualcomm Snapdragon and/or Intel Atom processor. Processor310may comprise at least one application-specific integrated circuit, ASIC. Processor310may comprise at least one field-programmable gate array, FPGA. Processor310may be means for performing method steps in device300. Processor310may be configured, at least in part by computer instructions, to perform actions.

Device300may comprise memory320. Memory320may comprise random-access memory and/or permanent memory. Memory320may comprise at least one RAM chip. Memory320may comprise solid-state, magnetic, optical and/or holographic memory, for example. Memory320may be at least in part accessible to processor310. Memory320may be at least in part comprised in processor310. Memory320may be means for storing information. Memory320may comprise computer instructions that processor310is configured to execute. When computer instructions configured to cause processor310to perform certain actions are stored in memory320, and device300overall is configured to run under the direction of processor310using computer instructions from memory320, processor310and/or its at least one processing core may be considered to be configured to perform said certain actions. Memory320may be at least in part comprised in processor310. Memory320may be at least in part external to device300but accessible to device300.

Device300may comprise a transmitter330. Device300may comprise a receiver340. Transmitter330and receiver340may be configured to transmit and receive, respectively, information in accordance with at least one cellular or non-cellular standard. Transmitter330may comprise more than one transmitter. Receiver340may comprise more than one receiver. Transmitter330and/or receiver340may be configured to operate in accordance with global system for mobile communication, GSM, wideband code division multiple access, WCDMA, 5G, long term evolution, LTE, IS-95, wireless local area network, WLAN, Ethernet and/or worldwide interoperability for microwave access, WiMAX, standards, for example.

Device300may comprise user interface, UI,360. UI360may comprise at least one of a display, a keyboard, a touchscreen, a vibrator arranged to signal to a user by causing device300to vibrate, a speaker and a microphone. A user may be able to operate device300via UI360, for example to accept incoming telephone calls, to originate telephone calls or video calls, to browse the Internet, to manage digital files stored in memory320or on a cloud accessible via transmitter330and receiver340, or via NFC transceiver350, and/or to play games.

Device300may comprise or be arranged to accept a user identity module370. User identity module370may comprise, for example, a subscriber identity module, SIM, card installable in device300. A user identity module370may comprise information identifying a subscription of a user of device300. A user identity module370may comprise cryptographic information usable to verify the identity of a user of device300and/or to facilitate encryption of communicated information and billing of the user of device300for communication effected via device300.

Processor310may be furnished with a transmitter arranged to output information from processor310, via electrical leads internal to device300, to other devices comprised in device300. Such a transmitter may comprise a serial bus transmitter arranged to, for example, output information via at least one electrical lead to memory320for storage therein. Alternatively to a serial bus, the transmitter may comprise a parallel bus transmitter. Likewise processor310may comprise a receiver arranged to receive information in processor310, via electrical leads internal to device300, from other devices comprised in device300. Such a receiver may comprise a serial bus receiver arranged to, for example, receive information via at least one electrical lead from receiver340for processing in processor310. Alternatively to a serial bus, the receiver may comprise a parallel bus receiver.

Device300may comprise further devices not illustrated inFIG.3. For example, where device300comprises a smartphone, it may comprise at least one digital camera. Some devices300may comprise a back-facing camera and a front-facing camera, wherein the back-facing camera may be intended for digital photography and the front-facing camera for video telephony. Device300may comprise a fingerprint sensor arranged to authenticate, at least in part, a user of device300. In some embodiments, device300lacks at least one device described above. For example, some devices300may lack a NFC transceiver350and/or user identity module370.

Processor310, memory320, transmitter330, receiver340, NFC transceiver350, UI360and/or user identity module370may be interconnected by electrical leads internal to device300in a multitude of different ways. For example, each of the aforementioned devices may be separately connected to a master bus internal to device300, to allow for the devices to exchange information. However, as the skilled person will appreciate, this is only one example and depending on the embodiment various ways of interconnecting at least two of the aforementioned devices may be selected without departing from the scope of the present invention.

