Bitcoin and double spending problem – What is it

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What is Double Spending & How Does Bitcoin Handle It?

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Bitcoin is gaining rapid popularity and adoption across the globe. It is redefining the way we use the money by being the world’s first fully functional digital currency.

You might be surprised to know that even before Bitcoin, there were attempts to create a sustainable digital monetary system. But all those attempts failed because an obvious problem with digital money is that transactions can be copied and spent twice.

Let me simply the concept…

Bitcoin has been able to survive and thrive because it solves the “double spending” problem.

What Does Double Spending Mean?

Double spending means spending the same money twice.

Let’s consider this example:

You go to Starbucks and order a cappuccino worth $10. You pay in cash. Now that $10 in cash is in the cash vault of Starbucks. By all means, you simply cannot spend the same $10 somewhere else to make another purchase.

Unless you steal it….

As you paid with your $10 bill, the service provider at Starbucks instantly confirmed that you have paid, and you received your coffee in exchange for the money.

But Bitcoin is digital money, not physical cash. Hence, Bitcoin transactions have a possibility of being copied and rebroadcasted. This opens up the possibility that the same BTC could be spent twice by its owner.

In our Starbucks example, you paid cash, so the payment was confirmed and verified instantly by another human. But with digital currency like BTC, if this verification mechanism is missing, it can lead to double spending.

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Anyone can just copy that digital money and pay somewhere else.

And here is where the unique invention lies…

Bitcoin, although being a digital currency, solves the problem of being copied and getting spent twice.

How Bitcoin Handles The Double Spending Problem

Bitcoin manages the double spending problem by implementing a confirmation mechanism and maintaining a universal ledger (called “blockchain”), similar to the traditional cash monetary system.

Bitcoin’s blockchain maintains a chronologically-ordered, time-stamped transaction ledger from the very start of its operation in 2009.

Every 10 mins, a block (i.e. a group of transactions) is added to the ledger. And all the nodes on the Bitcoin network keep a copy of this global ledger (the blockchain).

Let’s see how the Bitcoin network prevents double spending:

Let’s suppose you have 1 BTC which you try to spend twice.

You made the 1 BTC transaction to a merchant. Now, you again sign and send the same 1 BTC on another Bitcoin address to try and trick the merchant.

Both transactions go into the unconfirmed pool of transactions. But only your first transaction got confirmations and was verified by miners in the next block. Your second transaction could not get enough confirmations because the miners judged it as invalid, so it was pulled from the network.

But wait… what if both the transactions are taken simultaneously by the miners?

When miners pull the transactions simultaneously from the pool, then whichever transaction gets the maximum number of confirmations from the network will be included in the blockchain, and the other one will be discarded.

You might say that this is unfair for the merchant, as the transaction might fail in getting confirmations. Yeah, this can happen.

That’s why it is recommended for merchants to wait for a minimum of 6 confirmations.

Here, “6 confirmations” simply means that after a transaction was added to the blockchain, 6 more blocks containing several other transactions were added after it.

“Confirmations” are nothing but more blocks containing more transactions being added to the blockchain. Each transaction and blocks are mathematically related to the previous one.

All these confirmations and transactions are time-stamped on the blockchain, making them irreversible and impossible to tamper with.

So if a merchant receives his/her minimum number of confirmations, he/she can be positive it was not a double spend by the sender.

Why can the merchant be assured?

Because to be able to double spend that coin, the sender has to go back and reverse all transactions in the 6 blocks that have been added after their transaction, which is computationally impossible.

How Double-Spend Attacks Can Happen

  • Attack 51%

If somehow an attacker captures 51% of the hash power of the network, double spending can happen.

“Hash power” means the computational power which verifies transactions and blocks. If an attacker has this control, he/she can reverse any transaction and make a private blockchain which everyone will consider as real.

But so far, no such attack has happened because controlling 51% of the network is highly cost intensive. It depends on the present difficulty of mining, the hardware price, and the electricity cost, all of which is infeasible to acquire.

When an attacker sends the same coin in rapid succession to two different addresses, the obvious outcome is that only one of them will get included.

Now, if you as a merchant don’t wait for confirmations of payment, then in a case like this, there’s a 50% chance you got the double spent coin (and you won’t receive that money).

Your customer can trick you if he/she sends the same coins again to his/her address.

