Bitcoin : A Peer-to-Peer Electronic Cash System
Satoshi Nakamoto
satoshin@gmx.com
www.bitcoin.org

To find the original article, click here

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   non-reversible 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.

11. Calculation

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
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 :
λ=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.

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=0.1z=0
P=1.0000000z=1
P=0.2045873z=2
P=0.0509779z=3
P=0.0131722z=4
P=0.0034552z=5
P=0.0009137z=6
P=0.0002428z=7
P=0.0000647z=8
P=0.0000173z=9
P=0.0000046z=10
P=0.0000012q=0.3z=0
P=1.0000000z=5
P=0.1773523z=10
P=0.0416605z=15
P=0.0101008z=20
P=0.0024804z=25
P=0.0006132z=30
P=0.0001522z=35
P=0.0000379z=40
P=0.0000095z=45
P=0.0000024z=50
P=0.0000006

Solving for P less than 0.1%…

P < 0.001q=0.10
z=5q=0.15
z=8q=0.20
z=11q=0.25
z=15q=0.30
z=24q=0.35
z=41q=0.40
z=89q=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

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