Pick Your Poisson

This article is featured in Bitcoin Magazine’s“The Halving Issue” and is sponsored by HIVE Digital Technologies LTD as part of Bitcoin Magazine’s “Buy The Numbers” content series. Click here to get your Annual Bitcoin Magazine Subscription.

Calculated probabilities were calculated by Greg @learnmeabitcoin.com

Block 840,000 is not just another block in the blockchain; it triggers the Bitcoin halving where the block reward is reduced from 6.25 BTC to 3.125 BTC, cutting the amount of BTC mined each day in half. You don’t have to be a Princeton economist to understand the impact this will have on the supply and demand dynamics for bitcoin. Beyond the obvious halving of the block reward, a new market has developed around Ordinals which could have a significant impact on what happens to the first block of the halving. Contained within the first block of the halving is an extremely rare “epic sat”. While Ordinals have divided some Bitcoiners on their merit, there is no arguing the impact they have had on Bitcoin.and it raises an important question, could Ordinals cause a blockchain reorg? Through this article we will dig into the basics of a reorg, Ordinals demand, how mining probabilities work, and finally who could pull off a successful reorg.

Before we dig into this “epic sat”, let’s build an understanding of what a reorg is. The Bitcoin blockchain is a slow and dumb database that creates blocks of data every 10 minutes or so. It continues working as intended, but occasionally, things get tense. When two miners find blocks nearly simultaneously, it creates a temporary fork in the blockchain. This moment of overlap leads to a brief period of uncertainty. These forks are resolved by the network through the longest chain rule, which is when the fork tip of the blockchain with more proof-of-work (the longest chain or aka more blocks) will be adopted as the valid chain. Orphaned blocks from the shorter chain are not included in the longer one, and the transactions they contain are returned to the mempool to be included in future blocks. This process of one chain becoming longer than the other and becoming the accepted version is known as a reorganization, or reorg.

Due to the incentive structures built into Bitcoin mining, reorgs are usually resolved as soon as the next block is found and added to the tip of one of the forked chains. This is because finding a block is extremely difficult, and miners are incentivized to work on the longest chain in order to build the next block, and get paid. If they are mining on the short fork, the rest of the network will leave them behind and they will have invalid blocks. The last thing you would want is to build a block that is rejected by the network because you’ve built a block on a chain and are rejected by the network due to the longest chain rule. During the reorg period of a fork, miners build on whichever chain fork hits their node first and try to build a block to get the longest chain.

Now don’t get worried about reorgs. They happen every couple of months (on average) and typically involve one or two blocks. These short reorgs are part of the network’s regular operation and quickly resolve without any significant impact on the network and its users. It’s worth noting that deep reorgs that consist of many blocks are rare and, correspondingly, more disruptive. They can be triggered by a network split such as what happened in the Blocksize wars, or a new large miner coming to the network, or an attempt to double-spend transactions (this is very rare).

Most Recent Reorgs

The Bitcoin protocol and its incentives are designed so that there’s a low likelihood of deep reorgs occurring. Consensus rules and incentives are meant to keep the network stable and secure. For example, most exchanges and payment processors require that a transaction be confirmed by a set number of times—usually six or more—before a transaction can be considered final, thus greatly reducing the chances of it being unwound by a reorg. Small reorgs happen and are mundane and frequent operations within the Bitcoin blockchain, but large reorgs are notable and very irregular.

About That Epic Sat

You’ve probably heard the buzz about Ordinals, that is “a numbering scheme for satoshis that allows tracking and transferring individual sats”. Some argue that Ordinals are a scam and they have no place in Bitcoin, but here’s the thing, an emerging market is rapidly growing around Ordinals. For now, they are here, and they are getting attention from miners, devs, VC, collectors, scammers, and haters alike.

When it comes to Ordinals, they are classified by their “rarity” and markets determine value.

Ordinals rarity levels:

+ common: Any sat that is not the first sat of its block

+ uncommon: The first sat of each block

+ rare: The first sat of each difficulty adjustment period

+ epic: The first sat of each halving epoch

+ legendary: The first sat of each cycle

+ mythic: The first sat of the genesis block

If we consider the scenario where all Bitcoin has been mined, which implies that all 21 million bitcoins (or 2.1 quadrillion satoshis) are in circulation, we can calculate the total quantity of each level of Ordinals:

• Uncommon: There would be a total of 6,929,999 uncommon satoshis, corresponding to the first satoshi of each block.

