In [[cryptography]], we can divide [[cryptographic hash functions]] can be divided into two main categories. In the first category are those functions, where their design is based on a mathematical problem and thus their security follows rigorous mathematical proves, [[Computational complexity theory|complexity theory]] and [[Reduction (complexity)|formal reduction]]. These functions are called '''Provably Secure Cryptographic Hash Functions'''. However this does not mean that such a function could not be broken. To construct them is very difficult and only a few examples were introduced. The practical use is limited.
In the second category are functions that are not based on mathematical problems but on an ad hoc basis, where the bits of the message are nicely mixed to produce the hash. They are then believed to be hard to break, but no such formal proof is given. Almost all widely-spread hash functions fall in this category. Some of these functions are already broken and are no longer in use.