Once the customer does both transactions, both transactions go to an unconfirmed pool of transactions. Whichever transaction gets verified first and gets 6 confirmations will be accepted, and the other will be discarded.

As a merchant, you might get the 6 confirmations first, but if the attacker gets the confirmations first, then you won’t receive your funds. That’s why it is said to wait for a minimum of 6 confirmations.

So far, in the 8-year history of Bitcoin, no such attack has been successful. The Bitcoin mechanism of maintaining a universal transaction ledger based on confirmations has yet to be tricked.

I hope this knowledge about Bitcoin will help you use it safely.

Let me know your thoughts and feedback in the comments section below.

And don’t forget to share this post with your friends on Facebook and Twitter!

An award-winning blogger with a track record of 10+ years. An international speaker and author who loves blockchain and crypto world.

After discovering about decentralized finance and with his background of Information technology, he made his mission to help others learn and get started with it via CoinSutra.

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20 thoughts on “What is Double Spending & How Does Bitcoin Handle It?”

That’s very informative. Thanks

this is why Bitcoin Cash is NOT plausible! 51% attacks are much more possible with BCASH having a much smaller number of nodes, and mining hash power being distributed evenly to only 4 or 5 large mining pools.. only 4 or 5 of them need to collude with one another to perform 51% attacks on the network – BCASH is not a true representation of decentralization – support Bitcoin Core – run a full node, and hodl Litecoin while you’re at it too

Thanks for the article! I’m struggling to understand how it could be possible to double spend even without the blockchain. Why aren’t your funds simply subtracted from your wallet when you make a payment? It’s not like you could double spend via your Jaxx, Copay or mycelium wallet, so how would you go about copying your own transaction and sending the same bitcoin somewhere else again? Why is the double spend problem even a problem?

In November 2020 it was discovered that the GHash.io mining pool appeared to be engaging in repeated payment fraud against BetCoin Dice, a gambling site [https://bitcointalk.org/index.php?topic=327767.0] . Dice sites use one transaction per bet and don’t wait for confirmations. GHash.io claimed they had investigated and found a rogue employee who had been doing the double spending, who was fired. However no evidence supporting this was provided and the incident left a permanent cloud hanging over the pool. Regardless, it didn’t seem to hurt their market share much: most miners probably never heard about the incident at all.

Then why use bitcoin at all? Banks are doing the same. You are trusting a third party, and now it totally depends on jaxx and copay to handle your funds in whatever way you want. There is no decentralization

Well I believe I was tricked by an attaker like you say. I tried the doubler.block-chains.me to get double my investment of 0.01 Bitcoin that supposedly I would receive in 24 hrs. And never happened. Does anyone else got ripped off by this method in blockchain too. Please help me understand I’m pissed.

No, that is not a 51% attack. You just sent your bitcoin to a scammer, and they took it obviously.

How does one judge the second spend isn’t valid after he first is spent.

You will not be asked for judging that Blockahin will do that. Whichever transactions get added first to the longest blockchain version is the valid one.

Hello. On 6th od December I bought bitcoins in Bit2me.com company. I already bought before and I had to problem. But the thing is that on 6th I sent them two bank transaction with the same value 10.000 euro each. I bought them with 4 minutes difference. But they sent the same bitcoins amount at the same time to my wallet. Now I cannot do anything because they say they sent the bitcoins to my wallet but was busted as double spent. They told me to wait but I am very worry because it is a lot of money. �� I dont know what to do. Do you think I will loose all my money or it is just wait and I will receive it all?
Any help will be appreciated.

when both the addresses are yours you should not worry.

Thank you for your info!! I am in a deep trouble and confusion and need you advise urgently please! Someone gave me an wallet address/barcodes to send 10 Btc to. He told me to send 0.01 btc first and he got it and a min after I realized that the remaining btc was transferred to an “unknown wallet address” simultaneously?! How could this happened?? I need your professional advise please, thank you.

That’s impossible unless you showed that person your seed or private keys!!

Hi, I have just found this :

there is a way to save a broked transmission. A restart of the wallet and some patience typically fixes the issue.

