My first thought was to use linear transformations over Z_2 as a field, as that would create a natural interpretation of fixing certain bits as taking a linear subspace. Interestingly this leads to the property that XOR is preserved.
I implemented this in a very quick and hacky way for 32 bits. I generated a random boolean matrix M invertible in Z_2. To turn an input number x into a corresponding number y in an N-bit space, I convert x to binary and turn that into a vector of 1s and 0s, then multiply it by that randomized matrix to get y. Here are the first few y's corresponding to x=0, 1, ...:
00000000000000000000000000000000
0xx0x000xx00x0xx000xxx00xxx0x000
x0xx000xxxx0xxx00x000xxx0xx0xx0x
xx0xx00x00x00x0x0x0xx0xxx0000x0x
x0xx0x00xxx0xx0x00xx0x0xx0xx00x0
xx0xxx0000x00xx000x0x00x0x0xx0x0
00000x0x000000xx0xxx00x0xx0xxxxx
0xx0xx0xxx00x0000xx0xxx000xx0xxx
...
(Hoping the non-monospace font doesn't ruin the alignment too much.)
Which looks... random-ish? I expect that turning these into UUIDs may result in more random-looking sequences.
Not sure how to solve the problem of search with this, but the hope would be that the linear structure gives you what you need, since fixing bytes on UUIDs should correspond to considering linear subspaces of the y vectors. Perhaps this can also be used to apply lexicographic order on the corresponding x vectors (i.e.: ordering the indexes), so that you could jump to each UUID matching the search in order.
I implemented this in a very quick and hacky way for 32 bits. I generated a random boolean matrix M invertible in Z_2. To turn an input number x into a corresponding number y in an N-bit space, I convert x to binary and turn that into a vector of 1s and 0s, then multiply it by that randomized matrix to get y. Here are the first few y's corresponding to x=0, 1, ...:
00000000000000000000000000000000
0xx0x000xx00x0xx000xxx00xxx0x000
x0xx000xxxx0xxx00x000xxx0xx0xx0x
xx0xx00x00x00x0x0x0xx0xxx0000x0x
x0xx0x00xxx0xx0x00xx0x0xx0xx00x0
xx0xxx0000x00xx000x0x00x0x0xx0x0
00000x0x000000xx0xxx00x0xx0xxxxx
0xx0xx0xxx00x0000xx0xxx000xx0xxx
...
(Hoping the non-monospace font doesn't ruin the alignment too much.)
Which looks... random-ish? I expect that turning these into UUIDs may result in more random-looking sequences.
Not sure how to solve the problem of search with this, but the hope would be that the linear structure gives you what you need, since fixing bytes on UUIDs should correspond to considering linear subspaces of the y vectors. Perhaps this can also be used to apply lexicographic order on the corresponding x vectors (i.e.: ordering the indexes), so that you could jump to each UUID matching the search in order.