Persistence of quadratic-circuit lower bounds for general vector-valued Boolean functions

Determine whether quadratic-circuit lower bounds for vector-valued quadratic Boolean functions over finite fields, including , remain valid when computation is performed by unrestricted XOR--AND circuits with arbitrary nonlinear intermediate-wire reuse.

Background

The paper distinguishes quadratic Boolean circuits, in which every gate computes a Boolean function of algebraic degree at most two, from unrestricted Boolean circuits, which may multiply arbitrary XOR combinations of previously computed nonlinear wires. Boyar and Find formulate the broader question of whether quadratic circuits are always optimal for vector-valued quadratic Boolean functions; the paper notes that this is unresolved in the general (n,m) setting.

The paper resolves a particular instance of this broader problem by proving that four-term binary polynomial multiplication has unrestricted XOR--AND multiplicative complexity nine, matching the known quadratic lower bound. The general question for arbitrary vector-valued quadratic Boolean functions remains open.