• Rare: There would be a total of approximately 3,437 rare satoshis, corresponding to the first satoshi of each difficulty adjustment period.

• Epic: There would be a total of 32 epic satoshis, corresponding to the first satoshi of each halving epoch.

• Legendary: There would be approximately 5 legendary satoshis, corresponding to the first satoshi of each cycle (noting a slight approximation due to division).

• Mythic: There is 1 mythic satoshi, which is the first sat of the genesis block.

These figures give an overview of how the rarity classifications would distribute across the total supply of satoshis once all Bitcoin is mined, showcasing the unique and scarce nature of certain satoshis within the Bitcoin network.

The Ordinals Market and Beyond

Over the past 12 months we’ve seen rapid development in Ordinals technology and markets. Ordinals markets first emerged in Discord back channels where OTC deals were being made, but as demand has grown, digital marketplaces have developed for buying and selling Ordinals. US Based Magisat.io lists various types of Ordinals and has Rare sats listed for a staggering 3.49 BTC. This valuation has led to the creation of additional inventory of Ordinals beyond the category that was first described in the Ordinals documentation.

Current Market on Magisat.io for standard Ordinals

This data shows that there is a small but growing demand for Ordinals. You can see the volume for Rare amd Uncommon Ordinals are greater than 26 BTC at the time of writing this. Keep in mind that this is only one marketplace and there are a growing number of OTC deals that are happening between buyers and sellers not to mention demand and business happening in other parts of the world.

Looking beyond Ordinals marketplaces we are now seeing Ordinals make their way to legendary auction house Sotheby’s further propelling the phenomenon towards the mainstream. If you look across the Pacific Ocean there is also significant demand for Ordinals and BRC-20 tokens, which would not be possible without Ordinals. So the demand for Ordinals is real and it is growing, not waning.

The last significant item of note that could impact demand for this first block of the halving is the activation of Runes. Runes is another protocol released by the same creator of Ordinals, but the aim of Runes is to make a more efficient token protocol. The kicker on this is that with it going live in the first block of the halving, this alone will cause a significant demand to issue these new tokens as quickly as possible, presumably the first Runes issued will be more valuable than later issued Runes. “Yes there will be reorg incentive for block 840,000, but it’s not for epic sat — it’s for the 20btc in fees from Casey’s Runes.” said Charlie Spears on X. This fee revenue call is speculation but it comes from observation from previous Ordinals and BRC20 activity.

Sifting For Sats

In Bitcoin, “dust” refers to an amount of bitcoin so small that it cannot be spent because the cost of a transaction fee would be higher than the amount itself. The concept of a “dust limit” therefore varies depending on the transaction fee and the type of transaction being made. However, there are general guidelines for what is considered dust, based on the type of Bitcoin script or address being used.

The dust limit is calculated based on the size of the inputs and outputs that make up a transaction. For a transaction to be relayed by most nodes and mined, its outputs must be above the dust limit. The dust limit for a standard P2PKH (Pay-to-Public-Key Hash) transaction output is commonly considered to be 546 satoshis when using the default minimum relay fee of 1 satoshi per byte, but this can vary depending on the network conditions and the policies of individual nodes.

For different script types, the dust limit calculation takes into account the size of the script and therefore can vary:

• P2PKH (Pay-to-Public-Key Hash): This is the most common type, and its dust limit is usually around 546 satoshis.

• P2SH (Pay-to-Script Hash): Outputs for P2SH transactions can have a slightly higher dust limit because the script itself is more complex, requiring more data to be included in a transaction.

• P2WPKH (Pay-to-Witness-Public-Key Hash) and P2WSH (Pay-to-Witness-Script Hash): These SegWit (Segregated Witness) transactions have different weight calculations, leading to lower fees for the same amount of data. Consequently, the dust limit for SegWit transactions can be lower than for traditional P2PKH transactions. For P2WPKH, the dust limit might be closer to 294 satoshis.

• MultiSig: Transactions involving multiple signatures (MultiSig) have higher dust limits due to the increased data size required to accommodate multiple signatures.

The exact dust limit can vary because it depends on the transaction’s size and the current fee market. Additionally, changes in Bitcoin’s protocol or node policies can affect these thresholds. It’s also worth noting that some wallets and services might set their own dust limits based on their operational requirements.

