How can I hash an input into an arbitrary domain point?
Hash functions we use, e.g. Sha-1, Sha-256, Sha-512, usually don’t have a sufficiently large range. But we can construct full domain hash via repeated application of a hash function $h$: $FDH(m) = h(m||0)||h(m||1)||\cdots  $, then take the leading n-bit. This way the hash value is deterministic and the size is arbitrary.

This is something like MGF1 defined in PKCS #1 v2.2, B.2.1.