Would a simplified CBC-MAC be safe? Would using the ciphertext of the previous block as key instead of XOR:ing the block and encrypting with the normal key work?

Normally it's done like this:



What if we instead do this:



Seems like it should be more or less the same thing in the end?
I'll assume that you a prefix-free encoding (or fixed-length messages) to solve that problem of CBC-MAC. It's actually very possible that this would be secure since the proof of CBC-MAC pretty much works by showing that the input to each block is random. Thus, it's conceivable that it will be secure when using the result as a key (since it's also random). However, I say just "conceivable" because you would have to formally prove this.

I'll just note that in practice this would be a terrible idea since it would require a new key schedule for every block. That would be really inefficient (e.g., for AES).