How to stop/reverse a Bitcoin transaction without confirmations:

Run bitcoind and with -zapwallettxes.
This makes the wallet “forget” any unconfirmed transactions, thus enabling you to reuse their inputs.
Create a new transaction to make your payment and add an appropriate fee this time.
The unconfirmed transaction is still valid and may still be floating around in the network, so be sure to spend its inputs to yourself, or the transaction may be confirmed at a later time and you’ll end up double-paying.
The simplest way is to send all your bitcoins to yourself. Don’t forget an appropriate fee. ��

Hope you can get your coins back

This article appears to assume that any double-spending attempt is going to involve trying to send the same Bitcoin to TWO (2) different addresses. But what do you know about alleged double-spending attempts that were sent to the SAME address more than once?

In my case here just very recently, I attempted to “sweep” a paper wallet containing 0.5 BTC on it and send the full amount to my Coinbase account. I used the Mycelium Android wallet for this sweep attempt and have actually used the Mycelium wallet for this same purpose dozens of times in the past without any problems. However, this time was different; and when I did a secondary scanning, as well as third scanning, of the private key for the paper wallet into the Mycelium sweep utility, it did not register as having been swept, even though initially on both attempts a message appeared stating that the transaction had been “SENT” successfully.

In fact, though, it could not have been sent successfully for three (3) very good reasons:

1) Blockchain.INFO was still showing the funds fully available in the paper wallet.
2) Neither of the two initial send attempts registered through my Coinbase account.
3) And as noted above, the Mycelium wallet app was still showing the funds remaining.

So with good reason to believe that the funds were never successfully swept from the paper wallet, I attempted yet a THIRD sweep of the wallet — and on the third attempt, it worked. Blockchain.INFO was showing the funds being sent and currently registering as an “UNCONFIRMED” transaction, of which the unconfirmed part did not surprise me.

So about an hour later, I go back into my Coinbase account to see the status of the transaction. In this case, the Coinbase company uses the LIVE.BLOCKCYPHER.COM site as its choice of blockchain explorer, as opposed to using Blockchain.INFO or some of these other explorer sites. And when I clicked on the link into the explorer webpage, it gives the following ominous message:

“WARNING: This transaction has been double-spent by d46fe0c4cb4bbb0c2691d44628064…, be extremely careful when accepting this transaction!”

One point to clarify here is that in making the three attempts that I did at transferring the funds from the paper wallet to my Coinbase account, I definitely used ONLY ONE Bitcoin receiving address from my Coinbase account. I did not attempt to send the Bitcoin to more than one address on any of the three attempts made.

So NOW where is the double-spend logic coming into play here, at least regarding this article? Although I may be wrong, it occurs to me that the author of this article is not particular savvy when it comes to this blockchain technology stuff, and so maybe there is not going to be a knowledgeable answer to this question coming from this forum.

In any event, I am wondering if anybody knows the answer to the question about how, when or if these alleged double-spend attempts EVER end up correcting themselves on the blockchain…

Is there any good reason for me to believe that this apparent technical problem created by the Mycelium Android wallet itself is ever going to be corrected over a period of time? Or do these sorts of alleged double-spend transactions end up getting FOREVER stuck and lost in the “blockchain limbo” never to be rectified?

For one thing, my Coinbase account is continuing to show the transaction in a “PENDING” status. My guess is that beyond a certain period of time, whether it be a day, a week, a month or whatever the case, that my Coinbase account will end up effectively “timing out” on the transaction. If that ends up happening, then the other thing I know is that the only way for this situation to correct itself is for the 0.5 BTC to end up being deposited back into the original paper wallet address, where I could then presumably make another attempt at the sweep transaction.

I mean, we are talking about 0.5 Bitcoin in this case — a full HALF of a Bitcoin! If it takes a week, a month, or even a year for it to correct itself, I guess that frankly I have the time to wait and I’m not in a particularly big hurry. Obviously this would be a big chunk of money to find out that I ended up LOSING due to some unbelievably RETARDED scheme built into this so-called “blockchain technology” stuff that is unable to identify “after the fact” the legitimacy of whether any given amount of Bitcoin was ever truly spent successfully from any given Bitcoin address.

It will be interesting to see if and when I ever get access back to my half-a-Bitcoin. If like a month or two goes by or something like that and the funds are still locked into “blockchain limbo,” then I’m going to begin to largely assume the money is effectively irretrievable and irrevocably lost due to what would definitely amount to a VERY SERIOUS FLAW with this whole Bitcoin thing.

