Bitcoin: Un Sistema de Efectivo Electrónico Peer-to-Peer
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.
Abstract
Una version puramente peer-to-peer de dinero electronico permitiria enviar pagos en linea directamente de una parte a otra sin pasar por una institucion financiera. Las firmas digitales proporcionan parte de la solucion, pero los principales beneficios se pierden si todavia se requiere un tercero de confianza para prevenir el doble gasto. Proponemos una solucion al problema del doble gasto utilizando una red peer-to-peer. La red marca temporalmente las transacciones al incluirlas mediante hash en una cadena continua de proof-of-work basada en hash, formando un registro que no puede ser modificado sin rehacer el proof-of-work. La cadena mas larga no solo sirve como prueba de la secuencia de eventos presenciados, sino como prueba de que proviene del mayor conjunto de poder de CPU. Mientras la mayoria del poder de CPU este controlado por nodos que no cooperan para atacar la red, generaran la cadena mas larga y superaran a los atacantes. La red en si requiere una estructura minima. Los mensajes se transmiten con base en el mejor esfuerzo, y los nodos pueden abandonar y reincorporarse a la red a voluntad, aceptando la cadena de proof-of-work mas larga como prueba de lo que ocurrio mientras estuvieron ausentes.
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.
Introduction
El comercio en Internet ha llegado a depender casi exclusivamente de instituciones financieras que sirven como terceros de confianza para procesar pagos electronicos. Si bien el sistema funciona suficientemente bien para la mayoria de las transacciones, todavia adolece de las debilidades inherentes del modelo basado en la confianza. Las transacciones completamente irreversibles no son realmente posibles, ya que las instituciones financieras no pueden evitar mediar en disputas. El costo de la mediacion aumenta los costos de transaccion, limitando el tamano minimo practico de la transaccion y eliminando la posibilidad de pequenas transacciones casuales, y existe un costo mas amplio en la perdida de la capacidad de realizar pagos irreversibles por servicios irreversibles. Con la posibilidad de reversion, la necesidad de confianza se extiende. Los comerciantes deben desconfiar de sus clientes, solicitandoles mas informacion de la que de otro modo necesitarian. Un cierto porcentaje de fraude se acepta como inevitable. Estos costos e incertidumbres de pago pueden evitarse en persona utilizando moneda fisica, pero no existe ningun mecanismo para realizar pagos a traves de un canal de comunicacion sin un tercero de confianza.
Lo que se necesita es un sistema de pago electronico basado en prueba criptografica en lugar de confianza, que permita a dos partes dispuestas realizar transacciones directamente entre si sin la necesidad de un tercero de confianza. Las transacciones que son computacionalmente impracticas de revertir protegerian a los vendedores del fraude, y mecanismos rutinarios de deposito en garantia podrian implementarse facilmente para proteger a los compradores. En este documento, proponemos una solucion al problema del doble gasto utilizando un servidor de marcas de tiempo distribuido peer-to-peer para generar prueba computacional del orden cronologico de las transacciones. El sistema es seguro mientras los nodos honestos controlen colectivamente mas poder de CPU que cualquier grupo cooperante de nodos atacantes.
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.
Transactions
Definimos una moneda electronica como una cadena de firmas digitales. Cada propietario transfiere la moneda al siguiente firmando digitalmente un hash de la transaccion anterior y la clave publica del siguiente propietario, y anadiendo estos al final de la moneda. Un beneficiario puede verificar las firmas para verificar la cadena de propiedad.
El problema, por supuesto, es que el beneficiario no puede verificar que uno de los propietarios no haya gastado doblemente la moneda. Una solucion comun es introducir una autoridad central de confianza, o casa de moneda, que verifique cada transaccion en busca de doble gasto. Despues de cada transaccion, la moneda debe ser devuelta a la casa de moneda para emitir una nueva moneda, y solo las monedas emitidas directamente por la casa de moneda son confiables de no haber sido doblemente gastadas. El problema con esta solucion es que el destino de todo el sistema monetario depende de la empresa que administra la casa de moneda, y cada transaccion debe pasar por ella, igual que un banco.
Necesitamos una forma para que el beneficiario sepa que los propietarios anteriores no firmaron ninguna transaccion previa. Para nuestros propositos, la transaccion mas temprana es la que cuenta, por lo que no nos preocupan los intentos posteriores de doble gasto. La unica forma de confirmar la ausencia de una transaccion es estar al tanto de todas las transacciones. En el modelo basado en la casa de moneda, esta estaba al tanto de todas las transacciones y decidia cual llego primero. Para lograr esto sin un tercero de confianza, las transacciones deben ser anunciadas publicamente [^1], y necesitamos un sistema para que los participantes acuerden un unico historial del orden en que fueron recibidas. El beneficiario necesita prueba de que, en el momento de cada transaccion, la mayoria de los nodos acordo que fue la primera recibida.
