On the degree of univariate polynomials over the integers
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
ASIACRYPT'11 Proceedings of the 17th international conference on The Theory and Application of Cryptology and Information Security
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We study the following question What is the smallest t such that every symmetric boolean function on κ variables (which is not a constant or a parity function), has a non-zero Fourier coefficient of order at least 1 and at most t? We exclude the constant functions for which there is no such t and the parity functions for which t has to be κ. Let τ (κ) be the smallest such t. Our main result is that for large κ, τ (κ)≤4κ/logκ. The motivation for our work is to understand the complexity of learning symmetric juntas. A κ-junta is a boolean function of n variables that depends only on an unknown subset of κ variables. A symmetric κ-junta is a junta that is symmetric in the variables it depends on. Our result implies an algorithm to learn the class of symmetric κ-juntas, in the uniform PAC learning model, in time n o(κ) . This improves on a result of Mossel, O’Donnell and Servedio in [16], who show that symmetric κ-juntas can be learned in time n 2κ/3.