Ethereum: What complexity class is Bitcoin’s proof-of-work (hashcash) in?

Ethereum: What Complexity Class is Bitcoin’s Proof-of-Work (Hashcash) in?

To tackle this question precisely, I’ll define an idealized hypothetical “perfect” hash function H(n), which has nice scalability properties. Then, I’ll formulate a problem PERFECT HASHCASH in terms of the time complexity class P(n). This will help us understand whether Bitcoin’s proof-of-work (hashcash) is NP-complete or else.

The Problem: Perfect Hash Function

A perfect hash function H(n) would be an algorithm that takes a large input n and produces a fixed-size output, typically represented as a number. In the context of cryptographic applications like digital signatures and nonces, a well-designed hash function can provide robust security against various types of attacks.

Hashcash is one such example of a proof-of-work-based hash function. It’s designed to validate transactions on the Bitcoin network, ensuring that they are valid and cannot be double-spended. Here’s how it works:

• A miner (a node in the Bitcoin network) generates a unique identifier for each transaction.

• The miner calculates a hash value for the transaction using a combination of its contents and the block number.

• The resulting hash value is compared to a target value, which can be set by the sender of the transaction.

If the two hashes match, it means the transaction has been successfully validated (i.e., “found” in the database). In this case, the miner is rewarded with newly minted Bitcoin coins.

The Problem PERFECT HASHCASH

Now, let’s define a problem PERFECT HASHCASH that we can use to evaluate the computational complexity of hashcash:

Problem: Given an input n, find all possible outputs x such that H(n) = x modulo 2^64.

In other words, given a large number n, we need to generate all possible values of x in the range [0, 2^64] using the hash function H(n). The key insight here is that h(n) can be represented as:

h(n) ≡ x (mod 2^64)

where h(n) is the hash value for input n.

Time Complexity Analysis

To analyze the time complexity of PERFECT HASHCASH, we can use the following steps:

• Calculate the hash value H(n) using the given inputs and a cryptographic hash function (e.g., SHA-256).

• Find all possible outputs x in the range [0, 2^64] that satisfy the equation h(n) = x modulo 2^64.

• Count the number of valid solutions.

The time complexity of PERFECT HASHCASH can be analyzed by considering the number of possible inputs n and the number of iterations required to find a solution for each input.

Computational Complexity Class

Using the number of possible inputs n, we can estimate the computational complexity of PERFECT HASHCASH. In general, the time complexity of hash functions is classified into two main classes: P(n) and NP(n).

• P(n) problems are those that can be solved in polynomial time by a deterministic algorithm running in O(n^d) time (where d is the depth of the problem), where n is the input size.

• NP(n) problems are those that can be verified in polynomial time using a witness (e.g., a solution to the problem). If a solution exists, it must also be verifiable.

In this case, since PERFECT HASHCASH involves finding all possible outputs x for a given input n, we can estimate its computational complexity as follows:

• If n is relatively small (e.g., < 10^8), then PERFECT HASHCASH may take O(2^n) time.

• For larger inputs, the number of iterations required to find a solution grows polynomially with the size of the input.

Conclusion

In conclusion, while we have not explicitly proven that hashcash is NP-complete or else, our analysis suggests that it may be in P(n). However, this remains an open problem and requires further investigation.

CHOOSING RIGHT NEEDS

Categorised in: CRYPTOCURRENCY

This post was written by Munna