Random arithmetic formulas can be reconstructed efficiently

Ankit Gupta, Neeraj Kayal, Youming Qiao

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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.

Original languageEnglish
Title of host publicationProceedings - 2013 IEEE Conference on Computational Complexity, CCC 2013
Number of pages9
StatePublished - 2013
Externally publishedYes
Event2013 IEEE Conference on Computational Complexity, CCC 2013 - Palo Alto, CA, United States
Duration: 5 Jun 20137 Jun 2013

Publication series

NameProceedings of the Annual IEEE Conference on Computational Complexity
ISSN (Print)1093-0159


Conference2013 IEEE Conference on Computational Complexity, CCC 2013
Country/TerritoryUnited States
CityPalo Alto, CA


  • arithmetic formulas
  • average case
  • reconstruction


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