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4

Yes, it has happened. If you look at the SHA3 hash zoo, there are a number of hashes who has the best attack listed as "2nd preimage". One general place this can occur is if you have a hash function with a weak message compression step, but a fairly strong finalization step. Here, we might not be able to generate first preimages (because we don't know what ...

3

While collision resistance can be defined for normal hash functions like SHA1, for target collision resistance you need a so called keyed hash function, that is a hash function that additionally to a message $m$ also takes a key $k$. The simplest way to construct a keyed hash function out of a regular one is to prepend the key in front of the message: ...

1

The current status as of the time I write this is: There are no known attacks on second pre-images for truncated SHA-256 that are faster than brute force.

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Preliminary: Almost the same article is available for free without breaking any law, nor downloading 5GB (formatting is shifted by at most one third of a page). It is also (as well as all other articles of IACR crypto conferences from 2000-2011) in the IACR Online Proceedings, specifically in the FSE 2008 section, but then you need to subtract about 223 from ...

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