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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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Robustly Self-Ordered Graphs: Constructions and Applications to Property Testing
A graph G is called {\em self-ordered} (a.k.a asymmetric) if the identity permutation is its only automorphism. Equivalently, there is a unique isomorphism from G to any graph that is isomorphic to G. We say that G=(VE) is {\em robustly self-ordered}if the size of the symmetric difference between E and the edge-set of the graph obtained by permuting V using any permutation :VV is proportional to the number of non-fixed-points of . In this work, we initiate the study of the structure, construction and utility of robustly self-ordered graphs. We show that robustly self-ordered bounded-degree graphs exist (in abundance), and that they can be constructed efficiently, in a strong sense. Specifically, given the index of a vertex in such a graph, it is possible to find all its neighbors in polynomial-time (i.e., in time that is poly-logarithmic in the size of the graph). We provide two very different constructions, in tools and structure. The first, a direct construction, is based on proving a sufficient condition for robust self-ordering, which requires that an auxiliary graph, on {\em pairs} of vertices of the original graph, is expanding. In this case the original graph is (not only robustly self-ordered but) also expanding. The second construction proceeds in three steps: It boosts the mere existence of robustly self-ordered graphs, which provides explicit graphs of sublogarithmic size, to an efficient construction of polynomial-size graphs, and then, repeating it again, to exponential-size(robustly self-ordered) graphs that are locally constructible. This construction can yield robustly self-ordered graphs that are either expanders or highly disconnected, having logarithmic size connected components. We also consider graphs of unbounded degree, seeking correspondingly unbounded robustness parameters. We again demonstrate that such graphs (of linear degree)exist (in abundance), and that they can be constructed efficiently, in a strong sense. This turns out to require very different tools. Specifically, we show that the construction of such graphs reduces to the construction of non-malleable two-source extractors with very weak parameters but with some additional natural features. We actually show two reductions, one simpler than the other but yielding a less efficient construction when combined with the known constructions of extractors. We demonstrate that robustly self-ordered bounded-degree graphs are useful towards obtaining lower bounds on the query complexity of testing graph properties both in the bounded-degree and the dense graph models. Indeed, their robustness offers efficient, local and distance preserving reductions from testing problems on ordered structures (like sequences) to the unordered (effectively unlabeled) graphs. One of the results that we obtain, via such a reduction, is a subexponential separation between the query complexities of testing and tolerant testing of graph properties in the bounded-degree graph model. Changes to previous version: We retract the claims made in our initial posting regarding the construction of non-malleable two-source extractors (which are quasi-orthogonal) as well as the claims about the construction of relocation-detecting codes (see Theorems 1.5 and 1.6 in the original version). The source of trouble is a fundamental flaw in the proof of Lemma 9.7 (in the original version), which may as well be wrong. Hence, the original Section 9 was omitted, except that the original Section 9.3 was retained as a new Section 8.3. The original Section 8 appears as Section 8.0 and 8.1, and Section 8.2 is new.
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- Award ID(s):
- 1900460
- PAR ID:
- 10273422
- Date Published:
- Journal Name:
- Electronic colloquium on computational complexity
- Volume:
- ECCC TR20-149
- ISSN:
- 1433-8092
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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