 Separated Boolean decomposition

Every Boolean function of distributed inputs is an exclusive or of products of locally computable bits. The displayed identity follows because exactly one indicator is one. It uses at most $2^m$ terms, each a function of $x$ times a function of $y$; exchanging the two input roles gives at most $2^n$ terms. This elementary truth-table factorization is useful for distributed computation with PR boxes and does not depend on a probabilistic approximation.