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If I have 256 bits of handwavium "perfectly random data" and I hash this 256 bits of data with a secure hash function (possibly sha256) could the resulting hash be considered "perfectly random data" as well? I am assuming no, but don't know why.
What information / keywords would I use to find out more information about this?
Interesting seemingly related topics:
Suppose you take your 256 bits of handwavium "perfectly random data" and use it to seed a handwavium "perfect cryptographic seed expansion algorithm", such as the SHAKE XOF -- but it really makes no difference which algorithm you choose. You then generate 257 (or in general any $n > 256$) bits of "random data". In principle, if this was really random data, you'd have $2^n$, for $n > 256$, different bit strings.
However, suppose you cycled through all $2^{256}$ bit strings of "only" 256 bits. Evidently one of these would match the 256 bits of handwavium "perfectly random data" mentioned at the beginning, and having found it, then you'd be able to generate exactly the same $n$ supposedly random bits.
Thus, by brute-forcing "only" 256 bits (a computational effort of $2^{256}$ operations), you can find all $n$ bits, while if they were actually random, you'd need an actual computational effort of $2^n$ (for, again, $n > 256$), which is general is much greater than $2^{256}$.
So don't fool yourself: you only have 256 bits of randomness. Any bit coming after that is a 100% deterministic function of those 256 bits, and thus adds no randomness at all.
