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On Derandomized Composition of Boolean Functions

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Abstract

The (block-)composition of two Boolean functions \(f : \{0, 1\}^{m} \rightarrow \{0, 1\}, g : \{0, 1\}^{n} \rightarrow \{0, 1\}\) is the function \(f \diamond g\) that takes as inputs m strings \(x_{1}, \ldots , x_{m} \in \{0, 1\}^{n}\) and computes

$$(f \diamond g)(x_{1}, \ldots , x_{m}) = f (g(x_{1}), \ldots , g(x_{m})).$$

This operation has been used several times in the past for amplifying different hardness measures of f and g. This comes at a cost: the function \(f \diamond g\) has input length \(m \cdot n\) rather than m or n, which is a bottleneck for some applications.

In this paper, we propose to decrease this cost by “derandomizing” the composition: instead of feeding into \(f \diamond g\) independent inputs \(x_{1}, \ldots , x_{m},\) we generate \(x_{1}, \ldots , x_{m}\) using a shorter seed. We show that this idea can be realized in the particular setting of the composition of functions and universal relations (Gavinsky et al. in SIAM J Comput 46(1):114–131, 2017; Karchmer et al. in Computat Complex 5(3/4):191–204, 1995b). To this end, we provide two different techniques for achieving such a derandomization: a technique based on averaging samplers and a technique based on Reed–Solomon codes.

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Acknowledgements

We are grateful to Ronen Shaltiel for explaining to us his paper on derandomized parallel repetition (Shaltiel 2010) which served as an inspiration to this work, for pointers to the extractors’ literature, and for numerous valuable discussions. We would also like to thank anonymous referees for comments that improved the presentation of this work. This work was partially supported by the Israel Science Foundation (Grant No. 1445/16).

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Meir, O. On Derandomized Composition of Boolean Functions. comput. complex. 28, 661–708 (2019). https://doi.org/10.1007/s00037-019-00188-1

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