‘Trustless’ has become a rallying cry for Bitcoin evangelists, focusing public attention on the fact that Bitcoin enables P2P transactions without the participation of a trusted third party acting as an intermediary.
Bitcoin makes it possible to conduct money transfers without intermediaries. Intermediaries who could otherwise gain control over funds in a transaction, censor transactions or act as points of failure. Does this give grounds to assert that Bitcoin and other cryptocurrencies are a form of money that does not require trust? Is it possible to completely eliminate trust from monetary relations, and is there even a need to? This article is devoted to the study of the role of trust in monetary transactions (including cryptocurrencies), and its main conclusion is that this role is hard to overestimate.
The article is divided into four sections. First, we analyze the role of trust in monetary relations, including the role of trust in Bitcoin’s functioning, and show that trust is necessary for any money. Next, we proceed to justify the need for credit and credit money, which cannot exist without trust. After that, we will look at the limitations of trustless solutions for cryptocurrency-based off-chain transactions. Finally, we will propose a method to overcome those limitations, essentially by expanding the range of means of exchange that are used in off-chain transactions.
The role of trust in the functioning of money
In the book “Money between violence and trust”, Michel Aglietta and Andre Orlean formulated three forms of trust associated with the functioning of money .
Methodical trust. Money differs from ordinary goods by the characteristic of direct exchange. Individuals are willing to exchange their goods for money, only because they believe that the monetary form can then be exchanged for any other product on the market. That is why the functions of money can be manifested through such relatively useless things as scraps of paper or records in a database: a person doesn’t care what money is made of if they are sure that they can exchange it for any product or service.
This belief arises mainly from the previous experience of the successful exchange of goods for money by many individuals. The high frequency of such exchanges strengthens the payment community’s confidence, united by the use of money, that this unwritten rule of exchange will be observed in the future.
Will Bitcoin be able to perform the money functions if individuals do not believe that they can always exchange it for any product or service, or at least for other money? No. Before the appearance of market makers who began to buy and sell bitcoins, thus strengthening the public’s faith in its relatively stable exchangeability with money, Bitcoin was merely a plaything for a narrow circle of cypherpunks. If for some reason this belief is lost, Bitcoin will only retain the modest role of a digital collectible item.
Hierarchical trust. Methodical confidence in itself is quite fragile. In pursuit of profit, the rules of money exchange on which this form of trust is based could be violated. For example, bad actors could produce fake money or suddenly refuse to accept money as a means of payment, demanding something else in return. So, an authority with the ability to enforce monetary rules must support the honest members of the payment community. Hierarchical trust is the belief of individuals in the ability of this authority to maintain established monetary rules, including possible punishments for violators.
As a rule, this function is performed by the state. For example, the state has the capacity to punish counterfeiters. The harm that counterfeiters bring to the economy is not limited to the losses of those who received fake money in exchange for their goods or in repayment of a loan. Counterfeiters undermine the methodical confidence of all members of the payment community: confidence in the rules of exchange as well as in counter-parties. And confidence in the payment community itself, without which normal economic relations are impossible.
The state has legitimate instruments for coercing individuals to use money with the status of legal tender. A debtor can be sure that he/she will be able to pay off his debt with this money regardless of the whims or tricks of a creditor. A seller of goods can be sure that he/she will be able to use the money received for the goods for his or her future purchases. Thus, methodical trust is strengthened by hierarchical trust.
One of the slogans of Bitcoin evangelists is the formula “code is law”. In their opinion, government coercion to execute monetary rules is not necessary if these rules are hard-coded into Bitcoin. However, experience shows that the code regularly turns out to be imperfect and this can be used by intruders for the “illegal” emission of a cryptocurrency . “What blockchain does is shift some of the trust in people and institutions to trust in technology.” 
If we ignore the possibility of abuse of all sorts of bugs, then there is, for example, a 51% attack that allows a bad actor to perform a double-spend of bitcoins. Such an attack can also be compared to counterfeiting of a fiat currency.
