4n lower bound on the combinational complexity of certain symmetric boolean functions over the basis of unate dyadic boolean functions

A simple, and easy-to-check, property of a symmetric boolean function is shown to imply a 4n-O(1) lower bound on the circuit complexity of the function over U₂ = B₂ - {⊗, ≡}, the basis of unate dyadic boolean functions. Among the functions to which this lower bound applies are the modular functions MOD_k (n) for any fixed k ≥ 3 (MOD_k (n) is the function which returns 1 if and only if (Σ x_i) mod k = O). Finally, a 5n upper bound is obtained on the circuit complexity over U₂ of the function MOD₄ (n).