Raz-McKenzie simulation with the inner product gadget

In this note we show that the Raz-McKenzie simulation algorithm which lifts deterministic query lower bounds to deterministic communication lower bounds can be implemented for functions f composed with the Inner Product gadget 1ip(x, y) = ∑ i xiyi mod 2 of logarithmic size. In other words, given a function f : {0, 1}n → {0, 1} with deterministic query complexity D( f ), we show that the deterministic communication complexity of the composed function f ◦ 1ip is Θ(D( f ) log n), where f ◦ 1ip(x, y) = f (1ip(x, y1), . . . , 1ip(x, yn)) where x = (x1, . . . , xn), y = (y1, . . . , yn) and each xi and yi are O(log n) bit strings. In [RM97] and [GPW15], the simulation algorithm is implemented for functions composed with the Indexing gadget, where the size of the gadget is polynomial in the input length of the outer function f .