Let's say I want to rewrite the software behind the hardware of current asics. Would it still mine, for example, sha1 algorithms orders of magnitude faster than a normal cpu would?
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What do you mean by "the software behind the hardware"? The hardware cannot be changed, it is designed specifically for SHA256 – MeshCollider♦ Jun 19 '18 at 12:16
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Application Specific Integrated Circuits are hard set to only do one thing, and in the case of bitcoin ASICs, that's SHA256 hashes. You cannot re-write the software, you would need a new circuit built that only does SHA1 or whichever hash you're looking for.
https://en.wikipedia.org/wiki/Application-specific_integrated_circuit