Bitcoin, unlike national currencies, does not have the status of a legal tender. Nobody is obligated to exchange goods or money for bitcoins, nor to accept it as a mean of paying off debts.
Will the methodical trust of Bitcoin be strengthened if the state punishes anyone who attempts to profit by breaking the rules of this payment community? Will methodical trust in the Bitcoin be strengthened after one or more states declare it legal tender? You do not need to be a professional economist or sociologist to answer these questions.
Ethical trust. The state strengthens methodical confidence in money, but there is a third, more fundamental form of trust that limits political power over money. This is the trust of individuals in the social system offered by the government. If individuals lose faith in the project of the society proposed by the authorities, or the authorities violate the values of this society, then individuals can begin to introduce new rules of exchange, despite government efforts to save their existing monetary system.
The importance of the ethical form of trust is particularly pronounced during periods of social upheaval, especially during wars. Each of the belligerents can issue their own money, and individuals can accept money from one party or another based not only on coercion, but also on the project of society they support. A less tragic example of the role of ethical confidence in money is the confrontation between supporters of gold and fiat money. If gold money is supported by free market advocates, then fiat money is supported by those who believe that government should interfere in the economy.
Bitcoin emerged as an alternative to the project of a society determined by state authorities. This alternative ideology, another project of the society that gave birth to Bitcoin, is called crypto-anarchism. In this system, there is (by definition) no central authority that can win or lose the trust of individuals. But this does not mean that Bitcoin is not vulnerable to potential crises of ethical trust.
Even in Bitcoin’s short history there are examples of a kind of “civil war”, during which the payment community was split on the basis of ideology. The material result of such splits are forks. The most prominent “civil war” was a dispute over the size of the block. The party of crypto-anarchists were in favor of limiting the block size to protect the network from centralization, and the business-oriented party supported the increase of the block size for their commercial reasons. Crypto-anarchists who have lost faith in the privacy of the Bitcoin transactions, due to the development of services like Chainalysis , have switched to Bitcoin forks with increased privacy, or to alternative cryptocurrencies that use ring signatures, or zero-knowledge proof.
There is another example of ethical trust’s role in the history of the Bitcoin . In periods of economic and social crises, when state power and the money offered by it has lost the trust of individuals, Bitcoin has come to the rescue — precisely due to its stateless nature.
The role of credit and credit money in the functioning of the economy
Bitcoin was conceived as digital cash, allowing exchange without intermediaries. Today, transactions in the Bitcoin network are de facto carried out through intermediaries: mining pools and third-party wallet services that store the private keys of their users. But this does not change the essence of Bitcoin as digital cash. You can still run a full node, store your keys by yourself and try to add your own bitcoin transactions to the blockchain.
Whatever the form of a given type of money — gold, paper, or digital — it cannot serve the economy alone. For several centuries, cash has been working in solid lockstep with credit money; or, as it is also called, non-cash money.
Credit money, as the name implies, is based on credit. Therefore, before we talk about credit money, it is necessary to say a few words about credit itself.
The need for credit arises from the fact that different manufacturers and different products have different terms of production and sale. Thus, economies always contain, on the one hand, individuals with temporarily free resources; and, on the other hand, individuals experiencing a temporary need for additional resources. The former provide the latter with their resources for temporary use, thereby maintaining the continuity of the capitalist reproduction process. Capitalism cannot function and develop normally without credit. Without credit, the modern economy is simply unthinkable.
The first medium of exchange that was born out of credit relations is a promissory note. A buyer of goods on credit wrote out a promissory note to a seller, which was an obligation of the buyer to pay a fixed amount of cash within a prescribed period. The seller could transfer this bill to a third party in exchange for goods or cash, and the spread of such practices led to the fact that promissory notes of exchange were used as a medium of exchange along with cash.
The advantage of promissory notes in comparison with cash, (the functions of which were performed by precious metals), was that promissory notes were much more convenient and cheaper to store and transport. The disadvantage of promissory notes was their limited term of validity, and the absence of a fixed range of face values (that would allow any given amount to be paid with this kind of means of exchange).
