On Extractors and Exposure-Resilient Functions for Sublogarithmic Entropy

We study resilient functions and exposure-resilient functions in the low-entropy regime. A resilient function (a.k.a. deterministic extractor for oblivious bit-fixing sources) maps any distribution on n -bit strings in which k bits are uniformly random and the rest are fixed into an output distribution that is close to uniform. With exposure-resilient functions, all the input bits are random, but we ask that the output be close to uniform conditioned on any subset of n - k input bits. In this paper, we focus on the case that k is sublogarithmic in n. We simplify and improve an explicit construction of resilient functions for k sublogarithmic in n due to Kamp and Zuckerman (SICOMP 2006), achieving error exponentially small in k rather than polynomially small in k. Our main result is that when k is sublogarithmic in n, the short output length of this construction (O(log k) output bits) is optimal for extractors computable by a large class of space-bounded streaming algorithms. Next, we show that a random function is a resilient function with high probability if and only if k is superlogarithmic in n, suggesting that our main result may apply more generally. In contrast, we show that a random function is a static (resp. adaptive) exposure-resilient function with high probability even if k is as small as a constant (resp. loglog n). No explicit exposure-resilient functions achieving these parameters are known.Engineering and Applied SciencesMathematic

Two Source Extractors for Asymptotically Optimal Entropy, and (Many) More

A long line of work in the past two decades or so established close connections between several different pseudorandom objects and applications. 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: 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. 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. 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). 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 $2^{n-O(\log^2 n)}$.Comment: Fixed some minor error

Applications of Derandomization Theory in Coding

Randomized techniques play a fundamental role in theoretical computer science and discrete mathematics, in particular for the design of efficient algorithms and construction of combinatorial objects. The basic goal in derandomization theory is to eliminate or reduce the need for randomness in such randomized constructions. In this thesis, we explore some applications of the fundamental notions in derandomization theory to problems outside the core of theoretical computer science, and in particular, certain problems related to coding theory. First, we consider the wiretap channel problem which involves a communication system in which an intruder can eavesdrop a limited portion of the transmissions, and construct efficient and information-theoretically optimal communication protocols for this model. Then we consider the combinatorial group testing problem. In this classical problem, one aims to determine a set of defective items within a large population by asking a number of queries, where each query reveals whether a defective item is present within a specified group of items. We use randomness condensers to explicitly construct optimal, or nearly optimal, group testing schemes for a setting where the query outcomes can be highly unreliable, as well as the threshold model where a query returns positive if the number of defectives pass a certain threshold. Finally, we design ensembles of error-correcting codes that achieve the information-theoretic capacity of a large class of communication channels, and then use the obtained ensembles for construction of explicit capacity achieving codes. [This is a shortened version of the actual abstract in the thesis.]Comment: EPFL Phd Thesi

Deterministic Extractors for Small-Space Sources

We give polynomial-time, deterministic randomness extractors for sources generated in small space, where we model space s sources on n{0,1} as sources generated by width s2 branching programs. Specifically, there is a constant η>0 such that for any ζ>n−η, our algorithm extracts m=(δ−ζ)n bits that are exponentially close to uniform (in variation distance) from space s sources with min-entropy δn, where s=Ω(ζ3n). Previously, nothing was known for δ≤1/2, even for space 0. Our results are obtained by a reduction to the class of total-entropy independent sources. This model generalizes both the well-studied models of independent sources and symbol-fixing sources. These sources consist of a set of r independent smaller sources over ℓ{0,1}, where the total min-entropy over all the smaller sources is k. We give deterministic extractors for such sources when k is as small as polylog(r), for small enough ℓ.Engineering and Applied Science