Circuit Complexity of Powering in Fields of Odd Characteristic

By a careful analysis of the proofs of Kopparty in fields of characteristic 2, we show that the problem of powering in Fpn requires ACC(p) circuits of exponential size (in n), for any fixed prime p > 2. Similar bounds hold for quadratic residuosity. As a corollary, we obtain non-trivial bounds for exponential sums that express the correlation between the quadratic character of Fpn and n-variate polynomials over Fp of degree up to nε for some 0 < ε < 1.