Are fragments of hashes collision-resistent?

Question

If you only use the first 4 bytes of an MD5 hash, would that mean theoretically only 1 in 255^4 chance of collision? That is, are hashes designed such that you only have to use a small portion of the returned hash (say the hash is of a file of some size)?

Answer 1

Remember that, even without considering a smart attacker deliberately trying to cause collisions, you need to start worrying about accidental collisions once the number of objects you're hashing get comparable to the square root of the hash space... just a few tens of thousands of objects for a 32-bit hash key. This comes from the so-called birthday paradox.

Answer 2

It is 256, not 255.

Assuming that MD5 is a secure hash function (it turns out it is not secure, but, for the sake of the discussion, let's suppose that it is secure), then it should behave like a random oracle, a mythical object which outputs uniformly random values, under the sole constraint that it "remembers" its previous outputs and returns the same value again, given the same input.

Truncating the output of a random oracle yields another random oracle. Thus, if you keep 32 bits, then the probability of a collision with two distinct input messages is 1 in 2^32 (i.e. 1 in 256^4).

Now there is a thing known as the birthday paradox which says that, with about 2^16 distinct inputs, there are good chances that two of the 2^16 corresponding outputs collide.

MD5 has been shown to be insecure for some purposes -- in particular anything which is related to collisions. The current default recommendation is SHA-2 (a family of four functions, with output sizes 224, 256, 384 and 512 bits, respectively). A new (american) standard is currently being defined, through an open competition, under the code name SHA-3. This is a long process; the new function shall be chosen by mid-2012. Some of the remaining candidates (currently 14, out of an initial 51) are substantially faster than SHA-2, some approaching MD5 in performance, while being considerably more secure. But this is a bit new, so right now you shall use SHA-2 by default.