Pools Sifting

Based on the information above we can examine blocks found by pools to see if pools are actively sifting blocks. What we see is 44% of the network is or has sifted for sats in the past year. We have reason to believe that additional pools are in discussions with sifting technology developers in deploying the tech on their pools, but nothing has been made public at this time. Our findings reveal that a very significant percentage of the mining network sees value in sifting for these sats, otherwise this would be a hobbyist endeavor. When this many big players are participating, you know there is some market dynamics happening.

Beyond blockchain investigation outlined above, miners are developing new markets of their own in mining irregular transactions for various projects. Most notably is publicluy traded mining pool Marathon who launched a new service called Slipstream which will mine complex transactions such as 1 sat UTXO which is far below the SEGWIT dust threshold of 546 sats. I bring this up because as they are offering this service, you can’t help but assume that Marathon sees or will soon see value in Ordinals of they are willing to invest resources in serving Ordinal or Ordinal adjacent projects with this service. Afterall, the obligation of publicly traded companies is to maximize value for shareholders.

We know more than 40% of hashrate is sifting for sats, but what does that really mean in the grand scheme of things. Afterall, we are talking about one specific sat, the epic sat in block 840,000. We know that Ordinals have value according to the very small market, we also know that this sat will likely be sold for more than a single blocks reward, but the big question is who could forcibly win this block? Proof of Work is all about the longest chain and ethics don’t matter when it comes to Bitcoin and the blockchain. The chain is truth, even if you were to reorg. If you are hashing and following consensus and you build a longer chain then you are the victor. Based on the table from the previous section, we can see who the top pools are from the past six months. With that information we can model the probability of these pools forking and causing a reorg of the blockchain in order to win the epic sat but we need to run the numbers. For this we will explore the mining section of the Bitcoin Whitepaper.

Mining Explained in the Whitepaper

Bitcoin mining is a race to find a valid block by solving a cryptographic puzzle, known as proof of work. The difficulty of this puzzle is adjusted by the network so that, on average, a new block is found every 10 minutes, regardless of the total computing power of the network. Now the security of this is where things get interesting. Section 11 of the Bitcoin whitepaper discusses the mathematics behind the security of the blockchain against attackers who try to alter the transaction history.

The paper uses a comparison to a gambler’s “ruin problem” to explain how difficult it is for an attacker to catch up with the rest of the network once they fall behind in the race to add new blocks to the chain. Essentially, if honest nodes control more computational power, the probability that an attacker can catch up decreases rapidly as they fall further behind in the blockchain. The probability that an attacker can catch up becomes almost zero if they do not have a majority of the computational power.

The section outlines the process where transactions become more secure as new blocks are added to the blockchain, using a Poisson distribution to model the likelihood of an attacker catching up from being behind the chain tip. This framework provides the basis for understanding how blockchain achieves security through probabilistic means not absolute guarantees.

In the Bitcoin whitepaper, the Poisson distribution is used to model the security of mining. It’s used to quantify the probability that an attacker can catch up with the honest nodes after being z blocks behind, which is vital when considering the risk of a blockchain reorganization. It offers a statistical view of how likely it is for an attacker, with a certain percentage of the total network hash rate, to rewrite the blockchain history.

Converting to C code…

#include < math.h>

double AttackerSuccessProbability(double q, int z)

{

    double p = 1.0 – q;

    double lambda = z * (q / p);

    double sum = 1.0;

    int i, k;

    for (k = 0; k <= z; k++)

    {

        double poisson = exp(-lambda);

        for (i = 1; i <= k; i++)

        poisson *= lambda / i;

        sum -= poisson * (1 – pow(q / p, z – k));

    }

    return sum;

}

Who Could Pull This Off?

Ordinals introduces a new incentive to reorg. Before Ordinals, the threat of a reorg was focused around a double spend attack, but Ordinals introduced the demand for individual sats, in this case the demand to win a specific block. The question is this, does the value of a single Epic sat or block warrant abandoning the longest chain in hopes of finding a couple quick blocks and winning that epic sat? Pubco mining pools will have a hard time justifying such action to shareholders, it seems negligent. But for private mining pools, they have different incentives and have a bit more freedom in how they pursue revenue.

