Is there a checksum algorithm that also supports "subtracting" data from it?

Question

I have a system with roughly a 100 million documents, and I'd like to keep track of their modifications between mirrors. In order to exchange information about modifications effectively, I want to send information about modified documents by days, not by each separate document. Something like this:

[ 2012/03/26, cs26],
[ 2012/03/25, cs25],
[ 2012/03/24, cs24],
...

where each cs is the checksum of timestamps of all documents created on a particular day.

Now, the problem I'm running into is that I don't know of an algorithm that could "subtract" data from the checksum when a document is being deleted. None of the cryptographic hashes fit the need, for obvious reasons, and I couldn't find any algorithms for CRC that would do this.

One option I considered was to have deletes add extra information to the hash, but this would lead to even more problems, as nodes can receive delete requests in different order, and when a node would restart it would re-read all the timestamps from the documents, and thus the information about the deletes would be lost.

I also wouldn't like using a hash tree with all document hashes in-memory, as that would use roughly 8 gigs of memory, and I think it's a bit of overkill for just this need.

For now the best option seems to regenerate these hashes completely from time to time in background, but that is also a lot of needless overhead, and wouldn't provide immediate information on changes.

So, do you guys know of a checksum algorithm that would let me "remove" some data from the checksum? I need the algorithm to be somewhat fast and the checksum that would strongly indicate the smallest of changes (that's why I can't really use plain XOR).

Or maybe you have better ideas about the whole design?

Answer 1

How about

hash = X(documents, 0, function(document) { ... })

where X is an aggregate XOR (javascript-y pseudocode follows):

function X(documents, x, f)
{
   for each (var document in documents)
   {
      x ^= f(document);
   }
   return x;
}

and f() is a hash of individual document information? (whether timestamp or filename or ID or whatever)

The use of XOR would allow you to "subtract" out documents, but using a hash on a per-document basis allows you to preserve a hash-like quality of detecting small changes.