We construct explicit pseudorandom generators that fool n-variate polynomials of degree at most d over a finite field 𝔽_q. The seed length of our generators is O(d log n + log q), over fields of size exponential in d and characteristic at least d(d-1)+1. Previous constructions such as Bogdanov’s (STOC 2005) and Derksen and Viola’s (FOCS 2022) had either suboptimal seed length or required the field size to depend on n. Our approach follows Bogdanov’s paradigm while incorporating techniques from Lecerf’s factorization algorithm (J. Symb. Comput. 2007) and insights from the construction of Derksen and Viola regarding the role of indecomposability of polynomials.
@InProceedings{dwivedi_et_al:LIPIcs.APPROX/RANDOM.2024.44, author = {Dwivedi, Ashish and Guo, Zeyu and Volk, Ben Lee}, title = {{Optimal Pseudorandom Generators for Low-Degree Polynomials over Moderately Large Fields}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)}, pages = {44:1--44:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-348-5}, ISSN = {1868-8969}, year = {2024}, volume = {317}, editor = {Kumar, Amit and Ron-Zewi, Noga}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.44}, URN = {urn:nbn:de:0030-drops-210370}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.44}, annote = {Keywords: Pseudorandom Generators, Low Degree Polynomials} }
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