Looking at the top 10 mining pools by their percent of the network, we can model out who could pull off a reorg. One thing of note, the formulas described in the whitepaper only model catching up with the chain tip, however a reorg would require catching up to the tip +1 block, so our values below show that probability.

The first thing I noticed was Foundry and Antpool have a higher % probability of pulling off a reorg from 1 block behind than their own % hashrate of the network. How could this be possible? This is because A miner with 30% of the hashrate being 1 block behind and attempting a reorg is at a disadvantage because the rest of the network (70% hashrate) collectively has a higher probability of extending the current longest chain before the miner can catch up and surpass it. However, due to the randomness captured by the Poisson distribution, there’s always a non-zero probability that this miner could, through a streak of good luck, mine enough blocks in a row to take over the longest chain, even from one block behind. This is statistically unlikely but becomes possible with higher hashrate percentages and short reorg depths.

The next key takeaway is how successful reorgs become less likely for each block they are behind. It is remarkable how Foundry could still reorg from 5 blocks behind.

Conclusion

The Bitcoin space is weird (always has been) and bitcoin miners are the longest of long when it comes to outlook on Bitcoin. Based on the reorg probability and the potential value from the additional value on the first block of the halving, the probability of a reorg feels likely. If you take the BTC mined from this block, the epic sat, plus the projected amount of fees that will be earned from the release of Runes, bigger pools would be foolish to not try and make a move to win this block. The only real downside of a reorg would be by working on the old chain and NOT winning the reorg, so you would miss out on possibly winning 1-2 blocks by mining on the original longest chain. I do hope for fireworks. It will be legendary to hear the talk tracks from the new Wall Street financial bros trying to explain this. At the end of the day, miners will have to make a decision, that is to simply build on the longest chain or to try and build the longest chain with heavy amounts of luck. They must consider the tradeoffs and pick their poisson.

CODE

===========REORG-SUCCESS.RB=========================================

# —-

# Data

# —-

miners = {

‘    foundryusa’ => 30,

    ‘antpool’ => 25.64,

    ‘f2pool’ => 12.35,

    ‘viabtc’ => 10.98,

    ‘binancepool’ => 6.49,

    ‘marapool’ => 3.78,

    ‘luxor’ => 2.93,

    ‘sbicrypto’ => 1.90,

    ‘btcdotcom’ => 1.54,

    ‘braiinspool’ => 1.29,

    ‘unknown’ => 1.27,

}

# ——–

# Equation

# ——–

def attacker_success_probability(q, z)

    # p = probability honest node finds the next block

    # q = probability attacker finds the next block

    # z = number of blocks to catch up

    p = 1 – q

    lambda = z * (q / p) # expected number of occurrences in the poisson distribution

    sum = 1.0

    for k in 0..z

    poisson = Math.exp(-lambda) # exp() raises e (natural logarithm) to a number

    for i in 1..k

        poisson *= lambda / i

    end

    sum -= poisson * (1 – (q/p)**(z-k) )

    end

    return sum

end

# ——–

# Results

# ——–

# Run through each of the miners in the list

miners.each do |miner, percentage|

    # Print miner name

    puts “#{miner}”

    # Convert percentage to probability

    probability = percentage / 100.0

    # Calculate their success of replacing a different number of blocks near the top of the chain

    1.upto(5) do |blocks|

    # NOTE!

    # Add 1 to the number of blocks.

    # This is because we don’t want to calculate the probability of merely catching up to the tip of the chain (which is what the equation calculates).

    # To perform a successful attack, we want calculate the probability of building a chain that is ONE BLOCK LONGER than the current chain. That way, other nodes will be forced to adopt it and we will have successfully rewritten the blockchain.

    # Calculate success for specific number of blocks based on their hash share

    success = attacker_success_probability(probability, blocks+1)

   # Convert probability to percentage

    success_percentage = success * 100.0

    # show results

    puts ” #{blocks} = #{“%.8f” % success_percentage}%”

    # NOTE: The %.8f converts from scientific notation to decimal

    # Adjust the number (e.g. 8) to control how many decimal places you want to show

    end

    # Add gap between results for each miner

    puts

end

This article is featured in Bitcoin Magazine’s“The Halving Issue”. Click here to get your Annual Bitcoin Magazine Subscription.