These drawbacks are overcome by a banknote: paper money, representing credit money issued by banks. The banknote was a debt instrument with no fixed validity term, (i.e. it did not have an expiration date), and could be redeemed in metal money at any time. In addition, banks issued banknotes with a fixed number of face values.
Banks issued loans with their banknotes against the security of commercial promissory notes, thereby “re-minting“ the bills of merchants into their own debt obligations. This made it possible to grow the circulation of credit money to a new level, since banknotes coped with the role of ‘means of exchange’ much better than commercial promissory notes.
Often, a classic banknote is a banknote fully backed by a precious metal; but in reality, the full security of banknotes was an exception. The issuance of banknotes in excess of collateral satisfied demand when the market was hungry for cash. The limited supply of gold is a natural barrier to an increase in the supply of means of exchange, for which the growing capitalist economy has shown increasing demand. Credit money allows the economy to go beyond this natural limit.
The need to issue banknotes in excess of collateral is particularly pronounced in periods of economic crises, when cash rapidly leaves circulation in preference of savings and thereby causes a shortage of means of payment.
An example of an attempt to ban the issue of banknotes without full security is the Robert Peel Banking Act of 1844. This act gave the Bank of England a monopoly on the issue of banknotes, but at the same time limited its issuing activity by requiring a 100% security of banknotes in gold . Three years later, with the onset of an economic crisis, the government exercised its right to suspend this act. During the economic crises of 1857 and 1866, it did the same. The Bank of England was able to issue banknotes in excess of the available collateral to cope with the shortage of means of payment without aggravating the crisis. Choosing between full provision of banknotes and the rescue of the economy, the British government wisely chose the latter option.
The refusal of the US Federal Reserve to reduce the deficit aggravated the economic crisis, contributing to the Great Depression. The period of the Great Depression is replete with examples of how private credit money without full security can be a salvation from economic crises and cash shortages. During this period, thousands of local communities in the United States, Canada and Western Europe created their own credit money systems, including examples without full cash support. As a rule, these local means of exchange functioned successfully until the government prohibited them, (seeing them as a threat to their monopoly on the issuance of money ).
Despite the fact that governments maintain a monopoly to this day, the modern economy is dominated by private credit money issued by commercial banks. As before, banks issue credit money, “re-minting” the debt obligations of their borrowers into their own debt obligations. The share of cash in the money supply of developed countries is less than 10%. Commercial banks issue credit money without full provision of national currency, but the government does not see this as a threat to its monopoly on issuance. On the contrary, governments support the movement towards a cashless economy .
Vouchers are also debt obligations, but unlike banknotes are redeemed in goods rather than money. These debt instruments can solve various problems. The sale of gift cards and other vouchers allows the seller to receive a free loan, which will be repaid with the seller’s goods in the future. Bonus points that can be exchanged for goods can increase customer loyalty. Writing off small change to the buyer’s account, which they can use for future purchases, allows the seller to continue to trade during any shortages of small-face-value cash. The possibility of transferring vouchers to a third party, as in the case of commercial promissory notes, allows them to be used as a means of exchange.
Despite the popularity of the trustless principle in the cryptocurrency ecosystem, there remains a place for credit money. Bitcoin’s primary trading pair (as of December 2018) was stablecoin Tether (USDT) . USDT is electronic money secured by US dollars. Like the classic banknote, the USDT collateral is not stored in a smart contract, but in a bank. The user of USDT trusts the issuer of this stablecoin and the bank in which the collateral is kept: he/she believes that, if necessary, they will be able to redeem their USDT for USD at face value.
Despite rumors about the absence of full security , the lack of transparency and the absence of any guarantees of repayment at face value, the USDT maintains its undisputed leadership in the stablecoin market. USDT and four similar currencies (TUSD, USDC, GUSD, PAX), which are also electronic money backed by US dollars, account for more than 90% of the total capitalization and exchange turnover of stablecoins.
The role of credit money in a cryptocurrency ecosystem is not limited to the “replication” of national currencies on a blockchains. In fact, they solve the problem of Bitcoin’s scalability without increasing the block size.