FIG.4illustrates an exemplary process in accordance with at least some embodiments of the present invention. On the vertical axes are disposed, from the left to the right, first node110, second node120and third node130ofFIG.1. First node110, second node120and third node130may be participants in a blockchain. Time advances from the top toward the bottom.

At the start of the exemplary process, at step410, third node130may determine that there is a new transaction to be recorded to the blockchain. At step420, third node130may transmit information about the new transaction to the blockchain and consequently second node120may receive the information about the new transaction. Naturally any other node participating in the blockchain, such as first node110or second node120, may perform steps410and420as well.

Responsive to receiving the information about the new transaction to be recorded to a block in the blockchain, second node120may at step430determine, based on an identity of the block, that second node120is one of a subset of block managers in the blockchain to store the block. The subset of block managers may comprise all block managers of the blockchain that store the block. So for example fourth node140ofFIG.1(not shown inFIG.4) may also perform step430, if it is one of the subset of block managers that may store the block.

In some embodiments of the present invention, said determination at step430may also comprise determining that second node120is one of the subset of block managers based on a total number of nodes at a time when the block was created. Alternatively, or in addition, said determining that the second node is one of the subset of block managers may be further based on a number of block managers storing the block. For instance, said determination at step430may comprise determining that the second node is one of the subset of block managers using a mapping function. In some embodiments of the present invention, the mapping function in accordance with Equation (1) may be used.

In some embodiments of the present invention, the total number of nodes at a time when block K was created, denoted by nc, may be recorded in the blockchain, or in a header of a block, and hence second node120may retrieve nc. For instance, when a new block is created, nccan be found from the previous block, or from a header of the previous block, that has just been created. As there is no central authority, a new node joining the network may broadcast a message to the entire network to inform other nodes that the new node has joined, or is about to join, the network. Thus, in some embodiments of the present invention, the total number of nodes at a time when block K was created, nc, may be determined from the last nc, i.e., from the previous block, or a header of the previous block, that has just been created, and the number of nodes that have joined after the creation of the previous block.

Moreover, in some embodiments of the present invention, the number of block managers storing block K, nm, may be predefined, e.g., based on the total number of nodes, nc. For instance, the number of block managers storing block K, nm, may be set as as 10% of the total number of nodes nc.

If second node120determines at step430that it belongs to the subset of block managers in the blockchain that should store the block in question and second node120is selected for creating the block using a consensus mechanism, second node120may create a new block comprising the new transaction.

At step440, second node120may store, i.e., add, the new block to the blockchain in response to the determination. The updated blockchain comprising the new block may be stored to local block database155for example. At step440, second node120may also add information about the new block to a block list, such as block list160/260. In some embodiments, block list160/260may be stored in local block database155as well.

In some embodiments, the information about the new block may comprise an identity of the new block. Moreover, in some embodiments, the information about the new block may comprise a total number of nodes at a time when the new block was created and/or a number of block managers storing the new block, i.e., a size of the subset of the block managers.

At step450, second node120may transmit the block list, the block list comprising information about the new block, to the blockchain. Thus, at step450first node110and third node130may receive the block list that has been updated by second node120.

At some point after step450, first node110may determine that it would like to have a copy of the new block that was created by second node120. First node110may be a lightweight node and hence it may not have the entire blockchain, i.e., it may not have information about the new block but it may have the block list, provided by second node120. Alternatively, first node110may be a full node, i.e., a block manager, but it may not belong to the subset of block managers that have stored the block.

Thus, first node110may determine at step460, based on an identity of the new block, that second node120is one of the subset of block managers in the blockchain storing the new block. The subset of block managers may comprise all block managers of the blockchain that store the block. That is to say, said determination at step460may be based on the block list received at step450.