The Bitcoin Whitepaper

On October 31, 2008, an individual or group under the pseudonym Satoshi Nakamoto introduced the concept of Bitcoin through a whitepaper titled “Bitcoin: A Peer-to-Peer Electronic Cash System” to the cypherpunk mailing list. This whitepaper, drawing from decades of research and the works of individuals like Wei Dai, Nick Szabo, and Hal Finney, laid the foundation for the creation of a decentralized digital currency. The whitepaper is just 9 pages long and consists of 12 sections, each explaining the core components and the underlying philosophy of Bitcoin.

Bitcoin: A Peer-to-Peer Electronic Cash System Satoshi Nakamoto

satoshin@gmx.com

www.bitcoin.org

Abstract. A purely peer-to-peer version of electronic cash would allow online payments to be sent directly from one party to another without going through a financial institution. Digital signatures provide part of the solution, but the main benefits are lost if a trusted third party is still required to prevent double-spending. We propose a solution to the double-spending problem using a peer-to-peer network. The network timestamps transactions by hashing them into an ongoing chain of hash-based proof-of-work, forming a record that cannot be changed without redoing the proof-of-work. The longest chain not only serves as proof of the sequence of events witnessed, but proof that it came from the largest pool of CPU power. As long as a majority of CPU power is controlled by nodes that are not cooperating to attack the network, they’ll generate the longest chain and outpace attackers. The network itself requires minimal structure. Messages are broadcast on a best effort basis, and nodes can leave and rejoin the network at will, accepting the longest proof-of-work chain as proof of what happened while they were gone.

1. Introduction

Commerce on the Internet has come to rely almost exclusively on financial institutions serving as trusted third parties to process electronic payments. While the system works well enough for most transactions, it still suffers from the inherent weaknesses of the trust based model. Completely non-reversible transactions are not really possible, since financial institutions cannot avoid mediating disputes. The cost of mediation increases transaction costs, limiting the minimum practical transaction size and cutting off the possibility for small casual transactions, and there is a broader cost in the loss of ability to make non-reversible payments for nonreversible services. With the possibility of reversal, the need for trust spreads. Merchants must be wary of their customers, hassling them for more information than they would otherwise need. A certain percentage of fraud is accepted as unavoidable. These costs and payment uncertainties can be avoided in person by using physical currency, but no mechanism exists to make payments over a communications channel without a trusted party.

What is needed is an electronic payment system based on cryptographic proof instead of trust, allowing any two willing parties to transact directly with each other without the need for a trusted third party. Transactions that are computationally impractical to reverse would protect sellers from fraud, and routine escrow mechanisms could easily be implemented to protect buyers. In this paper, we propose a solution to the double-spending problem using a peer-to-peer distributed timestamp server to generate computational proof of the chronological order of transactions. The system is secure as long as honest nodes collectively control more CPU power than any cooperating group of attacker nodes.

2. Transactions

We define an electronic coin as a chain of digital signatures. Each owner transfers the coin to the next by digitally signing a hash of the previous transaction and the public key of the next owner and adding these to the end of the coin. A payee can verify the signatures to verify the chain of ownership.

The problem of course is the payee can’t verify that one of the owners did not double-spend the coin. A common solution is to introduce a trusted central authority, or mint, that checks every transaction for double spending. After each transaction, the coin must be returned to the mint to issue a new coin, and only coins issued directly from the mint are trusted not to be double-spent. The problem with this solution is that the fate of the entire money system depends on the company running the mint, with every transaction having to go through them, just like a bank.

We need a way for the payee to know that the previous owners did not sign any earlier transactions. For our purposes, the earliest transaction is the one that counts, so we don’t care about later attempts to double-spend. The only way to confirm the absence of a transaction is to be aware of all transactions. In the mint based model, the mint was aware of all transactions and decided which arrived first. To accomplish this without a trusted party, transactions must be publicly announced [1], and we need a system for participants to agree on a single history of the order in which they were received. The payee needs proof that at the time of each transaction, the majority of nodes agreed it was the first received.

3. Timestamp Server

The solution we propose begins with a timestamp server. A timestamp server works by taking a hash of a block of items to be timestamped and widely publishing the hash, such as in a newspaper or Usenet post [2-5]. The timestamp proves that the data must have existed at the time, obviously, in order to get into the hash. Each timestamp includes the previous timestamp in its hash, forming a chain, with each additional timestamp reinforcing the ones before it.

