PyBLOCK on Nostr: October 31, 2008 ## Abstract A purely peer-to-peer version of electronic cash would ...
October 31, 2008
## 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\. 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]:
$$ \begin{eqnarray*} \large p &=& \text{ probability an honest node finds the
next block}\\\ \large q &=& \text{ probability the attacker finds the next
block}\\\ \large q_z &=& \text{ probability the attacker will ever catch up
from $z$ blocks behind} \end{eqnarray*}$$
$$ \large q_z = \begin{Bmatrix} 1 & \textit{if}\; p \leq q\\\ (q/p)^z &
\textit{if}\; p > q \end{Bmatrix}$$
Given our assumption that $p \gt q$, 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:
$$\large \lambda = z \frac qp$$
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:
$$ \large \sum_{k=0}^{\infty} \frac{\lambda^k e^{-\lambda}}{k!} \cdot
\begin{Bmatrix} (q/p)^{(z-k)} & \textit{if}\;k\leq z\\\ 1 & \textit{if} \; k >
z \end{Bmatrix}$$
Rearranging to avoid summing the infinite tail of the distribution...
$$ \large 1 - \sum_{k=0}^{z} \frac{\lambda^k e^{-\lambda}}{k!} \left (
1-(q/p)^{(z-k)} \right )$$
Converting to C code...
#include
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.
## References
1. W. Dai, ["b-money,"](/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,"](/static/docs/secure-timestamping-service.pdf) In _20th Symposium on Information Theory in the Benelux_ , May 1999. ↩ ↩
3. S. Haber, W.S. Stornetta, ["How to time-stamp a digital document,"](/literature/time-stamp-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,"](/literature/improving-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,"](/static/docs/secure-names-bit-strings.pdf) 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,"](/static/docs/hashcash.pdf) <http://www.hashcash.org/papers/hashcash.pdf>;, 2002. ↩
7. R.C. Merkle, ["Protocols for public key cryptosystems,"](/literature/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,"](/literature/introduction-probability-theory-vol-i/) 1957. ↩
* * *
## 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\. 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]:
$$ \begin{eqnarray*} \large p &=& \text{ probability an honest node finds the
next block}\\\ \large q &=& \text{ probability the attacker finds the next
block}\\\ \large q_z &=& \text{ probability the attacker will ever catch up
from $z$ blocks behind} \end{eqnarray*}$$
$$ \large q_z = \begin{Bmatrix} 1 & \textit{if}\; p \leq q\\\ (q/p)^z &
\textit{if}\; p > q \end{Bmatrix}$$
Given our assumption that $p \gt q$, 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:
$$\large \lambda = z \frac qp$$
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:
$$ \large \sum_{k=0}^{\infty} \frac{\lambda^k e^{-\lambda}}{k!} \cdot
\begin{Bmatrix} (q/p)^{(z-k)} & \textit{if}\;k\leq z\\\ 1 & \textit{if} \; k >
z \end{Bmatrix}$$
Rearranging to avoid summing the infinite tail of the distribution...
$$ \large 1 - \sum_{k=0}^{z} \frac{\lambda^k e^{-\lambda}}{k!} \left (
1-(q/p)^{(z-k)} \right )$$
Converting to C code...
#include
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.
## References
1. W. Dai, ["b-money,"](/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,"](/static/docs/secure-timestamping-service.pdf) In _20th Symposium on Information Theory in the Benelux_ , May 1999. ↩ ↩
3. S. Haber, W.S. Stornetta, ["How to time-stamp a digital document,"](/literature/time-stamp-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,"](/literature/improving-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,"](/static/docs/secure-names-bit-strings.pdf) 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,"](/static/docs/hashcash.pdf) <http://www.hashcash.org/papers/hashcash.pdf>;, 2002. ↩
7. R.C. Merkle, ["Protocols for public key cryptosystems,"](/literature/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,"](/literature/introduction-probability-theory-vol-i/) 1957. ↩
* * *