I have been reading about various block cipher modes of operation, and all of them seem to treat the block cipher (including it's key and key schedule) as a sort of immutable black box.
I would like to know whether or not any one has designed a mode of operation where the effective key schedule of the cipher is changed between each block encrypted.
Suppose, purely for example, that we had a 256 bit block cipher, using a simple a 4 round Feistel structure. The expanded key, generated from a secure hash of the user-supplied key and an IV, would be 512 bits. In between each pair of blocks encrypted, we treat these 512 bits as the state of a 512-bit linear feedback shift register, advance the LFSR one step, then use the result as the scheduled key for the next block of data encrypted.
Like CTR mode, encryption and decryption are parallelizable, and it has the property of random read access.
Does anything remotely like that exist?
If not, is it because it would be obviously horribly secure in a way I'm not seeing, or for some other reason?