Ethereum: In what complexity is the proof of work Bitcoin (HashCash)?
To thoroughly solve this question, I will define an idealized hypothetical “ideal” abbreviation H (N) function, which has nice scalability. Then I formulate the PERFECT HASHCASH problem in terms of complexity of time p (n). This will help us understand whether the proof of work Bitcoin (HashCash) is e.g. comparative or.
Problem: an excellent shortcut
An ideal abbreviation function H (n) would be an algorithm that adopts a large input N and produces a fixed size output, usually represented as a number. In the context of cryptographic applications, such as digital signatures and nonces, a well -designed shortcut function can ensure solid safety against various types of attacks.
HashCash is one such examples of a shortcut function based on work. It has been designed to verify transactions in the Bitcoin network, ensuring that they are correct and cannot be doubled. Here’s how it works:
- MINER (node in the Bitcoin network) generates a unique identifier for each transaction.
- Miner calculates the abbreviation value for the transaction using a combination of its content and the number of blocks.
- The resulting value of the shortcut is compared with the target value, which transaction sender can determine.
If two abbreviations fit, it means that the transaction has been successfully approved (ie “found” in the database). In this case, Górnik is rewarded with newly broken Bitcoin coins.
Problem Perfect HashCash
Now let’s define the PERFECT HASHCASH problem, which we can use to assess the Computable complexity of HashCash:
Problem: Considering the entry n, find all possible outputs x such that h (n) = x modulo 2^64.
In other words, taking into account the large number N, we must generate all possible X values in the range [0, 2^64] using the H (n) abbreviation function. The key insight is that H (n) can be represented as:
H (n) ≡ x (mod 2^64)
where h (n) is a shortcut for the input N.
time complexity analysis
To analyze the complexity of the perfect time of HashCash, we can take the following steps:
1
- Find all possible x outputs in the range [0, 2^64] that meet the equation h (n) = x modulo 2^64.
- Count the number of important solutions.
The complexity of the excellent HashCash time can be analyzed, taking into account the number of possible input data N and the number of iterations required to find a solution for each input.
Class of computing complexity
By using the number of possible input data N, we can estimate the computing complexity of the excellent HashCash. In general, the complexity of the abbreviation function is divided into two main classes: P (n) and e.g. (n).
* P (n) Problems are those that can be solved during the multi -core time by a deterministic algorithm acting in time o (n^d) (where D is the depth of the problem), where n is the input size.
* NP (N) Problems are those that can be verified during a multi -core time using a witness (e.g. solving a problem). If there is a solution, it must also be verifiable.
In this case, because the excellent HashCash includes finding all possible outputs X for a given input N, we can estimate its computing complexity as follows:
- If N is relatively small (e.g. <10^8), then the excellent HashCash may take time with (2^n).
- In the case of larger input data, the number of iterations required to find the solution increases multiplely along with the input size.
Application
To sum up, although we have not clearly proved that HashCash is complete, e.g. or, our analysis suggests that it can be in p (n). However, this remains an open problem and requires further examination.