Abstract
Informally stated, we present here a randomized algorithm that given black box access to the polynomial Z_$f$_Z computed by an unknown/hidden arithmetic formula Z_$\phi$_Z? reconstructs, on the average, an equivalent or smaller formula ?Z_$\psi$_Z in time polynomial in the size of its output ?Z_$\psi$_Z. Specifically, we consider arithmetic formulas wherein the underlying tree is a complete binary tree, the leaf nodes are labelled by a ffine forms (i.e. degree one polynomials) over the input variables and where the internal nodes consist of alternating layers of addition and multiplication gates. We call these alternating normal form (ANF) formulas. If a polynomial Z_$f$_Z can be computed by an arithmetic formula of size Z_$s$_Z, it can also be computed by an ANF formula?, possibly of slightly larger size Z_$s^O(1)$_Z. Our algorithm gets as input black box access to the output polynomial Z_$f$_Z (i.e. for any point Z_$x$_Z in the domain, it can query the black box and obtain Z_$f(x)$_Z in one step) of a random ANF formula of size Z_$s$_Z (wherein the coefficients of the a ffine forms in the leaf nodes of are chosen independently and uniformly at random from a large enough subset of the underlying field). With high probability (over the choice of coefficients in the leaf nodes), the algorithm efficiently (i.e. in time Z_$s^O(1)$_Z) computes an ANF formula ?of size Z_$s$_Z computing Z_$f$_Z. This then is the strongest model of arithmetic computation for which a reconstruction algorithm is presently known, albeit efficient in a distributional sense rather than in the worst case.