Powering requires threshold depth 3

We study the circuit complexity of the powering function, defined as POWm(Z) = Zm for an n-bit integer input Z and an integer exponent m 6 poly(n). Let L̂Td denote the class of functions computable by a depth-d polynomial-size circuit of majority gates. We give a simple proof that POWm 6∈ L̂T2 for any m > 2. Specifically, we prove a 2Ω(n/ logn) lower bound on the size of any depth-2 majority circuit that computes POWm. This work generalizes Wegener’s earlier result that the squaring function (i.e., POWm for the special case m = 2) is not in L̂T2. Our depth lower bound is optimal due to Siu and Roychowdhury’s matching upper bound: POWm ∈ L̂T3. The second part of this research note presents several counterintuitive findings about the membership of arithmetic functions in the circuit classes L̂T1 and L̂T2. For example, we construct a function f (Z) such that f 6∈ L̂T1 but 5 f ∈ L̂T1. We obtain similar findings for L̂T2. This apparent brittleness of L̂T1 and L̂T2 highlights a difficulty in proving lower bounds for arithmetic functions.