Hamming distance on binary strings in SQL

I have a table in my DB where I store SHA256 hashes in a BINARY(32) column. I'm looking for a way to compute the Hamming distance of the entries in the column to a supplied value, i.e. something like:

SELECT * FROM table 
  ORDER BY HAMMINGDISTANCE(hash, UNHEX(<insert supplied sha256 hash here>)) ASC 
  LIMIT 10

(in case you're wondering, the Hamming distance of strings A and B is defined as BIT_COUNT(A^B), where ^ is the bitwise XOR operator and BIT_COUNT returns the number of 1s in the binary string).

Now, I know that both the ^ operator and BIT_COUNT function only work on INTEGERs and so I'd say that probably the only way to do it would be to break up the binary strings in substrings, cast each binary substring to integer, compute the Hamming distance substring-wise and then add them. The problem with this is that it sounds terribly complicated, not efficient and definitely not elegant. My question therefore is: could you suggest any better way? (please note that I'm on shared hosting and therefore I can't modify the DB server or load libraries)

edit(1): Obviously loading the whole table in PHP and doing the computations there would be possible but I'd rather avoid it because this table will probably grow quite large.

edit(2): The DB server is MySQL 5.1

edit(3): My answer below contains the code that I just described above.

edit(4): I just found out that using 4 BIGINTs to store the hash instead of a BINARY(32) yields massive speed improvements (more than 100 times faster). See the comments to my answer below.

It appears that storing the data in a BINARY column is an approach bound to perform poorly. The only fast way to get decent performance is to split the content of the BINARY column in multiple BIGINT columns, each containing an 8-byte substring of the original data.

In my case (32 bytes) this would mean using 4 BIGINT columns and using this function:

CREATE FUNCTION HAMMINGDISTANCE(
  A0 BIGINT, A1 BIGINT, A2 BIGINT, A3 BIGINT, 
  B0 BIGINT, B1 BIGINT, B2 BIGINT, B3 BIGINT
)
RETURNS INT DETERMINISTIC
RETURN 
  BIT_COUNT(A0 ^ B0) +
  BIT_COUNT(A1 ^ B1) +
  BIT_COUNT(A2 ^ B2) +
  BIT_COUNT(A3 ^ B3);

Using this approach, in my testing, is over 100 times faster than using the BINARY approach.

FWIW, this is the code I was hinting at while explaining the problem. Better ways to accomplish the same thing are welcome (I especially don't like the binary > hex > decimal conversions):

CREATE FUNCTION HAMMINGDISTANCE(A BINARY(32), B BINARY(32))
RETURNS INT DETERMINISTIC
RETURN 
  BIT_COUNT(
    CONV(HEX(SUBSTRING(A, 1,  8)), 16, 10) ^ 
    CONV(HEX(SUBSTRING(B, 1,  8)), 16, 10)
  ) +
  BIT_COUNT(
    CONV(HEX(SUBSTRING(A, 9,  8)), 16, 10) ^ 
    CONV(HEX(SUBSTRING(B, 9,  8)), 16, 10)
  ) +
  BIT_COUNT(
    CONV(HEX(SUBSTRING(A, 17, 8)), 16, 10) ^ 
    CONV(HEX(SUBSTRING(B, 17, 8)), 16, 10)
  ) +
  BIT_COUNT(
    CONV(HEX(SUBSTRING(A, 25, 8)), 16, 10) ^ 
    CONV(HEX(SUBSTRING(B, 25, 8)), 16, 10)
  );

Interesting question, I've found a way to do this for a binary(3) that might work as well for a binary(32):

drop table if exists BinaryTest;
create table  BinaryTest (hash binary(3));
insert BinaryTest values (0xAAAAAA);

set @supplied = cast(0x888888 as binary);

select  length(replace(concat(
            bin(ascii(substr(hash,1,1)) ^ ascii(substr(@supplied,1,1))),
            bin(ascii(substr(hash,2,1)) ^ ascii(substr(@supplied,2,1))),
            bin(ascii(substr(hash,3,1)) ^ ascii(substr(@supplied,3,1)))
        ),'0',''))
from    BinaryTest;

The replace removes any all zeroes, and the length of remainder is the number of ones. (The conversion to binary omits leading zeroes, so counting the zeroes would not work.)

This prints 6, which matches the number of ones in

0xAAAAAA ^ 0x888888 = 0x222222 = 0b1000100010001000100010

Hamming distance on binary strings in SQL

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