You can perform this operation without a database. The key is to reduce the size of A, since A is much larger than B. Here is how to do this:
Calculate 64-bit hashes using a decent hash function for the strings in the B file. Store these in memory (in a hash table), which you can do because B is small. Then hash all of the strings in your A file, line by line, and see if each one matches a hash for your B file. Any lines with matching hashes (to one from B), should be stored in a file C.
When this process is complete file C will have the small subset of A of potentially matching strings (to B). Now you have a much smaller file C that you need to compare lines of B with. This reduces the problem to a problem where you can actually load all of the lines from C into memory (as a hash table) and compare each line of B to see if it is in C.