In some embodiments of the present invention, said determination at step460may also comprise determining that second node120is one of the subset of block managers based on a total number of nodes at a time when the new block was created. Alternatively, or in addition, said determining that the second node is one of the subset of block managers may be further based on a number of block managers storing the new block, i.e., a size of the subset of the block managers.

For instance, said determination at step460may comprise determining that second node120is one of the subset of block managers using a mapping function. In some embodiments of the present invention, the mapping function in accordance with Equation (1) may be used.

Upon determining that second node120is one of the subset of block managers storing the new block, first node110may at step470transmit a request for a copy of the new block. Consequently, second node120may receive the request at step470and responsive to the request, transmit the copy of the new block to first node110at step480. First node110may thus receive the copy of the new block at step480, thereby ending the exemplary process ofFIG.4.

FIG.5illustrates a flow graph of a first method in accordance with at least some embodiments. The phases of the illustrated first method may be performed by second node120, or by a control device configured to control the functioning thereof, possibly when installed therein.

The first method may comprise, at step510, receiving, by second node120, a new transaction to be recorded to a block in a blockchain. The first method may also comprise, at step520, determining, by second node120, based on an identity of the block that second node120is one of a subset of block managers in the blockchain to store the block. Moreover, the first method may comprise, at step530, adding the block to the blockchain responsive to the determination. Finally, the first method may comprise, at step540, transmitting a block list, wherein the block list comprises information about the block.

FIG.6illustrates a flow graph of a second method in accordance with at least some embodiments. The phases of the illustrated second method may be performed by first node110, or by a control device configured to control the functioning thereof, possibly when installed therein.

The second method may comprise, at step610, receiving, by first node110, a block list, wherein the block list comprises information about a block in a blockchain. The second method may also comprise, at step620, determining, by first node110, based on an identity of the block that second node120is one of a subset of block managers in the blockchain storing the block. Finally, the second method may comprise, at step630, transmitting, to second node120, a request for a copy of the block.

Reference throughout this specification to one example embodiment or an example embodiment means that a particular feature, structure, or characteristic described in connection with the example embodiment is included in at least one example embodiment of the present invention. Thus, appearances of the phrases “in one example embodiment” or “in an example embodiment” in various places throughout this specification are not necessarily all referring to the same example embodiment. Where reference is made to a numerical value using a term such as, for example, about or substantially, the exact numerical value is also disclosed.

In an exemplary example embodiment, an apparatus, such as, for example, first node110or second node120, may comprise means for carrying out the example embodiments described above and any combination thereof.

In an exemplary example embodiment, a computer program may be configured to cause a method in accordance with the example embodiments described above and any combination thereof. In an exemplary example embodiment, a computer program product, embodied on a non-transitory computer readable medium, may be configured to control a processor to perform a process comprising the example embodiments described above and any combination thereof.

In an exemplary example embodiment, an apparatus, such as, for example, first node110or second node120, may comprise at least one processor, and at least one memory including computer program code, wherein the at least one memory and the computer program code are configured to, with the at least one processor, cause the apparatus at least to perform the example embodiments described above and any combination thereof.

INDUSTRIAL APPLICABILITY

At least some example embodiments of the present invention find industrial application in blockchain applications, wherein it is desirable to enable distributed storage of blocks.

ACRONYMS LIST

ASIC Application-specific integrated circuitFPGA Field-programmable gate arrayGSM Global system for mobile communicationIC Integrated CircuitIP Internet ProtocolIoT Internet of ThingsLTE Long term evolutionM2M Machine to machineNFC Near-field communicationUI User interfaceWCDMA Wideband code division multiple accessWLAN Wireless local area network

REFERENCE SIGNS LIST

110, 120, 130, 140Nodes150Block manager155Local block database160, 260Block list115, 125, 135Interfaces210Block220Header of a block300-370Structure of the apparatus of FIG. 3410-480Phases of the exemplary process in FIG. 4510-540Steps of the first method in FIG. 5610-630Steps of the second method in FIG. 6