What is the entropy of XORed CSPRNG bytes with low entropy hash?

Question

Let’s say I take 256 bits from a CSPRNG and assume it is perfectly 256 bits of entropy. Call this rand.

Then let’s say I take the sha256 of the ASCII text “password”. Call this hash.

Now we XOR rand and hash. Call this mixed.

Is the entropy of mixed less than that of rand?

If so, is there a formula for calculating its entropy?

Example below: What is the entropy of mixed as a function of rand and weak_hash

#!/usr/bin/python3
import hashlib, os

def main():
    rand = int(os.urandom(32).hex(),16)
    weak_hash = int(hashlib.sha256(b'password').digest().hex(),16)
    mixed = ("%064x" % (rand ^ weak_hash))
    print(mixed)

main()

Answer 1

You are describing a one-time-pad. If the key stream: the output of the CSPRNG is fully random then the ciphertext will be indistinguishable from random as well.

Of course the output of CSPRNG is not fully random. However, if the CSPRNG is well seeded with enough entropy then you'd have the same security as a stream cipher, which mimics a one time pad.

So the output (mixed) will be as random as the CSPRNG, as long as the CSPRNG doesn't get into a previously encountered state. That should basically only happen if the entropy source fails.