TY - GEN

T1 - Random arithmetic formulas can be reconstructed efficiently

AU - Gupta, Ankit

AU - Kayal, Neeraj

AU - Qiao, Youming

PY - 2013

Y1 - 2013

N2 - Informally stated, we present here a randomized algorithm that given blackbo× access to the polynomial f computed by an unknown/hidden arithmetic formula ℙ reconstructs, on average, an equivalent or smaller formula ℙ in time polynomial in the size of its output ℙ. Specifically, we consider arithmetic formulas wherein the underlying tree is a complete binary tree, the leaf nodes are labelled by affine 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 f can be computed by an arithmetic formula μ of size s, it can also be computed by an ANF formula ℙ, possibly of slightly larger size sO(1). Our algorithm gets as input blackbo× access to the output polynomial f (i.e. for any point × in the domain, it can query the blackbo× and obtain f(×) in one step) of a random ANF formula f of size s (wherein the coefficients of the affine forms in the leaf nodes of f 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 sO(1)) computes an ANF formula ℙ of size s computing f. 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.

AB - Informally stated, we present here a randomized algorithm that given blackbo× access to the polynomial f computed by an unknown/hidden arithmetic formula ℙ reconstructs, on average, an equivalent or smaller formula ℙ in time polynomial in the size of its output ℙ. Specifically, we consider arithmetic formulas wherein the underlying tree is a complete binary tree, the leaf nodes are labelled by affine 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 f can be computed by an arithmetic formula μ of size s, it can also be computed by an ANF formula ℙ, possibly of slightly larger size sO(1). Our algorithm gets as input blackbo× access to the output polynomial f (i.e. for any point × in the domain, it can query the blackbo× and obtain f(×) in one step) of a random ANF formula f of size s (wherein the coefficients of the affine forms in the leaf nodes of f 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 sO(1)) computes an ANF formula ℙ of size s computing f. 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.

KW - arithmetic formulas

KW - average case

KW - reconstruction

UR - http://www.scopus.com/inward/record.url?scp=84885675229&partnerID=8YFLogxK

U2 - 10.1109/CCC.2013.10

DO - 10.1109/CCC.2013.10

M3 - ???researchoutput.researchoutputtypes.contributiontobookanthology.conference???

AN - SCOPUS:84885675229

SN - 9780769549972

T3 - Proceedings of the Annual IEEE Conference on Computational Complexity

SP - 1

EP - 9

BT - Proceedings - 2013 IEEE Conference on Computational Complexity, CCC 2013

T2 - 2013 IEEE Conference on Computational Complexity, CCC 2013

Y2 - 5 June 2013 through 7 June 2013

ER -