Recover secret SHA-1 constants given Oracle There's a black box that's basically SHA-1, except that the constants (h0, h1,...,h4) are secret. We can pass it arbitrary inputs and get the corresponding outputs.

Given this, is it possible to recover the constants used?
No.  If it were, that would demonstrate a PRF distinguisher against the SHA-1 compression function, which would be an astonishing result showing that SHA-1 is far more broken than anyone has ever seriously anticipated.