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                                                                   v1.0 2009
 
 
 
                     Bitcoin: A Peer-to-Peer Electronic Cash System
 
 
                                   Satoshi Nakamoto
                                   [email protected]
                                   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 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.
 
 
 
 
                                             1
 
 
 
 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.
 
           ___________________            ___________________            ___________________
          |   Transaction     |          |   Transaction     |          |   Transaction     |
          |   _____________   |          |   _____________   |          |   _____________   |
          |  | Owner 1's   |  |          |  | Owner 2's   |  |          |  | Owner 3's   |  |
          |  | Public Key  |  |          |  | Public Key  |  |          |  | Public Key  |  |
          |   ``````|````|`   |          |   ``````|````|`   |          |   ``````|``````   |
      ````````````| |    |    |``````````````````| |    |    |``````````````````| |         |
          |       V V    |    |          |       V V    |    |          |       V V         |
          |      ______  |    |          |      ______  |    |          |      ______       |
          |     | Hash | |    |          |     | Hash | |    |          |     | Hash |      |
          |      ```|``  |    |          |      ```|``  |    |          |      ```|``       |
          |         |     \   |          |         |     \   |          |         |         |
          |         V      \  |          |         V      \  |          |         V         |
          |    ____________ \ |  Verify  |    ____________ \ |  Verify  |    ____________   |
          |   | Owner 0's  | `-------------->| Owner 1's  | `-------------->| Owner 2's  |  |
          |   | Signature  |  |    ,-------->| Signature  |  |    ,-------->| Signature  |  |
          |    ````````````   |   / Sign |    ````````````   |   / Sign |    ````````````   |
           ```````````````````   /        ```````````````````   /        ```````````````````
                                /                              /
              _____________    /             _____________    /              _____________
             | Owner 1's   |--'             | Owner 2's   |--'              | Owner 3's   |
             | Private Key |                | Private Key |                 | Private Key |
              `````````````                  `````````````                   `````````````
 
     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.
 
                         ________                        ________
            ----------->|  Hash  |--------------------->|  Hash  |
                   ,--->|        |                 ,--->|        |
                   |     ````````                  |     ````````
                 ___________________________     ___________________________
                |  Block                    |   |  Block                    |
                |  ______   ______   _____  |   |  ______   ______   _____  |
                | | Item | | Item | | ... | |   | | Item | | Item | | ... | |
                |  ``````   ``````   `````  |   |  ``````   ``````   `````  |
                 ```````````````````````````     ```````````````````````````
 
 
 
 
                                             2
 
 
 
 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.
 
                 _____________________________     _____________________________
                |  Block                      |   |  Block                      |
                |   ___________    _______    |   |   ___________    _______    |
             ----->| Prev Hash |  | Nonce |   |----->| Prev Hash |  | Nonce |   |
                |   ```````````    ```````    |   |   ```````````    ```````    |
                |   ______   ______   _____   |   |   ______   ______   _____   |
                |  |  Tx  | |  Tx  | | ... |  |   |  |  Tx  | |  Tx  | | ... |  |
                |   ``````   ``````   `````   |   |   ``````   ``````   `````   |
                 `````````````````````````````     `````````````````````````````
 
     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.
 
 
 
 
                                             3
 
 
 
     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.
 
