Ethereum: Why is the Poisson used instead of the negative binomial to calculate the potential progress of an attacker?
Chapter 11 of Satoshi Nakamoto’s seminal work “Bitcoin: A Peer-to-Peer Electronic Cash System” introduces a key concept that has significant implications for our understanding of the security and potential vulnerabilities of the Ethereum network. The paper discusses a scenario in which an attacker waits until a transaction is added to a block and then links to the blocks afterwards. However, instead of using the negative binomial distribution, which would be more appropriate to model this situation, Satoshi uses the Poisson distribution. In this article, we will explore why the Poisson distribution is used in this context.
What is the Poisson distribution?
The Poisson distribution is a discrete probability distribution that models the number of events that occur in a fixed interval of time or space, where these events occur with a known constant mean rate and independent of the time since the last event. The Poisson distribution is often used to model rare but unavoidable events, such as radioactive fallout or traffic jams.
Why not the negative binomial?
The negative binomial distribution is commonly used to model the number of failures until a certain event occurs, where each failure has a known probability and the time to the next failure is independent. However, in this scenario, we are concerned with the potential progress of an attacker after the transaction has been added to the block. The key difference between these two models lies in their parameters.
The negative binomial distribution requires that we know the rate parameter (λ) and the probability of success (p). In contrast, the Poisson distribution only requires us to know the mean velocity (μ), which is related to λ by the formula μ = λ / p. Therefore, we can use a single parameter (μ) to model the number of transactions until the block is connected and the attacker’s potential progress.
Potential Progress of a Computer Attacker
In the context of Ethereum, the receiver waits until the transaction is added to the block, and then joins z blocks after it. This creates a situation where we need to model the attacker’s potential progress beyond this point. The Poisson distribution is more suitable for modeling events that occur over time, such as transactions being verified or miner attempts.
For example, suppose the mean rate (μ) of transactions added to a block is 10 per hour. This means that approximately 3,600 new transactions are added to the block every hour. If we wait for a transaction to be added to a block and then chained from blocks afterwards, an attacker can potentially make progress by waiting more time before trying to mine or hijack the blockchain.
The Poisson distribution allows us to model this situation with a single parameter (μ) and calculate the probability of the attacker’s potential progression over time. This provides a clear and concise way to analyze the security of the Ethereum network.
Conclusion
In conclusion, Satoshi Nakamoto’s choice to use the Poisson distribution instead of the negative binomial distribution in Chapter 11 is not arbitrary. The Poisson distribution is more suitable for modeling events that occur over time, such as transactions being verified or miner attempts. By using a single parameter (μ) to model the number of transactions until a block is merged and the potential progress of an attacker, we can gain valuable insights into the security and vulnerabilities of the Ethereum network.
This article highlights the importance of considering the distribution used in modeling events that occur over time, such as transactions added to a block or miner attempts.