So to otherwise touch on the basis of your article here generally, which is to suggest that Bitcoin is the first ever successful digital currency ONLY because it has succeeded in solving the “DOUBLE-SPENDING” problem, then that may very well be true… HOWEVER, solving the double-spending is one thing; BUT then finding out that by virtue of solving the one problem it ends up leading to an equally serious problem of effectively permanently “freezing the assets” of people would still prove to show that the entire Bitcoin protocol is massively flawed from a technological standpoint.

What is Double Spending?

By: Ofir Beigel | Last updated: 11/14/19

One of the main issues any digital currency faces is double spending. This post explains exactly what the double spend problem is, and how to prevent it.

What is Double Spending Summary

Double spending is the act of trying to spend the same digital currency twice by creating duplicate transactions. This issue is one of the main obstacles a digital currency needs to solve in order to make sure it is not being abused, and that it maintains its value and trust.

Bitcoin solves the double spend problem through the use of a public ledger that is constantly monitored by network participants, and through the Proof of Work consensus mechanism.

That’s double spending in a nutshell. For a more detailed explanation keep on reading, here’s what I’ll cover:

1. Double Spending Explained

Since Bitcoin transactions are a digital file, it’s actually possible to duplicate transactions and spend the same Bitcoin twice. This issue of “copying and pasting” is a weakness any digital currency faces (even fiat currencies in their digital form).

Here are some examples of how you can double spend a digital currency:

  1. You can “copy” a coin and send it to someone while still retaining the original.
  2. You can simultaneously send the same coin to two different people.
  3. You can reverse a transaction that’s already been made after receiving the goods paid by it, hence keeping the goods and the money.

The double spend problem is a serious issue for any digital currency since it can create an inflated money supply which quickly erodes the value of the currency and the trust in it.

2. How to Prevent Double Spending?

There are two ways to stop double spending – a centralized way and a decentralized way.

The Centralized Solution

The centralized solution to prevent double spending is pretty simple. It usually involves a trusted authority that holds a record of everyone’s balance in the system.

For example, when Alice sends money to Bob, the transaction goes through the central authority (e.g. bank) that checks its ledger to makes sure Alice has the money she wants to spend. If everything is in order, the central authority authorizes the transaction and the money is transferred.

The Decentralized Solution

When you’re dealing with a decentralized currency like Bitcoin, you don’t have a central authority to validate transactions. Therefore, Bitcoin uses a mix of elements to solve the double spend problem.

Preventing Fraudulent Transactions

First of all, the Bitcoin ledger of transactions, known as the Blockchain, is public and visible to all. Every Bitcoin transaction ever made and every balance of every address can be inspected by anyone.

This means that if Alice sends one Bitcoin to Bob, every computer that holds a copy of the blockchain (also known as a full node) will verify the history of Alice’s transactions to make sure she indeed has that one Bitcoin to spend.

If Alice tries to cheat and creates a Bitcoin from thin air, she will quickly be exposed by one of the many nodes validating transactions.

Preventing Simultaneous Transactions

But what if Alice sends the same coin simultaneously to two people?

Assuming 50% of the nodes received transaction A first, and the other 50% received transaction B first. How do we know which transaction is valid, and which to discard? The answer is simple, the transaction that enters the ledger first will be the valid one.

That’s why it’s always recommended to wait for at least 1 confirmation before considering a Bitcoin transaction as complete.

In order to decide the order of transactions, Bitcoin uses a consensus mechanism called Proof of Work (POW). This mechanism describes the rules of who gets to update the ledger of Bitcoin transactions.

This is important since when there’s no central authority, you need to have a consensus about who gets to make changes in the transaction ledger. The whole process of updating the ledger of transactions is known as Bitcoin mining.

However, what happens if the mining process somehow happened simultaneously for both transaction A and transaction B?

Let’s assume that two unrelated miners managed to update the ledger at exactly the same time, each one with a different transaction that uses the same source of funds.

In this case, we will have 2 branches of the blockchain (also known as a fork) and the next block of transactions to be mined will determine which previous block was valid.

If the next block, in some magical way, is also mined simultaneously, we will have to wait until the next block, and so on.

If you want to be completely sure your Bitcoin transaction won’t be reversed in such a rare case of coincidences, you should wait until 6 confirmation arrives for your transaction.

