I'm reading FSB cryptographic hash function and the authors say that security of this function depends on NP-Complete Problem: "Decoding Linear Code Unlike most other cryptographic hash functions in use today" and I'm agree with the first part of claim. But I have a doubt respect to conventional hash functions for example the family SHA. The security of these functions depend of any NP-Complete Problem? According my understanding this hash function will be able to express than multivariate polynomials equations and solve multivariate polynomial equations is proved NP-Complete. Then a question is: Am I right?
That Wikipedia article is full of errors and false claims. Most importantly, FSB has not been proven to be as hard as an NP-complete problem. This is because the syndrome decoding problem is NP-hard in the worst case, but FSB uses random instances of the problem. Indeed, these random instances may be much easier to break than arbitrary instances. There is no guarantee that these random instances are actually hard, nor do we know how to generate NP-hard instances of decoding problems.
More generally, there is no known cryptographic primitive (of any kind) whose security is provably based on NP-hardness alone.