Payment channels: scalability within trustless limits
Banknotes successfully replaced gold as a medium of exchange because exchange using banknotes was faster and cheaper. Banknotes became the representatives of gold in the sphere of exchange. Gold sat, unmoving, in the bank’s vault; but at the same time it served the exchange of goods through a banknote as its representative.
If Bitcoin is digital gold, then ‘digital banknotes’ of some kind could act as ordinary banknotes did for ordinary gold. In order to make bitcoin-based transactions faster and cheaper, they must occur without progressing the blockchain. If transactions without actual cash movement are called cashless, then transactions without bitcoin movement on the Bitcoin blockchain are called off-chain transactions.
One can freeze bitcoins at a special address and, using these bitcoins as a collateral, issue a ‘digital banknote’ in a side-chain or other blockchain; but this, strictly speaking, is not an off-chain solution. Although the bitcoins with which such notes are secured are static, the notes themselves move in a side-chain or other blockchain.
But payment channels, (or rather, a network of bilateral payment channels), are a true off-chain solution to the problem of Bitcoin network scalability. The parties of a payment channel deposit bitcoins on the multi-signature address, after which they can transfer their deposit to the other side of the channel without moving bitcoins along the original Bitcoin blockchain. Nodes that are connected through payment channels to more than one node form a network of payment channels. By combining payment channels into a network, off-chain transactions can be carried out not only within one channel, but also along a chain of channels (hops).
a) balances of nodes before payment b) balances of nodes after payment
Figure 1. Multi-hop payment with one intermediary.
Multi-hop payments through a chain of payment channels allow for off-chain transactions between nodes that do not have a direct connection. Figure 1 shows the simplest payment scheme. Node A has a 0.1 BTC deposit in the channel with node B, and node B has deposited 0.05 BTC in the channel with node C. The presence of an intermediary (in the form of node B) allows node A to make an off-chain payment to node C in the amount of 0.05 BTS via the A-B-C chain, despite the fact that A and C do not have a direct payment channel with each other. Node B in this case is not a trusted party who can illegally seize the A’s bitcoins. It will get access to these bitcoins only after it makes payment to C. This property of multi-hop payments is called atomicity: the chain of payments is either fully implemented or not executed at all.
A related payment may occur through a chain in which there is more than one intermediary. Theoretically, one could build a network topology of bilateral payment channels in which it becomes possible to lay a chain of related payments between any two of its nodes.
The most well-known and most developed network of bilateral payment channels for Bitcoin is the Lightning Network . Despite significant progress in the development of the Lightning Network’s infrastructure , a considerable increase in the number of network nodes and payment channels, and in the amount of bitcoins deposited in these channels , the functionality of this network has a number of limitations. Some of these restrictions are related to the fact that the Lightning Network operates on the trustless principle, which requires full security of off-chain transactions. Let us consider the Lightning Network’s restrictions, and the current proposals for their mitigation, in more detail.
The funds throughput of the chains of payment channels. Let’s go back to the example of a multi-hop payment discussed above. If node A has deposited 0.1 BTC in the A-B channel, and node B has deposited only 0.05 BTC in the B-C channel, then node A will be able to pay node C only 0.05 BTC. The monetary throughput of a chain of payment channels is measured by the smallest deposit in this chain.
To overcome this limitation, the concept of atomic multi-path payments is being developed . Off-chain transaction can be divided into several chains. For example, if in addition to the A-B-C chain, which has a throughput capacity of 0.05 BTC, there is an A-D-C chain with a carrying capacity of 0.07 BTC, then payment from the node A to the node C of 0.1 BTC becomes possible. To do this, the payment is divided into two parts: one part of the payment passes through the A-B chain, the other through the A-D chain.
Freezing deposits of intermediaries. In the multi-hop payment A-B-C, participant B, who acts as an intermediary between A and C, will be able to manage bitcoins received from A only after C confirms receipt of payment from B. If C doesn’t confirm receipt for a while (for example, it could be offline), then B must wait until the waiting time for confirmation has expired. Only then will B be able to initiate the return of its bitcoins. The longer the chain through which the multi-hop payment is made, the more intermediaries in this chain freeze their bitcoins until the final recipient confirms receipt or the multi-hop payment is canceled.