It’s highly unlikely (like super highly unlikely) that this fork will happen more than 6 times. So, in the end, we will have a clear winner and only one transaction will be confirmed.

Preventing transaction reversal

There’s a special kind of double spend attack that is called a 51% attack. This happens when a single entity gains over 50% of the network’s mining power and can effectively control which version of the ledger is the legitimate one.

In this case, the attacker can spend coins on one version, obtaining good or services for this payment. Later on, the attacker creates a different version of the ledger where the original transaction doesn’t exist and retrieves the payment to his possession.

The only way to prevent a 51% attack is to make sure the network is decentralized enough and has enough computational power to make it basically impossible to amass more than 50% of its hashrate.

In Bitcoin’s case, a 51% is highly unlikely, but with smaller coins, this type of attack has happened.

3. RBF – “Legit” Double Spending

One form of “legit” double spending is Replace By Fee or RBF for short.

RBF is a function embedded in certain Bitcoin wallets (e.g. Electrum) that allows you to rebroadcast a transaction that is still unconfirmed, in order to get it confirmed faster.

In some cases, Bitcoin transactions are sent with a network fee that is too small to incentivize miners to pick them up. This can get your transaction stuck inside the mempool, waiting to be confirmed for a very long time.

RBF allows you to rebroadcast the transaction with a higher fee, effectively trying to double spend your money, so the miners will pick up the new transaction and the old one will get canceled.

4. Conclusion

Double spending is a major issue that needs to be addressed when dealing with digital currencies. Fortunately, it seems that Satoshi Nakamoto, the inventor of Bitcoin, managed to build a mechanism to prevent it from happening even in a decentralized network.

You may still have some comments or questions. If so, make sure to leave them in the comment section below.

Double-spending

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Double-spending problem is the successful use of the same funds twice. Double-spending of Bitcoin is not possible as Bitcoin is protected against a double-spending problem thanks to each transaction which is added to the blockchain being verified, and the majority of funds contained in this transaction cannot have been previously spent.

Double-spending is a potential flaw in a digital cash scheme in which the same single digital token can be spent more than once. This is possible because a digital token consists of a digital file that can be duplicated or falsified. [1] As with counterfeit money, such double-spending leads to inflation by creating a new amount of fraudulent currency that did not previously exist. This devalues the currency relative to other monetary units, and diminishes user trust as well as the circulation and retention of the currency. Fundamental cryptographic techniques to prevent double-spending while preserving anonymity in a transaction are blind signatures and particularly in offline systems, secret splitting.

Contents

Other numerical systems inhibit double-spending problem with the help of the authorized master source which follows certain trade rules for authorizing each transaction. In the case of Bitcoin, it uses a decentralized system where a large number of nodes following the same rules confirm the transaction without a central control node.

Bitcoin is vulnerable to double-spending problem during the initial period where a transaction is located on the network. The more transaction confirmations there are, the less risk there is that it will be used for fraud.

Double-spending in Blockchain [ edit ]

The following steps are required to perform a successful double-spending problem:

  • To execute a transaction that attacks carried out before payment.
  • Secretly mining using the block that includes this last transaction.
  • Wait for the transaction sending the money to the seller to receive enough confirming blocks, and the seller will hand over his goods, sure that the money is finally appropriated to him.
  • Continue to mine the secret alternative branch until it becomes more than public, after which it is broadcast to the network. Since the new branch is longer than all other known, it will be considered valid, and the btc transfer to the seller will be replaced by sending coins to the attacker.

(a) – network State prior to the attacker’s actions. (b) – the branch On the left includes a transaction to send btc to the seller. Has 2 confirmations. As a result, the seller handed over his product. At this time, the attacker generated a block that includes the attacking transaction. (c) if the attacker succeeds in creating a longer chain, he / she posts it to the network and the bitcoins are returned to him / her. The question arises: what is the probability that an attacker will be able to generate a branch that will be longer than the branch that is mined by everyone else.

To simulate the situation, let’s make a few simplifying assumptions, which we will use in the subsequent analysis: The total speed of mining in the General network and the attacker remains constant. The total mining speed will be H, of which part of pH refers to honest miners, and qH – to the attacker. At the same time: p + q = 1 . That is, the probability that the block will find an honest network is p , and that the attacker is q .