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] [^3] [^4] [^5]. The timestamp proves that the data must have existed at the time, obviously, in order to get into the hash. Each timestamp includes the previous timestamp in its hash, forming a chain, with each additional timestamp reinforcing the ones before it.
Timestamp Server
La solucion que proponemos comienza con un servidor de marcas de tiempo. Un servidor de marcas de tiempo funciona tomando un hash de un bloque de elementos a los que se les asignara una marca de tiempo y publicando ampliamente el hash, como en un periodico o una publicacion de Usenet [^2] [^3] [^4] [^5]. La marca de tiempo demuestra que los datos deben haber existido en ese momento, obviamente, para poder ser incluidos en el hash. Cada marca de tiempo incluye la marca de tiempo anterior en su hash, formando una cadena, donde cada marca de tiempo adicional refuerza las anteriores.
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.
Proof-of-Work
Para implementar un servidor de marcas de tiempo distribuido en una base peer-to-peer, necesitaremos utilizar un sistema de proof-of-work similar al Hashcash de Adam Back [^6], en lugar de publicaciones en periodicos o Usenet. El proof-of-work implica buscar un valor que, al ser hasheado, como con SHA-256, el hash comience con un numero de bits cero. El trabajo promedio requerido es exponencial en el numero de bits cero requeridos y puede verificarse ejecutando un unico hash.
Para nuestra red de marcas de tiempo, implementamos el proof-of-work incrementando un nonce en el bloque hasta que se encuentra un valor que le da al hash del bloque los bits cero requeridos. Una vez que el esfuerzo de CPU se ha gastado para satisfacer el proof-of-work, el bloque no puede ser cambiado sin rehacer el trabajo. A medida que se encadenan bloques posteriores, el trabajo para cambiar el bloque incluiria rehacer todos los bloques posteriores.
El proof-of-work tambien resuelve el problema de determinar la representacion en la toma de decisiones por mayoria. Si la mayoria se basara en una-direccion-IP-un-voto, podria ser subvertida por cualquiera capaz de asignar muchas IPs. El proof-of-work es esencialmente un-CPU-un-voto. La decision mayoritaria esta representada por la cadena mas larga, que tiene el mayor esfuerzo de proof-of-work invertido en ella. Si la mayoria del poder de CPU esta controlada por nodos honestos, la cadena honesta crecera mas rapido y superara a cualquier cadena competidora. Para modificar un bloque pasado, un atacante tendria que rehacer el proof-of-work del bloque y todos los bloques posteriores, y luego alcanzar y superar el trabajo de los nodos honestos. Mostraremos mas adelante que la probabilidad de que un atacante mas lento alcance a los demas disminuye exponencialmente a medida que se anaden bloques subsiguientes.
Para compensar el aumento de la velocidad del hardware y el interes variable en ejecutar nodos a lo largo del tiempo, la dificultad del proof-of-work se determina mediante un promedio movil que apunta a un numero promedio de bloques por hora. Si se generan demasiado rapido, la dificultad aumenta.
Network
The steps to run the network are as follows:
- New transactions are broadcast to all nodes.
- Each node collects new transactions into a block.
- Each node works on finding a difficult proof-of-work for its block.
- When a node finds a proof-of-work, it broadcasts the block to all nodes.
- Nodes accept the block only if all transactions in it are valid and not already spent.
- Nodes express their acceptance of the block by working on creating the next block in the chain, using the hash of the accepted block as the previous hash.
Nodes always consider the longest chain to be the correct one and will keep working on extending it. If two nodes broadcast different versions of the next block simultaneously, some nodes may receive one or the other first. In that case, they work on the first one they received, but save the other branch in case it becomes longer. The tie will be broken when the next proof-of-work is found and one branch becomes longer; the nodes that were working on the other branch will then switch to the longer one.
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.
Network
Los pasos para ejecutar la red son los siguientes:
- Las nuevas transacciones se transmiten a todos los nodos.
- Cada nodo recopila nuevas transacciones en un bloque.
- Cada nodo trabaja en encontrar un proof-of-work dificil para su bloque.
- Cuando un nodo encuentra un proof-of-work, transmite el bloque a todos los nodos.
