Preimage awareness proofs of two compression functions

Hash functions have to be with many security properties to meet the demands of various applications. The preimage awareness (PrA), proposed by Dodis et al., is a security property of hash function that lies between being a random oracle (RO) and collision resistance (CR). The virtue of PrA is that Merkle-Damgard (MD) paradigm is PrA preserving and one can realized a PRO hash function through composing a PRO compression function and a PrA hash function. In this paper we prove Stam´s two compression functions having PrA property, where one proof is new and another is difference from Lee et al.´s proof.