How does this affect network performance? While the intermediary’s deposit is frozen, it cannot be used for other payments. The longer the deposit is frozen, the greater the costs of lost profits for the intermediary. The longer the average freeze of deposits, the greater the deposit needed to ensure the specified monetary throughput of its node per unit of time:
IМB = ID / ATF
where IMB is the intermediary’s monetary throughput per unit of time, D is the intermediary deposit (deposit), and ATF is the average freezing time of the intermediary’s deposit (average time of freezing). The higher the deposit required to maintain a given node capacity, the smaller the number of participants who can deposit such an amount; and, therefore, the higher the centralization of the network.
The possibility of prolonged freezing of the intermediary deposit is a potential vector of attack on the network. To create interruptions in the network, the attacker will seek to freeze the maximum amount of deposits in the maximum number of channels for the maximum period.
The authors of the Sprites  off-chain network project have proposed an approach to organizing multi-hop payments which shortens the maximum time for freezing deposits. This reduces the costs of lost profits for intermediaries, mitigates the tendency to centralize the network and the consequences of an attack in the mentioned scenario. However, this decision “… requires the flexibility of Ethereum-style smart contracts … and therefore cannot (we conjecture) be implemented in Bitcoin. “ 
Replenish and partial withdrawal of funds from payment channels. If a channel participant has exhausted its deposit and its current balance in the channel is zero, then it has two options for further use of the channel. First: wait for an incoming payment from another channel member, thanks to which its balance will be replenished. Second: close the channel and open a new one in which it will make a new deposit. Both of these options are not very convenient. Further, withdrawing bitcoins from the channel is only possible by closing it. Permanent closure and opening of channels, in addition to the additional costs of channel participants, leads to temporary loss of nodes from the network and makes it difficult to make related payments.
Splicing  will allow on-chain transactions to replenish payment channels, as well as enabling the withdrawal of bitcoins without the need to close and re-open a channel. It is assumed that the development and implementation of this solution will make possible both on-chain payments from off-chain balances on the Lightning Network, and on-chain payments on off-chain balances.
Payment channels re-balancing. There could be distortions in the balance of payment channels, which complicate their usage in the on-chain payment network.
Figure 2. The imbalance in the balance of payment channels.
Figure 2 illustrates an example of balance skewing, when payments from one participant to another move cyclically in one direction: A only pays B, B pays C only, and C pays A only. Such a payment pattern causes the payer’s balance to be exhausted in each channel, and the balance of the payee to accumulate.
In this situation, participants have two options to continue using payment channels. The first one is to use multi-hop payments instead of direct channels, which can be more expensive if the intermediary charges a commission for this. Second: to replenish the channel with the help of on-chain transactions, which also requires additional costs.
Strictly speaking, in the simplest case with one user, there is a third option that does not require additional costs. The user can move the balance from one channel to another by performing a circular off-chain payment to himself . “However, in a broader context of different users, different payment channels, different objectives and different payment amounts, it was not clear how a set of re-balancing transactions could be created.” 
The Revive protocol  alleviates the problem of balance skewing in payment channels in the aforementioned “broader context”. This protocol allows partial re-balancing of states in channels without the need for on-chain transactions. We say “partial” because Revive does not allow transferral of deposits, made at the opening of one channel, to another channel. Rather, it allows the clearing of off-chain balances of network participants whose payment channels form cycles. In other words, “this means that, when modeling the participants as nodes and the payment channels among them as edges, any such graph that contains no cycles is not re-balanceable.” . Re-balancing using Revive does not happen automatically: for this to happen, participants in the off-chain payment network need to be combined into special subnets and take an active part in each rebalancing round .
High entry threshold for intermediaries. The trend towards network centralization has already been mentioned above in the context of the maximum term for freezing intermediary deposits. A much more significant factor determining this tendency is the business model of intermediaries itself, which consists of charging fees for an off-chain transaction.
