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Pseudorandom generators for combinatorial checkerboards

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Abstract

We define a combinatorial checkerboard to be a function f : {1, . . . , m}d → {1,−1} of the form \({f(u_1,\ldots,u_d)=\prod_{i=1}^df_i(u_i)}\) for some functions f i : {1, . . . , m} → {1,−1}. This is a variant of combinatorial rectangles, which can be defined in the same way but using {0, 1} instead of {1,−1}. We consider the problem of constructing explicit pseudorandom generators for combinatorial checkerboards. This is a generalization of small-bias generators, which correspond to the case m = 2. We construct a pseudorandom generator that \({\epsilon}\)-fools all combinatorial checkerboards with seed length \({O\bigl(\log m+\log d\cdot\log\log d+\log^{3/2} \frac{1}{\epsilon}\bigr)}\). Previous work by Impagliazzo, Nisan, and Wigderson implies a pseudorandom generator with seed length \({O\bigl(\log m+\log^2d+\log d\cdot\log\frac{1}{\epsilon}\bigr)}\). Our seed length is better except when \({\frac{1}{\epsilon}\geq d^{\omega(\log d)}}\).

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  1. Computer Science Division, University of California, Berkeley, Berkeley, CA, 94720, USA

    Thomas Watson

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Watson, T. Pseudorandom generators for combinatorial checkerboards. comput. complex. 22, 727–769 (2013). https://doi.org/10.1007/s00037-012-0036-6

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