- Los nodos aceptan el bloque solo si todas las transacciones en el son validas y no han sido gastadas previamente.
- Los nodos expresan su aceptacion del bloque trabajando en crear el siguiente bloque en la cadena, utilizando el hash del bloque aceptado como el hash anterior.
Los nodos siempre consideran la cadena mas larga como la correcta y continuaran trabajando para extenderla. Si dos nodos transmiten diferentes versiones del siguiente bloque simultaneamente, algunos nodos pueden recibir una u otra primero. En ese caso, trabajan en la primera que recibieron, pero guardan la otra rama en caso de que se vuelva mas larga. El empate se rompera cuando se encuentre el siguiente proof-of-work y una rama se vuelva mas larga; los nodos que estaban trabajando en la otra rama cambiaran entonces a la mas larga.
Las transmisiones de nuevas transacciones no necesariamente necesitan llegar a todos los nodos. Mientras lleguen a muchos nodos, entraran en un bloque en poco tiempo. Las transmisiones de bloques tambien son tolerantes a mensajes perdidos. Si un nodo no recibe un bloque, lo solicitara cuando reciba el siguiente bloque y se de cuenta de que le falta uno.
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.
Incentive
Por convencion, la primera transaccion en un bloque es una transaccion especial que inicia una nueva moneda propiedad del creador del bloque. Esto anade un incentivo para que los nodos apoyen la red, y proporciona una forma de distribuir inicialmente monedas en circulacion, ya que no existe una autoridad central para emitirlas. La adicion constante de una cantidad fija de nuevas monedas es analoga a los mineros de oro que gastan recursos para anadir oro a la circulacion. En nuestro caso, es el tiempo de CPU y la electricidad lo que se gasta.
El incentivo tambien puede financiarse con tarifas de transaccion. Si el valor de salida de una transaccion es menor que su valor de entrada, la diferencia es una tarifa de transaccion que se anade al valor del incentivo del bloque que contiene la transaccion. Una vez que un numero predeterminado de monedas ha entrado en circulacion, el incentivo puede transicionar completamente a tarifas de transaccion y estar completamente libre de inflacion.
El incentivo puede ayudar a alentar a los nodos a mantenerse honestos. Si un atacante codicioso es capaz de reunir mas poder de CPU que todos los nodos honestos, tendria que elegir entre usarlo para defraudar a las personas robando sus pagos, o usarlo para generar nuevas monedas. Deberia encontrar mas rentable jugar segun las reglas, reglas que lo favorecen con mas monedas nuevas que todos los demas combinados, que socavar el sistema y la validez de su propia riqueza.
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.
Reclaiming Disk Space
Una vez que la ultima transaccion en una moneda esta enterrada bajo suficientes bloques, las transacciones gastadas anteriores pueden descartarse para ahorrar espacio en disco. Para facilitar esto sin romper el hash del bloque, las transacciones se hashean en un Merkle Tree [^7] [^2] [^5], con solo la raiz incluida en el hash del bloque. Los bloques antiguos pueden entonces compactarse eliminando ramas del arbol. Los hashes interiores no necesitan ser almacenados.
Un encabezado de bloque sin transacciones seria de aproximadamente 80 bytes. Si suponemos que los bloques se generan cada 10 minutos, 80 bytes * 6 * 24 * 365 = 4.2MB por ano. Con los sistemas informaticos que tipicamente se vendian con 2GB de RAM en 2008, y la Ley de Moore prediciendo un crecimiento actual de 1.2GB por ano, el almacenamiento no deberia ser un problema incluso si los encabezados de bloque deben mantenerse en memoria.
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.
Simplified Payment Verification
Es posible verificar pagos sin ejecutar un nodo completo de la red. Un usuario solo necesita mantener una copia de los encabezados de bloque de la cadena de proof-of-work mas larga, que puede obtener consultando los nodos de la red hasta estar convencido de que tiene la cadena mas larga, y obtener la rama del Merkle Tree que vincula la transaccion al bloque en el que se le asigno la marca de tiempo. No puede verificar la transaccion por si mismo, pero al vincularla a un lugar en la cadena, puede ver que un nodo de la red la ha aceptado, y los bloques anadidos despues de ella confirman aun mas que la red la ha aceptado.
