Separated Boolean decomposition ($f(x,y)={⨁}_{\xi \in \{0,1{\}}^m}{1}_{\{x=\xi \}}f(\xi ,y)$)

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.