# Difference between revisions of "Essay:Bitcoin: A Peer-to-Peer Electronic Cash System"

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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 <ref name="ref_8">W. Feller, "An introduction to probability theory and its applications," 1957.</ref>: | 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 <ref name="ref_8">W. Feller, "An introduction to probability theory and its applications," 1957.</ref>: | ||

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p = probability an honest node finds the next block | p = probability an honest node finds the next block | ||

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

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[[File:bitcoin-calculations.png|frame|center|alt=Bitcoin Calculations]] | [[File:bitcoin-calculations.png|frame|center|alt=Bitcoin Calculations]] |

## Revision as of 10:40, 26 March 2014

## The original bitcoin white-paper by Satoshi Nakamoto

## Contents

- 1 The original bitcoin white-paper by Satoshi Nakamoto
- 2 Bitcoin: A Peer-to-Peer Electronic Cash System
- 3 Introduction
- 4 Transactions
- 5 Timestamp Server
- 6 Proof-of-Work
- 7 Network
- 8 Incentive
- 9 Reclaiming Disk Space
- 10 Simplified Payment Verification
- 11 Combining and Splitting Value
- 12 Privacy
- 13 Calculations
- 14 Conclusion
- 15 References

## Bitcoin: A Peer-to-Peer Electronic Cash System

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.

## Introduction

## Transactions

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.

^{[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.

## Timestamp Server

^{[2]}

^{[3]}

^{[4]}

^{[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.

## Proof-of-Work

^{[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.

## Network

- New transactions are broadcast to all nodes.
- Each node collects new transactions into a block.
- Each node works on finding a difficult proof-of-work for its block.
- When a node finds a proof-of-work, it broadcasts the block to all nodes.
- Nodes accept the block only if all transactions in it are valid and not already spent.
- 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.

## Incentive

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.

## Reclaiming Disk Space

^{[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.

## Simplified Payment Verification

## Combining and Splitting Value

## Privacy

## Calculations

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
```

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:

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:

Rearranging to avoid summing the infinite tail of the distribution...

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## Conclusion

## References

- ↑ W. Dai, "b-money," http://www.weidai.com/bmoney.txt, 1998.
- ↑
^{2.0}^{2.1}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. - ↑ S. Haber, W.S. Stornetta, "How to time-stamp a digital document," In Journal of Cryptology, vol 3, no 2, pages 99-111, 1991.
- ↑ 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.
- ↑
^{5.0}^{5.1}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. - ↑ A. Back, "Hashcash - a denial of service counter-measure," http://www.hashcash.org/papers/hashcash.pdf, 2002.
- ↑ R.C. Merkle, "Protocols for public key cryptosystems," In Proc. 1980 Symposium on Security and Privacy, IEEE Computer Society, pages 122-133, April 1980.
- ↑ W. Feller, "An introduction to probability theory and its applications," 1957.