Como tal, la verificacion es confiable mientras los nodos honestos controlen la red, pero es mas vulnerable si la red es dominada por un atacante. Aunque los nodos de la red pueden verificar las transacciones por si mismos, el metodo simplificado puede ser enganado por transacciones fabricadas de un atacante mientras este pueda continuar dominando la red. Una estrategia para protegerse contra esto seria aceptar alertas de los nodos de la red cuando detecten un bloque invalido, solicitando al software del usuario descargar el bloque completo y las transacciones alertadas para confirmar la inconsistencia. Los negocios que reciben pagos frecuentes probablemente aun querran ejecutar sus propios nodos para una seguridad mas independiente y una verificacion mas rapida.
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.
Combining and Splitting Value
Aunque seria posible manejar monedas individualmente, seria poco practico hacer una transaccion separada por cada centavo en una transferencia. Para permitir que el valor se divida y combine, las transacciones contienen multiples entradas y salidas. Normalmente habra una unica entrada de una transaccion previa mayor o multiples entradas que combinan cantidades menores, y como maximo dos salidas: una para el pago, y una devolviendo el cambio, si lo hay, al remitente.
Cabe senalar que la ramificacion, donde una transaccion depende de varias transacciones, y esas transacciones dependen de muchas mas, no es un problema aqui. Nunca es necesario extraer una copia completa e independiente del historial de una transaccion.
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.
Privacy
El modelo bancario tradicional logra un nivel de privacidad limitando el acceso a la informacion a las partes involucradas y al tercero de confianza. La necesidad de anunciar todas las transacciones publicamente excluye este metodo, pero la privacidad aun puede mantenerse rompiendo el flujo de informacion en otro lugar: manteniendo las claves publicas anonimas. El publico puede ver que alguien esta enviando una cantidad a alguien mas, pero sin informacion que vincule la transaccion a nadie. Esto es similar al nivel de informacion publicado por las bolsas de valores, donde el tiempo y tamano de las operaciones individuales, la "cinta", se hace publica, pero sin revelar quienes fueron las partes.
Como cortafuegos adicional, se deberia usar un nuevo par de claves para cada transaccion para evitar que se vinculen a un propietario comun. Cierto grado de vinculacion es aun inevitable con transacciones de multiples entradas, que necesariamente revelan que sus entradas pertenecian al mismo propietario. El riesgo es que si se revela el propietario de una clave, la vinculacion podria revelar otras transacciones que pertenecieron al mismo propietario.
Calculations
We consider the scenario of an attacker trying to generate an alternate chain faster than the honest chain. Even if this is accomplished, it does not throw the system open to arbitrary changes, such as creating value out of thin air or taking money that never belonged to the attacker. Nodes are not going to accept an invalid transaction as payment, and honest nodes will never accept a block containing them. An attacker can only try to change one of his own transactions to take back money he recently spent.
The race between the honest chain and an attacker chain can be characterized as a Binomial Random Walk. The success event is the honest chain being extended by one block, increasing its lead by +1, and the failure event is the attacker's chain being extended by one block, reducing the gap by -1.
The probability of an attacker catching up from a given deficit is analogous to a Gambler's Ruin problem. Suppose a gambler with unlimited credit starts at a deficit and plays potentially an infinite number of trials to try to reach breakeven. We can calculate the probability he ever reaches breakeven, or that an attacker ever catches up with the honest chain, as follows [^8]:
p = probability an honest node finds the next block
q = probability the attacker finds the next block
qz = probability the attacker will ever catch up from z blocks behind
\[ qz = \begin{cases} 1 & \text{if } p \leq q \\ \left(\frac{q}{p}\right) z & \text{if } p > q \end{cases} \]
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:
\[ \lambda = z\frac{q}{p} \]
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:
\[ \sum_{k=0}^{\infty} \frac{\lambda^k e^{-\lambda}}{k!} \cdot \left\{ \begin{array}{cl} \left(\frac{q}{p}\right)^{(z-k)} & \text{if } k \leq z \\ 1 & \text{if } k > z \end{array} \right. \]
Rearranging to avoid summing the infinite tail of the distribution...
\[ 1 - \sum_{k=0}^{z} \frac{\lambda^k e^{-\lambda}}{k!} \left(1-\left(\frac{q}{p}\right)^{(z-k)}\right) \]
Converting to C code...
