Would it make sense to have some partial/incremental SHA256 implementation? E.g. one that gives the first character only (in ASCII representation). So that you could have a cheaper money miner to narrow down the search space?
Is it even possible to have cheaper first character SHA256 algorithm?
SHA256 doesn't generate characters, it generates bytes that are usually rendered in hex form, which gives you the "first character".
There isn't a known way to calculate just the first bytes of a SHA256 hash faster. Each step in SHA256 depends on the value of the previous step, so you have to calculate the whole thing through. SHA256 is meant for cryptography and would be considered vulnerable to attacks if a shortcut were known.
Well, such an algorithm existing would be a massive breakthrough in cryptography research. The answer should be no since that make it easier to do a preimage attack than brute-force.
No. Hash functions are specifically designed to make any shortcuts impossible. Essentially, they consist of many steps of scrambling the input bits, each step taking as its input the output of the previous one.
If any such shortcut would be found, it would be considered a vulnerability, and the use of SHA256 for anything involving cryptography would be strongly discouraged.
Is it even possible to have cheaper first character SHA256 algorithm?