4. Proof-of-Work

To implement a distributed timestamp server on a peer-to-peer basis, we will need to use a proof-of-work system similar to Adam Back’s Hashcash [6], rather than newspaper or Usenet posts. The proof-of-work involves scanning for a value that when hashed, such as with SHA-256, the hash begins with a number of zero bits. The average work required is exponential in the number of zero bits required and can be verified by executing a single hash.

For our timestamp network, we implement the proof-of-work by incrementing a nonce in the block until a value is found that gives the block’s hash the required zero bits. Once the CPU effort has been expended to make it satisfy the proof-of-work, the block cannot be changed without redoing the work. As later blocks are chained after it, the work to change the block would include redoing all the blocks after it.

The proof-of-work also solves the problem of determining representation in majority decision making. If the majority were based on one-IP-address-one-vote, it could be subverted by anyone able to allocate many IPs. Proof-of-work is essentially one-CPU-one-vote. The majority decision is represented by the longest chain, which has the greatest proof-of-work effort invested in it. If a majority of CPU power is controlled by honest nodes, the honest chain will grow the fastest and outpace any competing chains. To modify a past block, an attacker would have to redo the proof-of-work of the block and all blocks after it and then catch up with and surpass the work of the honest nodes. We will show later that the probability of a slower attacker catching up diminishes exponentially as subsequent blocks are added.

To compensate for increasing hardware speed and varying interest in running nodes over time, the proof-of-work difficulty is determined by a moving average targeting an average number of blocks per hour. If they’re generated too fast, the difficulty increases.

5. Network

The steps to run the network are as follows:

1) New transactions are broadcast to all nodes.

2) Each node collects new transactions into a block.

3) Each node works on finding a difficult proof-of-work for its block.

4) When a node finds a proof-of-work, it broadcasts the block to all nodes.

5) Nodes accept the block only if all transactions in it are valid and not already spent.

6) Nodes express their acceptance of the block by working on creating the next block in the chain, using the hash of the accepted block as the previous hash.

Nodes always consider the longest chain to be the correct one and will keep working on extending it. If two nodes broadcast different versions of the next block simultaneously, some nodes may receive one or the other first. In that case, they work on the first one they received, but save the other branch in case it becomes longer. The tie will be broken when the next proof-of-work is found and one branch becomes longer; the nodes that were working on the other branch will then switch to the longer one.

New transaction broadcasts do not necessarily need to reach all nodes. As long as they reach many nodes, they will get into a block before long. Block broadcasts are also tolerant of dropped messages. If a node does not receive a block, it will request it when it receives the next block and realizes it missed one.

6. Incentive

By convention, the first transaction in a block is a special transaction that starts a new coin owned by the creator of the block. This adds an incentive for nodes to support the network, and provides a way to initially distribute coins into circulation, since there is no central authority to issue them. The steady addition of a constant of amount of new coins is analogous to gold miners expending resources to add gold to circulation. In our case, it is CPU time and electricity that is expended.

The incentive can also be funded with transaction fees. If the output value of a transaction is less than its input value, the difference is a transaction fee that is added to the incentive value of the block containing the transaction. Once a predetermined number of coins have entered circulation, the incentive can transition entirely to transaction fees and be completely inflation free.

The incentive may help encourage nodes to stay honest. If a greedy attacker is able to assemble more CPU power than all the honest nodes, he would have to choose between using it to defraud people by stealing back his payments, or using it to generate new coins. He ought to find it more profitable to play by the rules, such rules that favour him with more new coins than everyone else combined, than to undermine the system and the validity of his own wealth.

7. Reclaiming Disk Space

Once the latest transaction in a coin is buried under enough blocks, the spent transactions before it can be discarded to save disk space. To facilitate this without breaking the block’s hash, transactions are hashed in a Merkle Tree [7][2][5], with only the root included in the block’s hash. Old blocks can then be compacted by stubbing off branches of the tree. The interior hashes do not need to be stored.

A block header with no transactions would be about 80 bytes. If we suppose blocks are generated every 10 minutes, 80 bytes * 6 * 24 * 365 = 4.2MB per year. With computer systems typically selling with 2GB of RAM as of 2008, and Moore’s Law predicting current growth of 1.2GB per year, storage should not be a problem even if the block headers must be kept in memory.