#include math.h
double AttackerSuccessProbability(double q, int z)
{
double p = 1.0 - q;
double lambda = z * (q / p);
double sum = 1.0;
int i, k;
for (k = 0; k = z; k++)
{
double poisson = exp(-lambda);
for (i = 1; i = k; i++)
poisson *= lambda / i;
sum -= poisson * (1 - pow(q / p, z - k));
}
return sum;
}
Running some results, we can see the probability drop off exponentially with z.
q=0.1
z=0 P=1.0000000
z=1 P=0.2045873
z=2 P=0.0509779
z=3 P=0.0131722
z=4 P=0.0034552
z=5 P=0.0009137
z=6 P=0.0002428
z=7 P=0.0000647
z=8 P=0.0000173
z=9 P=0.0000046
z=10 P=0.0000012
q=0.3
z=0 P=1.0000000
z=5 P=0.1773523
z=10 P=0.0416605
z=15 P=0.0101008
z=20 P=0.0024804
z=25 P=0.0006132
z=30 P=0.0001522
z=35 P=0.0000379
z=40 P=0.0000095
z=45 P=0.0000024
z=50 P=0.0000006
Solving for P less than 0.1%...
P 0.001
q=0.10 z=5
q=0.15 z=8
q=0.20 z=11
q=0.25 z=15
q=0.30 z=24
q=0.35 z=41
q=0.40 z=89
q=0.45 z=340
Calculations
Consideramos el escenario de un atacante que intenta generar una cadena alternativa mas rapido que la cadena honesta. Incluso si esto se logra, no abre el sistema a cambios arbitrarios, como crear valor de la nada o tomar dinero que nunca pertenecio al atacante. Los nodos no van a aceptar una transaccion invalida como pago, y los nodos honestos nunca aceptaran un bloque que las contenga. Un atacante solo puede intentar cambiar una de sus propias transacciones para recuperar dinero que gasto recientemente.
La carrera entre la cadena honesta y la cadena de un atacante puede caracterizarse como un Paseo Aleatorio Binomial. El evento de exito es que la cadena honesta se extienda un bloque, aumentando su ventaja en +1, y el evento de fracaso es que la cadena del atacante se extienda un bloque, reduciendo la brecha en -1.
La probabilidad de que un atacante alcance desde un deficit dado es analoga al problema de la Ruina del Jugador. Supongamos que un jugador con credito ilimitado comienza con un deficit y juega potencialmente un numero infinito de intentos para tratar de alcanzar el punto de equilibrio. Podemos calcular la probabilidad de que alguna vez alcance el punto de equilibrio, o de que un atacante alguna vez alcance a la cadena honesta, de la siguiente manera [^8]:
p = probabilidad de que un nodo honesto encuentre el siguiente bloque
q = probabilidad de que el atacante encuentre el siguiente bloque
q = probabilidad de que el atacante alguna vez alcance desde z bloques detras
``````
\[
qz =
\begin{cases}
1 & \text{if } p \leq q \\
\left(\frac{q}{p}\right) z & \text{if } p > q
\end{cases}
\]
Dada nuestra suposicion de que p q, la probabilidad cae exponencialmente a medida que aumenta el numero de bloques que el atacante tiene que alcanzar. Con las probabilidades en su contra, si no logra un avance afortunado temprano, sus posibilidades se vuelven infinitesimalmente pequenas a medida que queda mas atras.
Ahora consideramos cuanto tiempo necesita esperar el destinatario de una nueva transaccion antes de estar suficientemente seguro de que el remitente no puede cambiar la transaccion. Asumimos que el remitente es un atacante que quiere hacer creer al destinatario que le pago durante un tiempo, y luego cambiarlo para pagarse a si mismo despues de que haya pasado algun tiempo. El receptor sera alertado cuando eso suceda, pero el remitente espera que sea demasiado tarde.
El receptor genera un nuevo par de claves y entrega la clave publica al remitente poco antes de firmar. Esto evita que el remitente prepare una cadena de bloques con anticipacion trabajando en ella continuamente hasta que tenga la suerte de adelantarse lo suficiente, y luego ejecutar la transaccion en ese momento. Una vez que la transaccion es enviada, el remitente deshonesto comienza a trabajar en secreto en una cadena paralela que contiene una version alternativa de su transaccion.
El destinatario espera hasta que la transaccion se haya anadido a un bloque y z bloques se hayan vinculado despues de el. No conoce la cantidad exacta de progreso que el atacante ha hecho, pero asumiendo que los bloques honestos tomaron el tiempo promedio esperado por bloque, el progreso potencial del atacante sera una distribucion de Poisson con valor esperado:
\[
\lambda = z\frac{q}{p}
\]
Para obtener la probabilidad de que el atacante aun pueda alcanzar, multiplicamos la densidad de Poisson para cada cantidad de progreso que podria haber hecho por la probabilidad de que pueda alcanzar desde ese punto:
\[
\sum_{k=0}^{\infty} \frac{\lambda^k e^{-\lambda}}{k!} \cdot \left\{
\begin{array}{cl}
\left(\frac{q}{p}\right)^{(z-k)} & \text{if } k \leq z \\
1 & \text{if } k > z
\end{array}
\right.
