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The three pillars of PoW
Asymmetry
Easy to verify, hard to compute
Puzzle/Challenge
Have to solve a very challenging(computationally) problem
Nonce
A value used to vary inputs until the puzzle solution meets a requirement.
Common Puzzles/Challenges for PoW
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Bitcoin's SHA256: Find a nonce such that
hash(block + nonce)starts with a specific number of zeros. This is usually only used for blockchains, hence there is the block. However, this could be used for other applications as well. -
ProgPoW: Heavily GPU and CPU intensive, as it creates a psuedo-random program that does several seemingly random operations using the nonce to generate a number that one can "mine" in order to get.
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RandomX (Used by Monero) Is heavily CPU-dependent, as it doesn't translate onto the GPU that well, and it works by using Just-In-Time(JIT) to create another psuedo-random program that can be mined as well. However, this uses all kinds of operations, like floating point, AES, and memory loads.