The correlation between parity and quadratic polynomials mod 3

We prove exponentially small upper bounds on the correlation between parity and quadratic polynomials mod 3. One corollary of this is that in order to compute parity, circuits consisting of a threshold gate at the top, mod 3 gates in the middle, and AND gates of fan-in two at the inputs must be of size 2^Ω(n). This is the first result of this type for general mod subcircuits with ANDs of fan-in greater than 1. This yields an exponential improvement over a recent result of Alon and Beigel (2001). The proof uses a novel inductive estimate of the relevant exponential sums introduced by Cai et al. (1996). The exponential sum bounds are tight