Fuzzy Extractor for Binary Sequence
You can use pin sketch. Consider the set of number [0,1,2,3,4,5,6,7]. The presence of a number can represent the presence of a 1 at that bit position in an 8 bit number.

E.G.

[0,2,4,6] == 10101010 == 0x55

[0,1,2,3] == 11110000 == 0xF0

Extend to a useful size (like 2040 bits which maps nicely to a BCH code) and you are comparing sets that have on average 1020 unique values, each showing the positions of the ones.