Abstract
It is a classical result that the inner product function cannot be computed by an \({\rm AC}^0\) circuit. It is conjectured that this holds even if we allow arbitrary preprocessing of each of the two inputs separately. We prove this conjecture when the preprocessing of one of the inputs is limited to output \(n + n/(\log^{\omega(1)}n)\) bits and obtain a tight correlation bound. Our methods extend to many other functions, including pseudorandom functions, and imply a---weak yet nontrivial---limitation on the power of encoding inputs in low-complexity cryptography. Finally, under cryptographic assumptions, we relate the question of proving variants of the above conjecture with the question of learning \({\rm AC}^0\) under simple input distributions.
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Acknowledgements
We thank Andrej Bogdanov, Mika Göös, Sajin Koroth, Dinesh Krishnamoorthy, Srikanth Srinivasan, Avishay Tal, and anonymous reviewers for helpful discussions, comments, and pointers. A previous version of this paper has appeared as Filmus et al.(2020). This research was supported by the European Union's Horizon 2020 research and innovation programme under grant agreement No 802020-ERC-HARMONIC, ERC Project NTSC (742754), NSF-BSF grant 2015782, and BSF grant 2018393.
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Filmus, Y., Ishai, Y., Kaplan, A. et al. Limits of Preprocessing. comput. complex. 33, 5 (2024). https://doi.org/10.1007/s00037-024-00251-6
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DOI: https://doi.org/10.1007/s00037-024-00251-6
Keywords
- Circuit complexity
- communication complexity
- IPPP
- pseudorandom function
- simultaneous messages
- constant-depth circuit