If each intermediary in a chain of multi-hop payments charges a fee, then a decrease in the number of intermediaries in the chain leads to a decrease in the commission that the user pays for sending an off-chain transaction. The more payment channels the intermediary has opened, the more users can make a multi-hop payment with the minimum number of intermediaries equal to one. A larger number of channels requires a larger amount of deposits. The greater the amount of deposits, the smaller the number of participants who can deposit such an amount and, consequently, the higher the centralization of the network.
The predominance of unidirectional payments that do not form cycles requires the intermediary to spend additional time and money on deposit management. If that entails tens or hundreds of thousands of small payments of buyers to large sellers, then the costs of deposit management can be substantial.
To reduce intermediary costs, the Liquidity Network project  proposed an alternative to two-way payment channels — an n-party non-custodial payment hub. This solution reduces the deposit requirement of an intermediary acting as a hub operator. The intermediary does not need to deposit cryptocurrency in the channel with each user of the hub, since Liquidity Network does not use two-way payment channels. In order to carry out a trustless off-chain transaction, an intermediary needs only to deposit a pledge in the smart contract equal to “the volume of transactions during the period of time allocated for transactions litigation.”  User balances are updated every 36 hours and another 36 hours is given to execute transactions. Thus, the intermediary must deposit funds to the amount of off-chain transactions that await final confirmation within 72 hours.
The number of users that can join the hub is unlimited. If the user wants to join the hub only for receiving off-chain payments, without making a deposit, then he/she does not need to send on-chain transaction for this. Instead he/she can do it off-chain. If the user wants to make a deposit, then he/she must send the deposit by an on-chain transaction to the hub’s smart contract address.
Liquidity Network operates on the basis of Ethereum and the direct implementation of this solution on the Bitcoin network is problematic. According to the authors of the project, “our scheme also requires an environment for the execution of smart contracts, similar to that provided by Ethereum”.  Bitcoins cannot be directly deposited in the Liquidity Network smart contract, but WBTC (Wrapped Bitcoin), for example, can be deposited .
Unlike the Liquidity Network payment hub, channel factories  can be implemented in the Bitcoin network, and their functioning does not require an intermediary operator to deposit collateral to ensure trustless transactions.
Channel factories are designed to reduce the costs of opening payment channels and moving deposits between channels. This solution, which acts as an intermediate layer between the blockchain and payment channels, will relieve users from having to send on-chain transactions whenever they want to open a new two-way payment channel. Instead, users will be able to open an unlimited number of payment channels within the group they have created in an off-chain mode. Creating a group requires each participant to make one on-chain transaction to deposit bitcoins at a multisig address. After creating a group, participants will be able to move their balance from a channel with one group member to a channel with another group member without on-chain transactions and in trustless mode.
Another difference between the channel factory and the Liquidity Network hub is that the number of factory participants cannot be increased, since it is set at the moment the multisig address is created.
It is not yet known when channel factories will be launched on the Bitcoin network. The concept of channel factories is at an initial stage of development and its implementation in the Bitcoin network will require a soft fork.
So, we looked at solutions that are designed to overcome the Lightning Network’s monetary throughput limitations in various ways, and to reduce the costs of using and maintaining an off-chain network. However, all these solutions have a general limitation, (which imposes a trustless principle on them). That limitation could be stated as: “a payment can only be made if the sender and all intermediaries have a sufficient value of deposits”.
Trustless + Trusted > Trustless
A multi-hop payment of a sum x in the chain of channels requires that the deposit of each intermediary in the chain be not less than x. Thus, the total minimum amount of deposits of intermediaries for the multi-hop payment is:
TMID = x * n
where TMID is the total minimum deposit of intermediaries required to make a related payment of x (total minimum intermediaries’ deposit), x is the amount of payment, n is the number of intermediaries participating in a multi-hop payment.
