|Abstrakt:||Cryptographic hash functions are corner-stones of current cryptography. Recently, NIST (National Institute for Standards and Technology) has announced a public competition to develop a new hash standard called AHS (Advanced Hash Standard). We summarize the basic properties that the new cryptographic hash standard should preserve, give formal definitions of them and work out all of the implications or separations among these definitions. Some of the implications/separations have been proven before, others appear to be new. We provide two types of the implication and separation, conventional and provisional. While the conventional implication (separation) carries the usual semantics of the word implication (separation), the strength of the provisional implication or separation depends on a particular hash function. We show that a property pseudo-random oracle introduced by Coron, Dodis, Malinaud and Puniya is (as expected) the strongest one, since it implies almost all of the other properties. We also discuss the practical use of the pseudo-random oracle and multi-property preserving transforms introduced by Bellare and Ristenpart.