8. Simplified Payment Verification

It is possible to verify payments without running a full network node. A user only needs to keep a copy of the block headers of the longest proof-of-work chain, which he can get by querying network nodes until he’s convinced he has the longest chain, and obtain the Merkle branch linking the transaction to the block it’s timestamped in. He can’t check the transaction for himself, but by linking it to a place in the chain, he can see that a network node has accepted it, and blocks added after it further confirm the network has accepted it.

As such, the verification is reliable as long as honest nodes control the network, but is more vulnerable if the network is overpowered by an attacker. While network nodes can verify transactions for themselves, the simplified method can be fooled by an attacker’s fabricated transactions for as long as the attacker can continue to overpower the network. One strategy to protect against this would be to accept alerts from network nodes when they detect an invalid block, prompting the user’s software to download the full block and alerted transactions to confirm the inconsistency. Businesses that receive frequent payments will probably still want to run their own nodes for more independent security and quicker verification.

9. Combining and Splitting Value

Although it would be possible to handle coins individually, it would be unwieldy to make a separate transaction for every cent in a transfer. To allow value to be split and combined, transactions contain multiple inputs and outputs. Normally there will be either a single input from a larger previous transaction or multiple inputs combining smaller amounts, and at most two outputs: one for the payment, and one returning the change, if any, back to the sender.

It should be noted that fan-out, where a transaction depends on several transactions, and those transactions depend on many more, is not a problem here. There is never the need to extract a complete standalone copy of a transaction’s history.

10. Privacy

The traditional banking model achieves a level of privacy by limiting access to information to the parties involved and the trusted third party. The necessity to announce all transactions publicly precludes this method, but privacy can still be maintained by breaking the flow of information in another place: by keeping public keys anonymous. The public can see that someone is sending an amount to someone else, but without information linking the transaction to anyone. This is similar to the level of information released by stock exchanges, where the time and size of individual trades, the “tape”, is made public, but without telling who the parties were.

As an additional firewall, a new key pair should be used for each transaction to keep them from being linked to a common owner. Some linking is still unavoidable with multi-input transactions, which necessarily reveal that their inputs were owned by the same owner. The risk is that if the owner of a key is revealed, linking could reveal other transactions that belonged to the same owner.

As an additional firewall, a new key pair should be used for each transaction to keep them from being linked to a common owner. Some linking is still unavoidable with multi-input transactions, which necessarily reveal that their inputs were owned by the same owner. The risk is that if the owner of a key is revealed, linking could reveal other transactions that belonged to the same owner.

11. Calculations

We consider the scenario of an attacker trying to generate an alternate chain faster than the honest chain. Even if this is accomplished, it does not throw the system open to arbitrary changes, such as creating value out of thin air or taking money that never belonged to the attacker. Nodes are not going to accept an invalid transaction as payment, and honest nodes will never accept a block containing them. An attacker can only try to change one of his own transactions to take back money he recently spent.

The race between the honest chain and an attacker chain can be characterized as a Binomial Random Walk. The success event is the honest chain being extended by one block, increasing its lead by +1, and the failure event is the attacker’s chain being extended by one block, reducing the gap by -1.

The probability of an attacker catching up from a given deficit is analogous to a Gambler’s Ruin problem. Suppose a gambler with unlimited credit starts at a deficit and plays potentially an infinite number of trials to try to reach breakeven. We can calculate the probability he ever reaches breakeven, or that an attacker ever catches up with the honest chain, as follows [8]:

p = probability an honest node finds the next block

q = probability the attacker finds the next block

qz = probability the attacker will ever catch up from z blocks behind

qz={ 1 if p≤q q/ p z if pq}

Given our assumption that p > q, the probability drops exponentially as the number of blocks the

attacker has to catch up with increases. With the odds against him, if he doesn’t make a lucky

lunge forward early on, his chances become vanishingly small as he falls further behind.

We now consider how long the recipient of a new transaction needs to wait before being

sufficiently certain the sender can’t change the transaction. We assume the sender is an attacker who wants to make the recipient believe he paid him for a while, then switch it to pay back to himself after some time has passed. The receiver will be alerted when that happens, but the sender hopes it will be too late.

The receiver generates a new key pair and gives the public key to the sender shortly before

signing. This prevents the sender from preparing a chain of blocks ahead of time by working on

it continuously until he is lucky enough to get far enough ahead, then executing the transaction at that moment. Once the transaction is sent, the dishonest sender starts working in secret on a

parallel chain containing an alternate version of his transaction.