The complexity of mining remains constant. Let’s denote z = n – m the number of blocks in which an honest network has an advantage over an attacker. After each discovery of a new block, z changes, increasing by 1 if it is found by an honest network, and decreasing by 1 if the attacker. Mathematically, this is a Markov chain. If z reaches the value -1, then the attack fails. If this never happens, the attack is failed. Since we are interested in whether z=-1 will ever become, and when it will happen, we can use Markov’s chain theory to solve the problem, where each step represents the fact of finding the block by someone. zi + 1 can be either (zi+ 1) with probability p or (zi – 1) with probability q .

Prevention of the double-spend [ edit ]

The prevention of double-spending has taken two general forms: centralized and decentralized.

The value of the “double-spending” problem is that bitcoin or parts of it can be spent more than once, that is, it is possible to carry out a theoretically impossible operation. A transaction verification mechanism has been developed to combat this problem. Confirmation occurs in the following sequence:

  • Carrying out the transaction.
  • The inclusion of transactions in the block.
  • Confirmation of the transaction. The status of a” legitimate ” transaction is assigned after six confirmations. This number of confirmations is based on the probability theory, according to which the risk of such an operation is insignificant (less than 0.1%). Founders of the developed mechanism of protection considered that one person won’t be able to own more than ten percent of miners of a network. As you know, miners store all information about the network and confirm transactions.
  • The transaction is recognized as legitimate.

Centralized [ edit ]

This is usually implemented using an online central trusted third party that can verify whether a token has been spent. [1] This normally represents a single point of failure from both availability and trust viewpoints.

Decentralized [ edit ]

By 2007, a number of distributed systems for double-spending prevention had been proposed.

The cryptocurrency Bitcoin implemented a solution in early 2009. It uses a cryptographic protocol called a proof-of-work to avoid the need for a trusted third party to validate transactions. Instead, transactions are recorded in a public ledger called a blockchain. A transaction is considered valid when it is included in the blockchain that contains the most amount of computational work. This makes double-spending impossibly difficult, and more infeasible as the size of the overall network grows. Other cryptocurrencies also have similar features [2] .

Types of Bitcoin attacks [ edit ]

Race attack [ edit ]

If the transaction has no confirmations, shops and services which accept payment can be exposed to a so-called ‘race attack’. For example, two transactions are created from the same funds and are then sent to different shops/services. In this case, only one of those shops will receive the funds – a transaction from this shop will appear first in the blockchain.

Shops can take numerous precautions to reduce this type of attack but it is always good to remember should you accept transactions without any confirmation.

Finney Attack [ edit ]

Another type of attack. Shops or services which accept transactions without any confirmation are affected. “Finney Attack” is an attack which requires the participation of the mining expert to add repeated transactions to the block. The risk of such an attack cannot be reduced to nothing regardless of the preventative measures taken by shops or services, but it does require the participation of a mining expert and an ideal combination of contributing factors. It costs a lot of money and is no mean feat. Just as with the other type of attack, the shop or service must seriously consider its politics concerning transactions without any confirmation.

Vector76 Attack [ edit ]

Also called an “attack with confirmation”. This is a combination of the 2 aforementioned attacks which gives the perpetrator the ability to spend funds twice simply with a confirmation.

Brute Force Attack [ edit ]

This attack is possible even if the shop or service is expecting several transaction confirmations. It requires the attacker to be in possession of relatively high-performance hardware (hash frequency).

The perpetrator sends a transaction to the shop paying for a product/service and at the same time continues looking for a connection in the blockchain (blockchain fork) which recognizes this transaction. After a certain number of confirmations, the shop sends the product. If the perpetrator has found more than n blocks at this point, he breaks his blockchain fork and regains his money, but if the perpetrator has not succeeded in doing this, the attack can be deemed a failure and the funds are sent to the shop, as should be the case.

The success of this attack depends on the speed (hash frequency) of the attacker and the number of confirmations for the shop/service. For example, if the attacker possesses 10% of the calculation power of the Bitcoin network and the shop expects 6 confirmations for a successful transaction, the probability of success of such an attack will be 0.1%.

>50% Attack [ 50% Attack”>edit ]

If the perpetrator controls more than 50% of the Bitcoin network power, the probability of success of the aforementioned attack will be 100%. By virtue of the fact that the perpetrator can generate blocks more often than the other part of the network, he can create his own blockchain until it becomes longer than the “integral” part of the network.

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