Note: I'm looking at this question after this edit was made, and taking it into account:
Note: I specifically mean the multi-round password hashes described by the linked documents and marked with the codes $5$ and $6$ in crypt hashes, not a single round of the plain SHA256 or SHA512 hash functions.
Looking at the long, 22-step algorithm in this link you provided, I'd rather flip a question around: why would you prefer to use this instead of PBKDF2 with HMAC-SHA2? Because, at least as presented:
- The definition of PBKDF2 looks much simpler. This is because it is more modular—it defers most of its work to an externally-supplied pseudo-random function. This is normally instantiated with HMAC, which in turn defers most of its work to an external hash function like SHA-1 or SHA-2.
- This means that the security of PBKDF2 should be easier to analyze.
In contrast, the algorithm in the document you provide lists a ton of steps whose motivation is harder to understand. For example:
11. For each bit of the binary representation of the length of the
password string up to and including the highest 1-digit, starting
from to lowest bit position (numeric value 1):
a) for a 1-digit add digest B to digest A
b) for a 0-digit add the password string
NB: this step differs significantly from the MD5 algorithm. It
adds more randomness.
It adds more randomness? How does it do this? Why does it this step exist at all—is SHA-2 not adding sufficient randomness? If SHA-2 isn't random enough, why use it in the first place? And doesn't this step introduce secret-dependent branching into the algorithm, raising the question of possible timing attacks against it?
I'm not by any means saying that the algorithms you link are insecure. It's just that:
- The work factor they introduce comes down to the same thing that PBDKF2--HMAC-SHA2 would do (a large number of SHA2 iterations);
- They look very broadly similar to what you'd have if you unrolled a PBKDF2-HMAC-SHA2 implementation, but with additional complexity whose purpose I don't understand;
- So at least as presented in those documents, I find it harder to gain confidence on their design than I do for PBKDF2.