\]
Reorganizando para evitar sumar la cola infinita de la distribucion...
\[
1 - \sum_{k=0}^{z} \frac{\lambda^k e^{-\lambda}}{k!} \left(1-\left(\frac{q}{p}\right)^{(z-k)}\right)
\]
Convirtiendo a codigo C...
```c
#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;
}
Ejecutando algunos resultados, podemos ver que la probabilidad cae exponencialmente con 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
Resolviendo para P menor que 0.1%...
P 0.001
q=0.10 z=5
q=0.15 z=8
q=0.20 z=11
q=0.25 z=15
q=0.30 z=24
q=0.35 z=41
q=0.40 z=89
q=0.45 z=340
Conclusion
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.
Conclusion
Hemos propuesto un sistema para transacciones electronicas sin depender de la confianza. Comenzamos con el marco habitual de monedas hechas de firmas digitales, que proporciona un fuerte control de propiedad, pero es incompleto sin una forma de prevenir el doble gasto. Para resolver esto, propusimos una red peer-to-peer que utiliza proof-of-work para registrar un historial publico de transacciones que rapidamente se vuelve computacionalmente impractico de cambiar para un atacante si los nodos honestos controlan la mayoria del poder de CPU. La red es robusta en su simplicidad no estructurada. Los nodos trabajan todos a la vez con poca coordinacion. No necesitan ser identificados, ya que los mensajes no se enrutan a ningun lugar particular y solo necesitan ser entregados con base en el mejor esfuerzo. Los nodos pueden abandonar y reincorporarse a la red a voluntad, aceptando la cadena de proof-of-work como prueba de lo que ocurrio mientras estuvieron ausentes. Votan con su poder de CPU, expresando su aceptacion de bloques validos al trabajar en extenderlos y rechazando bloques invalidos al negarse a trabajar en ellos. Cualquier regla e incentivo necesario puede ser aplicado con este mecanismo de consenso.
References
-
H. Massias, X.S. Avila, and J.-J. Quisquater, "Design of a secure timestamping service with minimal trust requirements," In 20th Symposium on Information Theory in the Benelux, May 1999.
-
S. Haber, W.S. Stornetta, "How to time-stamp a digital document," In Journal of Cryptology, vol 3, no 2, pages 99-111, 1991.
-
D. Bayer, S. Haber, W.S. Stornetta, "Improving the efficiency and reliability of digital time-stamping," In Sequences II: Methods in Communication, Security and Computer Science, pages 329-334, 1993.
-
S. Haber, W.S. Stornetta, "Secure names for bit-strings," In Proceedings of the 4th ACM Conference on Computer and Communications Security, pages 28-35, April 1997.
-
A. Back, "Hashcash - a denial of service counter-measure," http://www.hashcash.org/papers/hashcash.pdf, 2002.
-
R.C. Merkle, "Protocols for public key cryptosystems," In Proc. 1980 Symposium on Security and Privacy, IEEE Computer Society, pages 122-133, April 1980.
-
W. Feller, "An introduction to probability theory and its applications," 1957.
References
-
H. Massias, X.S. Avila, and J.-J. Quisquater, "Design of a secure timestamping service with minimal trust requirements," In 20th Symposium on Information Theory in the Benelux, May 1999.
-
S. Haber, W.S. Stornetta, "How to time-stamp a digital document," In Journal of Cryptology, vol 3, no 2, pages 99-111, 1991.
-
D. Bayer, S. Haber, W.S. Stornetta, "Improving the efficiency and reliability of digital time-stamping," In Sequences II: Methods in Communication, Security and Computer Science, pages 329-334, 1993.
-
S. Haber, W.S. Stornetta, "Secure names for bit-strings," In Proceedings of the 4th ACM Conference on Computer and Communications Security, pages 28-35, April 1997.
-
A. Back, "Hashcash - a denial of service counter-measure," http://www.hashcash.org/papers/hashcash.pdf, 2002.
-
R.C. Merkle, "Protocols for public key cryptosystems," In Proc. 1980 Symposium on Security and Privacy, IEEE Computer Society, pages 122-133, April 1980.
-
W. Feller, "An introduction to probability theory and its applications," 1957.
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