Ethereum mining has used two mining algorithms, Dagger Hashimoto and Ethash. Dagger Hashimoto was never used to mine Ethereum, being superseded by Ethash before mainet launched. It was a R&D mining algorithm that paved the way for Ethash. However, it has historical significance as an important innovation in Ethereum's development. Proof-of-work mining itself will be deprecated in favor of proof-of-stake during The Merge, which is forecast to happen in Q3-Q4 2022.
The fundamental idea of both mining algorithms is that a miner tries to find a nonce input using brute force computation so that the result is below a certain difficulty threshold. This difficulty threshold can be dynamically adjusted, allowing block production to happen at a regular interval.
Dagger Hashimoto was a precursor research algorithm for Ethereum mining that Ethash superseded. It was an amalgamation of two different algorithms: Dagger and Hashimoto.
Dagger involves the generation of a Directed Acyclic Graph, random slices of which get hashed together. The core principle is that each nonce only requires a small portion of a large total data tree. Recomputing the subtree for each nonce is prohibitive for mining - hence the need to store the tree - but okay for a single nonce’s worth of verification. Dagger was designed to be an alternative to existing algorithms like Scrypt, which are memory-hard but difficult to verify when their memory-hardness increases to genuinely secure levels. However, Dagger was vulnerable to shared memory hardware acceleration and dropped in favor of other avenues of research.
Hashimoto is an algorithm that adds ASIC-resistance by being I/O bound (i.e. memory reads are the limiting factor in the mining process). The theory is that RAM is more available than computation; billions of dollars worth of research have already investigated optimizing RAM for different use cases, which often involve near-random access patterns (hence “random access memory”). As a result, existing RAM is likely to be moderately close to optimal for evaluating the algorithm. Hashimoto uses the blockchain as a source of data, simultaneously satisfying (1) and (3) above.
Dagger-Hashimoto used amended versions of the Dagger and Hashimoto algorithms. The difference between Dagger Hashimoto and Hashimoto is that, instead of using the blockchain as a data source, Dagger Hashimoto uses a custom-generated data set, which updates based on block data every N blocks. The data set is generated using the Dagger algorithm, allowing for efficiently calculating a subset specific to every nonce for the light client verification algorithm. The difference between Dagger Hashimoto and Dagger is that, unlike in the original Dagger, the dataset used to query the block is semi-permanent, only being updated at occasional intervals (e.g. once per week). This means that the portion of the effort of generating the dataset is close to zero, so Sergio Lerner’s arguments regarding shared memory speedups become negligible.
More on Dagger-Hashimoto.
Ethash is Ethereum's current mining algorithm. Ethash was effectively a new name given to a specific version of Dagger-Hashimoto after the algorithm got significantly updated, whilst still inheriting the fundamental principles of its predecessor. Ethereum mainnet has only ever used Ethash - Dagger Hashimoto was an R&D version of the mining algorithm that was superseded before mining started on Ethereum mainnet.
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