A reduction in the average number of intermediaries of multi-hop payments, while the total amount of intermediary deposits is constant, increases the cash flow of the network per time unit. Abstracting from direct payments, we can present the simplest formula for the monetary network throughput as follows:
NMB = TID / (ATF * n)
where NMB is the network’s monetary throughput, TID is the total intermediaries deposit, ATF is the average time of freezing of the middleman’s deposit, and n is the average number of intermediaries servicing the multi-hop payment.
NMB can be improved by minimizing the ATF, by reducing the number of intermediaries in the network to one, and by optimising the deposit management practices of intermediaries. But otherwise, NMB can only be improved through an increase in the TID. The limit of the TID growth is the intermediary’s own capital. There is only one method to go beyond the limitations imposed by the intermediary’s own capital, the method any entrepreneur would use — credit. Credit provides the possibility of continuous commercial activity in the face of recurring shortages of capital.
Let’s consider an example. Node A, (a client of the intermediary node B), wants to make an off-chain payment to its counterpart node C, (another client of node B), in the amount of 0.1 BTC. But B has only deposited 0.05 BTC in the B-C channel. If B does not currently have funds available in its other channels to replenish the B-C channel, it will not be able to process this payment. If node C agrees to accept 0.05 BTC from the intermediary node B as a trustless ‘banknote’, and another 0.05 BTC as a trusted one issued by B, then the payment can be made. In turn, C will be able to use the trusted banknote for payments through B or to repay B in case of a need for ‘cash’ bitcoins.
This is the simplest example. A chain of related payments can be longer and contain various combinations of trustless and trusted links. In particular, credit can be used not only by the intermediary, but also by the sender of a payment.
Expansion of the range of exchange tools used in the off-chain network of bilateral payment channels, through trusted tools, allows us to increase its cash flow capacity and reduce the costs of intermediaries for deposit management. Reducing costs, in turn, allows intermediaries to reduce fees for off-chain transaction.
The GEO Protocol  makes it possible to perform off-chain transactions not only with the help of trustless banknotes (IOUs), but also with the help of various means of exchange based on trust. In terms of securities, the means of exchange that the GEO Protocol supports can be classified into three types:
1. Trustless. Means of exchange, fully secured by cryptographic assets deposited in payment channels.
2. Trusted with full collateral. Means of exchange, fully secured by off-chain non-digital assets. For example, electronic money secured by national currencies or precious metals.
3. Trusted without full collateral. Such means of exchange include, primarily, promissory notes and vouchers denominated in various units of value.
Means of exchange, such as electronic money, promissory notes and vouchers, have a long history of successful operation around the world, and the rules governing their use have been developed and tested in detail. On this basis, it can be argued that these means of exchange deserve even greater methodological, hierarchical and ethical trust than cryptocurrency.
GEO nodes can use any of the listed media in any combination for transactions with each other. If they want to conduct trustless transactions with each other, then they can open a channel by depositing cryptocurrency. If they want to use means of exchange based on trust, then they can open a channel (trustline) without depositing crypto assets and denominate their debt obligations in any convenient accounting unit. At the same time, each node of the GEO network independently sets a credit limit for each of its counterparties, within which the node agrees to accept one or other medium of exchange.
Multi-hop payments in the GEO Protocol can occur in chains that contain any available trustless and trusted combination of channels.
Increasing the throughput on the GEO network is not only achieved through the principle of “trustless + trusted”, described above. Atomic multi-path payments  and automatic cycled clearing  have already been implemented in the GEO Protocol.
A fixed supply of bitcoins restricts the throughput of the Lightning Network as a superstructure over the Bitcoin network. Spreading the Lightning Network to other blockchains would overcome this limitation. However, the Lightning Network will only be able to support cross-chain off-chain payments between blockchains with a compatible hashing algorithm. At the GEO network there are no such restrictions on cross-chain transactions; and, therefore, there are virtually no restrictions on monetary throughput.
In addition, the Lightning Network has technical limitations on network scalability. The GEO Protocol does not have them.
Ultimately, the GEO Protocol allows one to create not only payment channels, but also generalized state channels. This extends the functionality of the GEO Protocol far beyond the simple movement of cryptocurrency between nodes of the off-chain network.
