Many public-key cryptosystems have some kind of provable security. For example, the Rabin cryptosystem is provably as hard as factoring.
I wonder whether such kind of provable security exists for secret-key cryptosystems, such as AES. If not, what is the evidence that breaking such cryptosystems is hard? (other than resistance to trial-and-error attacks)
Remark: I'm familiar with AES operations (AddRoundKey, SubBytes, ShiftRows, and MixColumns). It seems that the hardness of AES stems from the MixColumns operation, which in turn must inherit its difficulty from some hard problem over Galois Fields (and thus, algebra). In fact, I can restate my question as: "Which hard algebraic problem guarantees the security of AES?"