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Truncating a hash function's output should, in theory, not increase the likelihood of a collision more than you would expect based on the truncated output size. This makes sense when thinking about the the ideal hash function, a random function.

Since real-world hashes are not random functions, what properties of algorithms like sha2 make us comfortable with truncating, and what (popular) hashes exist out there that are known to have issues?

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You really should never be truncating a hash. Really you're talking about the difference between traditional hashing algorithms and cryptographic hashing algorithms. A cryptographic hash algorithm has to not be predictable, while a traditional hash algorithm doesn't have that concern. You aren't guaranteed perfect random distribution, so really you shouldn't ever be truncating a hash in any situation.

Could you specify a situation where people "feel comfortable truncating a SHA2 hash"? I don't know what you're referring to.

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    $\begingroup$ I feel comfortable truncating SHA2 as long as the truncated size is sufficient for my desired security level. NIST does so as well (consider SHA-512/256 as an example) $\endgroup$ – CodesInChaos Apr 25 '16 at 7:30
  • $\begingroup$ Then there you go. While cryptographic hashes don't create a perfect uniform distribution, they do a 'good enough' job where it's not a concern, and truncation is acceptable. $\endgroup$ – Daisetsu Apr 25 '16 at 7:33

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