Money can not exist without trust and Bitcoin is no exception. Even traditional cash, which is used in P2P transactions, requires methodical, hierarchical and ethical trust for its normal functioning.
Credit and credit money do not arise by chance, but by necessity. Capitalism needs credit, and cash needs credit money. Economic history provides many examples of how credit money, including that without full collateral, helps economies to function, develop, and mitigate the effects of crises.
Banknotes made it possible to reduce the costs of, and speed up, transactions that were previously conducted with precious metals. Digital gold transactions can also be sped up and cheapened thanks to digital banknotes that circulate outside blockchains. Off-chain transactions in networks of bilateral payment channels allow Bitcoins to be stored without movement in the blockchain, but at the same time maintain economic turnover through digital banknotes.
The trustless principle limits the cash throughput of the off-chain network (NMB). This restriction can be formulated as follows: a payment can be made only under the condition that the sender and all intermediaries of the payment have a sufficient value of deposits. This limitation can be overcome by combining trustless transactions with trusted transactions.
The driving force behind the evolution of networks of bilateral payment channels from trustless, to trustless + trusted, is the increase in monetary throughput (even when the supply of cryptocurrency is limited).
The logic of this evolution is as follows: the NMB rises by increasing the number of nodes, the number of their payment channels and the amount of their deposits. When this type of growth reaches its limit, growth in NMB occurs due to the centralization of the network. After centralization of the network, further growth in NMB is possible due to an increase in the intermediary deposits. The limiting factor of this stage of growth is the stock of intermediary cryptocurrency. The intermediary may use various methods of deposit management to more effectively distribute their deposits through channels, but these solutions still have an upper limit — the value of intermediary deposits.
A further increase in the monetary throughput of a network of bilateral payment channels is possible only through credit, and therefore through transactions based on trust .
In the GEO network, nodes can use three classes of exchanges: trustless (fully secured with a crypto asset deposited in the payment channel), trusted with full collateral (fully secured off-chain non-digital assets), trusted without full collateral (promissory notes and vouchers).
The GEO Protocol can be used both by those who support the ideology of crypto-anarchism, and by those who adhere to a more traditional project of society. Moreover, the ability to use trustless and trust-based tools in the same off-chain network simplifies the interaction of the former with the latter.
The introduction of credit-based means of exchange makes the monetary throughput of an off-chain network unlimited only in theory. In practice, each network participant sets a trust limit for each counterparty with whom he or she enters into a credit relationship. Expansion of the range of media operating in a network of bilateral payment channels, through trusted tools, allows us to achieve results that are unattainable for off-chain networks limited by the trustless principle.
 M. Aglietta A. Orlean, La violence de la monnaie
 Zcash Discloses Vulnerability That Could Have Allowed ‘Infinite Counterfeit’ Cryptocurrency // http://fortune.com/2019/02/05/zcash-vulnerability-cryptocurrency/
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 Ibid, p. 4.
 Ibid, p. 5–6.
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 Khalil R., Gervais A. NOCUST — A Non-Custodial 2nd-Layer Financial Intermediary, c. 12 // https://liquidity.network/NOCUST_Liquidity_Network_Paper.pdf
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 Khalil R., Gervais A. NOCUST — A Non-Custodial 2nd-Layer Financial Intermediary, c. 4 // https://liquidity.network/NOCUST_Liquidity_Network_Paper.pdf
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 GEO Protocol // https://www.geoprotocol.io
 LINK TO THE ARTICLE ABOUT PAYMENT ALGO
 LINK TO THE ARTICLE ABOUT CYCLED CLEARING ALGO
 A further increase in network liquidity within the framework of the trustless approach is possible only by eliminating bilateral payment channels. We are talking about the elimination of the intermediary thanks to trustless solutions that allow the use of an off-chain deposit for direct payments to any member of a payment hub or multilateral payment channel, and ideally to any member of the network. However, in this case, the need for credit does not disappear. The liquidity of the n-party payment channels/hubs network still comes up against a limited cryptocurrency offer, and this restriction is overcome by credit.
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