The recipient waits until the transaction has been added to a block and z blocks have been

linked after it. He doesn’t know the exact amount of progress the attacker has made, but

assuming the honest blocks took the average expected time per block, the attacker’s potential

progress will be a Poisson distribution with expected value:

入 = z q/p

To get the probability the attacker could still catch up now, we multiply the Poisson density for each amount of progress he could have made by the probability he could catch up from that point:

∑ k=0 ∞ k e − k! ⋅{ q/ p z−k if k≤z 1 if kz}

Rearranging to avoid summing the infinite tail of the distribution…

1−∑ k=0 z k e − k! 1−q/ p z−k

Converting to C code…

#include <math.h>

double AttackerSuccessProbability(double q, int z)

{

double p = 1.0 – q;

double lambda = z * (q / p);

double sum = 1.0;

int i, k;

for (k = 0; k <= z; k++)

{

double poisson = exp(-lambda);

for (i = 1; i <= k; i++)

poisson *= lambda / i;

sum -= poisson * (1 – pow(q / p, z – k));

}

return sum;

}

Running some results, we can see the probability drop off exponentially with z.

q=0.1

z=0 P=1.0000000

z=1 P=0.2045873

z=2 P=0.0509779

z=3 P=0.0131722

z=4 P=0.0034552

z=5 P=0.0009137

z=6 P=0.0002428

z=7 P=0.0000647

z=8 P=0.0000173

z=9 P=0.0000046

z=10 P=0.0000012

q=0.3

z=0 P=1.0000000

z=5 P=0.1773523

z=10 P=0.0416605

z=15 P=0.0101008

z=20 P=0.0024804

z=25 P=0.0006132

z=30 P=0.0001522

z=35 P=0.0000379

z=40 P=0.0000095

z=45 P=0.0000024

z=50 P=0.0000006

Solving for P less than 0.1%…

P < 0.001

q=0.10 z=5

q=0.15 z=8

q=0.20 z=11

q=0.25 z=15

q=0.30 z=24

q=0.35 z=41

q=0.40 z=89

q=0.45 z=340

12. Conclusion

We have proposed a system for electronic transactions without relying on trust. We started with the usual framework of coins made from digital signatures, which provides strong control of ownership, but is incomplete without a way to prevent double-spending. To solve this, we proposed a peer-to-peer network using proof-of-work to record a public history of transactions that quickly becomes computationally impractical for an attacker to change if honest nodes control a majority of CPU power. The network is robust in its unstructured simplicity. Nodes work all at once with little coordination. They do not need to be identified, since messages are not routed to any particular place and only need to be delivered on a best effort basis. Nodes can leave and rejoin the network at will, accepting the proof-of-work chain as proof of what happened while they were gone. They vote with their CPU power, expressing their acceptance of valid blocks by working on extending them and rejecting invalid blocks by refusing to work on them. Any needed rules and incentives can be enforced with this consensus mechanism. 8

References

[1] W. Dai, “b-money,” http://www.weidai.com/bmoney.txt, 1998.

[2] H. Massias, X.S. Avila, and J.-J. Quisquater, “Design of a secure timestamping service with minimal trust requirements,” In 20th Symposium on Information Theory in the Benelux, May 1999.

[3] S. Haber, W.S. Stornetta, “How to time-stamp a digital document,” In Journal of Cryptology, vol 3, no 2, pages 99-111, 1991.

[4] D. Bayer, S. Haber, W.S. Stornetta, “Improving the efficiency and reliability of digital time-stamping,” In Sequences II: Methods in Communication, Security and Computer Science, pages 329-334, 1993.

[5] S. Haber, W.S. Stornetta, “Secure names for bit-strings,” In Proceedings of the 4th ACM Conference on Computer and Communications Security, pages 28-35, April 1997.

[6] A. Back, “Hashcash – a denial of service counter-measure,” http://www.hashcash.org/papers/hashcash.pdf, 2002.

[7] R.C. Merkle, “Protocols for public key cryptosystems,” In Proc. 1980 Symposium on Security and Privacy, IEEE Computer Society, pages 122-133, April 1980.

[8] W. Feller, “An introduction to probability theory and its applications,” 1957.

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