TY - JOUR

T1 - Random arithmetic formulas can be reconstructed efficiently

AU - Gupta, Ankit

AU - Kayal, Neeraj

AU - Qiao, Youming

N1 - Funding Information:
Y. Q. is partially funded by the Singapore Ministry of Education and the National Research Foundation, and by the Tier 3 Grant “Random numbers from quantum processes.”

PY - 2014/5

Y1 - 2014/5

N2 - Informally stated, we present here a randomized algorithm that given black-box access to the polynomial f computed by an unknown/hidden arithmetic formula φ reconstructs, on the average, an equivalent or smaller formula φ̌ in time polynomial in the size of its output φ̌. Specifically, we consider arithmetic formulas wherein the underlying tree is a complete binary tree, the leaf nodes are labeled 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 φ, possibly of slightly larger size s O(1). Our algorithm gets as input black-box access to the output polynomial f (i.e., for any point x in the domain, it can query the black box and obtain f(x) in one step) of a random ANF formula φ of size s (wherein the coefficients of the affine forms in the leaf nodes of φ 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 s O(1)) computes an ANF formula φ̌ 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 black-box access to the polynomial f computed by an unknown/hidden arithmetic formula φ reconstructs, on the average, an equivalent or smaller formula φ̌ in time polynomial in the size of its output φ̌. Specifically, we consider arithmetic formulas wherein the underlying tree is a complete binary tree, the leaf nodes are labeled 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 φ, possibly of slightly larger size s O(1). Our algorithm gets as input black-box access to the output polynomial f (i.e., for any point x in the domain, it can query the black box and obtain f(x) in one step) of a random ANF formula φ of size s (wherein the coefficients of the affine forms in the leaf nodes of φ 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 s O(1)) computes an ANF formula φ̌ 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=84901635455&partnerID=8YFLogxK

U2 - 10.1007/s00037-014-0085-0

DO - 10.1007/s00037-014-0085-0

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AN - SCOPUS:84901635455

SN - 1016-3328

VL - 23

SP - 207

EP - 303

JO - Computational Complexity

JF - Computational Complexity

IS - 2

ER -