Limits of Preprocessing

It is a classical result that the inner product function cannot be computed by an AC⁰ circuit. It is conjectured that this holds even if we allow arbitrary preprocessing of each of the two inputs separately. We prove this conjecture when the preprocessing of one of the inputs is limited to output n+n/(log^ω(1)n) bits and obtain a tight correlation bound. Our methods extend to many other functions, including pseudorandom functions, and imply a---weak yet nontrivial---limitation on the power of encoding inputs in low-complexity cryptography. Finally, under cryptographic assumptions, we relate the question of proving variants of the above conjecture with the question of learning AC⁰ under simple input distributions.