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oai:drops-oai.dagstuhl.de:19719 2024-03-11T06:37:03Z ddc:004 open_access

Testing Equivalence to Design Polynomials Baraskar, Omkar Dewan, Agrim Saha, Chandan Polynomial equivalence design polynomials graph isomorphism vector space decomposition An n-variate polynomial g of degree d is a (n,d,t) design polynomial if the degree of the gcd of every pair of monomials of g is at most t-1. The power symmetric polynomial PSym_{n,d} : = ∑_{i = 1}ⁿ x^d_i and the sum-product polynomial SP_{s,d} : = ∑_{i = 1}^{s}∏_{j = 1}^{d} x_{i,j} are instances of design polynomials for t = 1. Another example is the Nisan-Wigderson design polynomial NW, which has been used extensively to prove various arithmetic circuit lower bounds. Given black-box access to an n-variate, degree-d polynomial f(𝐱) ∈ 𝔽[𝐱], how fast can we check if there exist an A ∈ GL(n, 𝔽) and a 𝐛 ∈ 𝔽ⁿ such that f(A𝐱+𝐛) is a (n,d,t) design polynomial? We call this problem "testing equivalence to design polynomials", or alternatively, "equivalence testing for design polynomials". In this work, we present a randomized algorithm that finds (A, 𝐛) such that f(A𝐱+𝐛) is a (n,d,t) design polynomial, if such A and 𝐛 exist, provided t ≤ d/3. The algorithm runs in (nd)^O(t) time and works over any sufficiently large 𝔽 of characteristic 0 or > d. As applications of this test, we show two results - one is structural and the other is algorithmic. The structural result establishes a polynomial-time equivalence between the graph isomorphism problem and the polynomial equivalence problem for design polynomials. The algorithmic result implies that Patarin’s scheme (EUROCRYPT 1996) can be broken in quasi-polynomial time if a random sparse polynomial is used in the key generation phase. We also give an efficient learning algorithm for n-variate random affine projections of multilinear degree-d design polynomials, provided n ≥ d⁴. If one obtains an analogous result under the weaker assumption "n ≥ d^ε, for any ε > 0", then the NW family is not VNP-complete unless there is a VNP-complete family whose random affine projections are learnable. It is not known if random affine projections of the permanent are learnable. The above algorithms are obtained by using the vector space decomposition framework, introduced by Kayal and Saha (STOC 2019) and Garg, Kayal and Saha (FOCS 2020), for learning non-degenerate arithmetic circuits. A key technical difference between the analysis in the papers by Garg, Kayal and Saha (FOCS 2020) and Bhargava, Garg, Kayal and Saha (RANDOM 2022) and the analysis here is that a certain adjoint algebra, which turned out to be trivial (i.e., diagonalizable) in prior works, is non-trivial in our case. However, we show that the adjoint arising here is triangularizable which then helps in carrying out the vector space decomposition step. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Omkar Baraskar and Agrim Dewan and Chandan Saha 2024 Is Part Of LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.STACS.2024.9 urn:nbn:de:0030-drops-197193 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.9 eng https://creativecommons.org/licenses/by/4.0/legalcode