We give a randomized polynomial time algorithm for polynomial identity testing for the class of n-variate poynomials of degree bounded by d over a field 𝔽, in the blackbox setting. Our algorithm works for every field 𝔽 with | 𝔽 | ≥ d+1, and uses only d log n + log (1/ ε) + O(d log log n) random bits to achieve a success probability 1 - ε for some ε > 0. In the low degree regime that is d ≪ n, it hits the information theoretic lower bound and differs from it only in the lower order terms. Previous best known algorithms achieve the number of random bits (Guruswami-Xing, CCC'14 and Bshouty, ITCS'14) that are constant factor away from our bound. Like Bshouty, we use Sidon sets for our algorithm. However, we use a new construction of Sidon sets to achieve the improved bound. We also collect two simple constructions of hitting sets with information theoretically optimal size against the class of n-variate, degree d polynomials. Our contribution is that we give new, very simple proofs for both the constructions.
@InProceedings{blaser_et_al:LIPIcs.APPROX/RANDOM.2020.8, author = {Bl\"{a}ser, Markus and Pandey, Anurag}, title = {{Polynomial Identity Testing for Low Degree Polynomials with Optimal Randomness}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {8:1--8:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.8}, URN = {urn:nbn:de:0030-drops-126112}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.8}, annote = {Keywords: Algebraic Complexity theory, Polynomial Identity Testing, Hitting Set, Pseudorandomness} }
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