Brute-force attack given small search-space and hash prefix
If you have an $m-$bit output decent hash function (such as SHA1) and you're hashing $k-$bit values, and your list $L$ of candidates is not tiny ($2^{56}$ which is the size of possible candidates, isn't) the function will behave like a random mapping on any substring of its' output. 

If your prefix consists of 8 4-bit characters, then you'll 
be filtering roughly one out of every $2^{32}$ candidates during precomputation.

Thus, at the end of precomputation, you'll have $2^{24}$ candidates left.

In a large enough list is expected that whatever the correct input is, it has shown up as part of an unknown subset of roughly $2^{32}$ inputs in your filtered candidate list. Since you only have $2^{24}$ candidates left, it's likely that most of them are actually the unique candidate.