SHA256 circuits can't do anything but SHA256. True.
But is it possible to transform a hash loop from a SHA unit into a useful mathematical process even when it isn't necessarily useful for a SHA256 process?
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I know the human brain will automatically scream no, but that automatic response may not always be the case when looked at critically and slowly.
For example, in radio and television, a (cosine? I forget the exact term at the moment) transform is used to convert a TV picture into something easier to transmit and at the end point is then transformed back into another format for display on a TV set.
You have a hashing engine that won't take anything but a SHA256 instruction. Is there any part of the Scrypt algorithm that might benefit (even marginally) from the computation process found in a SHA256 engine? The only idea that comes to mind is invoking a partial transform that is compatible with SHA256 mathematical capability and then using a general processor to wait on the result and finish the rest. I have heard that scrypt uses a simple algorithm that requires memory and high speed access to previous iteration of a result.
http://en.wikipedia.org/wiki/Scrypt#Algorithm
http://www.righto.com/2014/09/mining-bitcoin-with-pencil-and-paper.html
The analogy with the TV picture example does not work here, the way a TV signal is packed is known and all the element necessary to decode the signal are present.
However the miner does not work that way, it doesn't try to decode a signal. If we keep the TV analogy it is rather trying to find the all the pixel corresponding to an image by looking at a partially filled image and using a very incomplete signal... (or the inverse)
And the only way to do that, is to generate all the possibilities until it finds something which fits both the image and the incomplete signal.
Hopefully for you your TV receptor doesn't do that.
A sha256 miner, is just a pseudo-random generator of sha256 function using as seed the block header (trying to find a solution which match a given target)
So it isn't possible the use on scrypt because a scrypt miner is a pseudo-random generator of scrypt function using as seed the block header.
and the 2 functions have absolutely nothing in common.