( 29th July 2019 )
It should be difficult to find two different messages m1 and m2 such that hash(m1) = hash(m2). Such a pair is called a cryptographic hash collision. This property is sometimes referred to as strong collision resistance. It requires a hash value at least twice as long as that required for pre-image resistance; otherwise collisions may be found by a birthday attack.
Collision resistance implies second pre-image resistance, but does not imply pre-image resistance. The weaker assumption is always preferred in theoretical cryptography, but in practice, a hash-function which is only second pre-image resistant is considered insecure and is therefore not recommended for real applications.
Informally, these properties mean that a malicious adversary cannot replace or modify the input data without changing its digest. Thus, if two strings have the same digest, one can be very confident that they are identical. Second pre-image resistance prevents an attacker from crafting a document with the same hash as a document the attacker cannot control. Collision resistance prevents an attacker from creating two distinct documents with the same hash.
A function meeting these criteria may still have undesirable properties. Currently popular cryptographic hash functions are vulnerable to length-extension attacks: given hash(m) and len(m) but not m, by choosing a suitable m' an attacker can calculate hash (m || m') where || denotes concatenation. This property can be used to break naive authentication schemes based on hash functions. The HMAC construction works around these problems.
In practice, collision resistance is insufficient for many practical uses. In addition to collision resistance, it should be impossible for an adversary to find two messages with substantially similar digests; or to infer any useful information about the data, given only its digest. In particular, a hash function should behave as much as possible like a random function (often called a random oracle in proofs of security) while still being deterministic and efficiently computable. This rules out functions like the SWIFFT function, which can be rigorously proven to be collision resistant assuming that certain problems on ideal lattices are computationally difficult, but as a linear function, does not satisfy these additional properties.
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