.NET flavour (52506 bytes)
Note: If you opened the regex within a few minutes of me posting this answer, I've changed the regex due to a last-minute bug. Sorry.
Subset sum, deluxe edition.
Regex here, expanded version here, tested on RegExLib and Regex Hero
Surely with these numbers there's a high chance that there's a unique solution, right? (statistically speaking)
Edit: Now that I think about it: 2256 is a lot of subsets, so that's probably false.
Match:
1000010001000000001101011000001101110101001010011101000101010011011101000001010101001000010010000111011101100101001101001111000111010101100000101000101010110001010101001100100001110010001101010101100010110011000000110110000000011111101000001000011111100010
Mismatch:Huzzah for NP-complete regexes
This regex is one giant subset sum problem, and uses 16 groups to store data. Each1
in the string represents 16 10-bit numbers, which together represent a 160-bit integer. The last few lines of the regex carry the values in the groups so that groups 2-16 go up to 1023 (e.g. 11023 + 1024 becomes 21023 + 1), as otherwise we'd only be solving 16 simultaneous mini subset sum problems as opposed to one big one.