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A long line of work in the past two decades or so established close connections between several different pseudorandom objects and applications, including seeded or seedless non-malleable extractors, two source extractors, (bipartite) Ramsey graphs, privacy amplification protocols with an active adversary, non-malleable codes and many more. These connections essentially show that an asymptotically optimal construction of one central object will lead to asymptotically optimal solutions to all the others. However, despite considerable effort, previous works can get close but still lack one final step to achieve truly asymptotically optimal constructions. In this paper we provide the last missing link, thus simultaneously achieving explicit, asymptotically optimal constructions and solutions for various well studied extractors and applications, that have been the subjects of long lines of research. Our results include: 1. Asymptotically optimal seeded non-malleable extractors, which in turn give two source extractors for asymptotically optimal min-entropy of $$O(\log n)$$, explicit constructions of $$K$$-Ramsey graphs on $$N$$ vertices with $$K=\log^{O(1)} N$$, and truly optimal privacy amplification protocols with an active adversary. 2. Two source non-malleable extractors and affine non-malleable extractors for some linear min-entropy with exponentially small error, which in turn give the first explicit construction of non-malleable codes against $$2$$-split state tampering and affine tampering with constant rate and \emph{exponentially} small error. 3. Explicit extractors for affine sources, sumset sources, interleaved sources, and small space sources that achieve asymptotically optimal min-entropy of $$O(\log n)$ or $$2s+O(\log n)$$ (for space $$s$$ sources). 4. An explicit function that requires strongly linear read once branching programs of size $$2^{n-O(\log n)}$$, which is optimal up to the constant in $$O(\cdot)$$. Previously, even for standard read once branching programs, the best known size lower bound for an explicit function is $$2^{n-O(\log^2 n)}$$.
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Extractors for Images of Varieties
We construct explicit deterministic extractors for polynomial images of varieties, that is, distributions sampled by applying a low-degree polynomial map 𝑓 to an element sampled uniformly at random from a 𝑘-dimensional variety 𝑉. This class of sources generalizes both polynomial sources, studied by Dvir, Gabizon and Wigderson (FOCS 2007, Comput. Complex. 2009), and variety sources, studied by Dvir (CCC 2009, Comput. Complex. 2012). Assuming certain natural non-degeneracy conditions on the map 𝑓 and the variety 𝑉 , which in particular ensure that the source has enough min-entropy, we extract almost all the min-entropy of the distribution. Unlike the Dvir–Gabizon–Wigderson and Dvir results, our construction works over large enough finite fields of arbitrary characteristic. One key part of our construction is an improved deterministic rank extractor for varieties. As a by-product, we obtain explicit Noether normalization lemmas for affine varieties and affine algebras. Additionally, we generalize a construction of affine extractors with exponentially small error due to Bourgain, Dvir and Leeman (Comput. Complex. 2016) by extending it to all finite prime fields of quasipolynomial size.
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- PAR ID:
- 10484417
- Publisher / Repository:
- ACM
- Date Published:
- Journal Name:
- Conference proceedings of the annual ACM Symposium on Theory of Computing
- ISSN:
- 0734-9025
- ISBN:
- 9781450399135
- Page Range / eLocation ID:
- 46 to 59
- Format(s):
- Medium: X
- Location:
- Orlando FL USA
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1145/3564246.3585109
