Discrete Algebraic Methods

Here, ⊕ is the bitwise exclusive-or. All these mappings can easily be computed (assuming that this is the case for ck), but they are difficult to invert if we have a good encryption function. More robust statements are usually not available. On the one hand, this is because such statements about the security of encryption functions are not known; however on the other hand, it is also important how ⊕ and ck behave relative to each other. If, for example, ck(x) = x ⊕ k, then all of the above compression functions are easy to invert.

Merkle–Damgård construction

We now assume that we have a collision resistant compression function g : 2n →n; for example, ...