      ___________________________________________     ___________________________________________
     | Block  ____________________________       |   | Block  ____________________________       |
     |       | Block Header (Block Hash)  |      |   |       | Block Header (Block Hash)  |      |
     |       |   ___________    _______   |      |   |       |   ___________    _______   |      |
     |       |  | Prev Hash |  | Nonce |  |      |   |       |  | Prev Hash |  | Nonce |  |      |
     |       |   ```````````    ```````   |      |   |       |   ```````````    ```````   |      |
     |       |      _______________       |      |   |       |      _______________       |      |
     |       |     |   Root Hash   |      |      |   |       |     |   Root Hash   |      |      |
     |       |      `+```````````+`       |      |   |       |      `+```````````+`       |      |
     |        ``````/`````````````\```````       |   |        ``````/`````````````\```````       |
     |             /               \             |   |             /               \             |
     |       _____/__             __\_____       |   |       _____/__             __\_____       |
     |      | Hash01 |           | Hash23 |      |   |      | Hash01 |           | Hash23 |      |
     |       `+````+`             `+````+`       |   |       ````````             `+````+`       |
     |       /      \             /      \       |   |                            /      \       |
     |      /        \           /        \      |   |                           /        \      |
     |   _____      _____      _____      _____  |   |                         _____      _____  |
     |  |Hash0|    |Hash1|    |Hash2|    |Hash3| |   |                        |Hash2|    |Hash3| |
     |   ``+``      ``+``      ``+``      ``+``  |   |                         ``+``      ``+``  |
     |     |          |          |          |    |   |                           |          |    |
     |   __|__      __|__      __|__      __|__  |   |                         __|__      __|__  |
     |  | Tx0 |    | Tx1 |    | Tx2 |    | Tx3 | |   |                        | Tx2 |    | Tx3 | |
     |   `````      `````      `````      `````  |   |                         `````      `````  |
      ```````````````````````````````````````````     ```````````````````````````````````````````
         Transactions Hashed in a Merkle Tree              After Pruning Tx0-2 from the Block
 
 
     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.
 
 
 
 
                                             4
 
 
 
 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.
 
   Longest Proof-of-Work Chain
       _____________________________     _____________________________     _____________________________
      |  Block                      |   |  Block                      |   |  Block                      |
      |   ___________    _______    |   |   ___________    _______    |   |   ___________    _______    |
   ----->| Prev Hash |  | Nonce |   |----->| Prev Hash |  | Nonce |   |----->| Prev Hash |  | Nonce |   |--->
      |   ```````````    ```````    |   |   ```````````    ```````    |   |   ```````````    ```````    |
      |       _______________       |   |      _______________        |   |       _______________       |
      |      |  Merkle Root  |      |   |     |  Merkle Root  |       |   |      |  Merkle Root  |      |
      |       ```````````````       |   |      `+```````````+`        |   |       ```````````````       |
       `````````````````````````````     ``````/`````````````\````````     `````````````````````````````
                                              /               \
                                        _____/__             __\_____
                                       | Hash01 |           | Hash23 |
                                        ````````             ``````+`
                                                                    \   Merkle Branch for Tx3
                                                                     \
                                                                    _____
                                                                   |Hash3|
                                                                    ``+``
                                                                      |
                                                                    __|__
                                                                   | Tx3 |
                                                                    `````
 
     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.
 
                                   ________________
                                  | Transaction    |
                                  |  ____   _____  |
                              ----->| In | | Out |----->
                                  |  ````   `````  |
                                  |  ____   _____  |
                              ----->| In | | ... |----->
                                  |  ````   `````  |
                                  |  _____         |
                              ----->| ... |        |
                                  |  `````         |
                                   ````````````````
 
     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.
 
 
 
 
                                             5
 
 
 
 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.
 
  Traditional Privacy Model             _____________     ______________  |  ______________
      ____________     ____________    |             |   |              | | |              |
     | Identities |---|Transactions|-->|  Trusted    |-->| Counterparty | | |    Public    |
      ````````````     ````````````    | Third Party |   |              | | |              |
                                        `````````````     ``````````````  |  ``````````````
 
  New Privacy Model                     ______________
      ____________  |  ____________    |              |
     | Identities | | |Transactions|-->|    Public    |
      ````````````  |  ````````````    |              |
                                        ``````````````
 
     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
 
         ,                .
        /   1      if p<=q \
     qz=|                  |
        \ (q/p)^z  if p>q  /
         `                `
 
 
 
 
                                             6
 
 
 
 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:
 
          q
    A = z---
          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:
 
    oo
   _____
   \                 /                    \
    \     A^k e^-k  |  (q/p)^(z-k) if k<=z |
     |   ---------- |                      |
    /        k!     |     1        if k>z  |
   /                 \                    /
   ~~~~~
    k=0
 
 Rearranging to avoid summing the infinite tail of the distribution...
 
     z
   _____
   \
    \     A^k e^-k   /             \
 1 - |   ---------- | 1-(q/p)^(z-k) |
    /        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;
     }
 
 
 
 
                                             7
 
 
 
 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=0.1
     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
 
 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.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
                                             9