Better short-seed quantum-proof extractors

We construct a strong extractor against quantum storage that works for every min-entropy $k$, has logarithmic seed length, and outputs $\Omega(k)$ bits, provided that the quantum adversary has at most $\beta k$ qubits of memory, for any \beta < \half. The construction works by first condensing the source (with minimal entropy-loss) and then applying an extractor that works well against quantum adversaries when the source is close to uniform. We also obtain an improved construction of a strong quantum-proof extractor in the high min-entropy regime. Specifically, we construct an extractor that uses a logarithmic seed length and extracts $\Omega(n)$ bits from any source over \B^n, provided that the min-entropy of the source conditioned on the quantum adversary's state is at least $(1-\beta) n$, for any \beta < \half.Comment: 14 page

A Hypercontractive Inequality for Matrix-Valued Functions with Applications to Quantum Computing and LDCs

The Bonami-Beckner hypercontractive inequality is a powerful tool in Fourier analysis of real-valued functions on the Boolean cube. In this paper we present a version of this inequality for matrix-valued functions on the Boolean cube. Its proof is based on a powerful inequality by Ball, Carlen, and Lieb. We also present a number of applications. First, we analyze maps that encode $n$ classical bits into $m$ qubits, in such a way that each set of $k$ bits can be recovered with some probability by an appropriate measurement on the quantum encoding; we show that if $m<0.7 n$, then the success probability is exponentially small in $k$. This result may be viewed as a direct product version of Nayak's quantum random access code bound. It in turn implies strong direct product theorems for the one-way quantum communication complexity of Disjointness and other problems. Second, we prove that error-correcting codes that are locally decodable with 2 queries require length exponential in the length of the encoded string. This gives what is arguably the first ``non-quantum'' proof of a result originally derived by Kerenidis and de Wolf using quantum information theory, and answers a question by Trevisan.Comment: This is the full version of a paper that will appear in the proceedings of the IEEE FOCS 08 conferenc

Two-Source Condensers with Low Error and Small Entropy Gap via Entropy-Resilient Functions

In their seminal work, Chattopadhyay and Zuckerman (STOC\u2716) constructed a two-source extractor with error epsilon for n-bit sources having min-entropy {polylog}(n/epsilon). Unfortunately, the construction\u27s running-time is {poly}(n/epsilon), which means that with polynomial-time constructions, only polynomially-small errors are possible. Our main result is a {poly}(n,log(1/epsilon))-time computable two-source condenser. For any k >= {polylog}(n/epsilon), our condenser transforms two independent (n,k)-sources to a distribution over m = k-O(log(1/epsilon)) bits that is epsilon-close to having min-entropy m - o(log(1/epsilon)). Hence, achieving entropy gap of o(log(1/epsilon)). The bottleneck for obtaining low error in recent constructions of two-source extractors lies in the use of resilient functions. Informally, this is a function that receives input bits from r players with the property that the function\u27s output has small bias even if a bounded number of corrupted players feed adversarial inputs after seeing the inputs of the other players. The drawback of using resilient functions is that the error cannot be smaller than ln r/r. This, in return, forces the running time of the construction to be polynomial in 1/epsilon. A key component in our construction is a variant of resilient functions which we call entropy-resilient functions. This variant can be seen as playing the above game for several rounds, each round outputting one bit. The goal of the corrupted players is to reduce, with as high probability as they can, the min-entropy accumulated throughout the rounds. We show that while the bias decreases only polynomially with the number of players in a one-round game, their success probability decreases exponentially in the entropy gap they are attempting to incur in a repeated game

A New Approach for Constructing Low-Error, Two-Source Extractors

Our main contribution in this paper is a new reduction from explicit two-source extractors for polynomially-small entropy rate and negligible error to explicit t-non-malleable extractors with seed-length that has a good dependence on t. Our reduction is based on the Chattopadhyay and Zuckerman framework (STOC 2016), and surprisingly we dispense with the use of resilient functions which appeared to be a major ingredient there and in follow-up works. The use of resilient functions posed a fundamental barrier towards achieving negligible error, and our new reduction circumvents this bottleneck. The parameters we require from t-non-malleable extractors for our reduction to work hold in a non-explicit construction, but currently it is not known how to explicitly construct such extractors. As a result we do not give an unconditional construction of an explicit low-error two-source extractor. Nonetheless, we believe our work gives a viable approach for solving the important problem of low-error two-source extractors. Furthermore, our work highlights an existing barrier in constructing low-error two-source extractors, and draws attention to the dependence of the parameter t in the seed-length of the non-malleable extractor. We hope this work would lead to further developments in explicit constructions of both non-malleable and two-source extractors

Structural Insight into Regulation of the Proteasome Ub-Receptor Rpn10

Ubiquitylation is a posttranslational modification that determines protein fate. The ubiquitin code is written by enzymatic cascades of E1 and E2 and E3 enzymes. Ubiquitylation can be edited or erased by deubiquitylating enzymes. Ub-receptors are proteins that read and decipher the ubiquitin codes into cellular response. They harbor a ubiquitin-binding domain and a response element. Interestingly, Ub-receptors are also regulated by ubiquitylation and deubiquitylation. However, until recently, the molecular details and the significance of this regulation remained enigmatic. Rpn10 is a Ub-receptor that shuttles ubiquitylated targets to the proteasome for degradation. Here we review recent data on Rpn10, with emphasis on its regulation by ubiquitylation

The Compact Chromatin Structure of a Ty Repeated Sequence Suppresses Recombination Hotspot Activity in Saccharomyces cerevisiae

Recombination between repeated DNA sequences can have drastic consequences on the integrity of the genome. Repeated sequences are abundant in most eukaryotes, yet the mechanism that prevents recombination between them is currently unknown. Ty elements, the main family of dispersed repeats in Saccharomyces cerevisiae, exhibit low levels of exchange. Other regions in the genome have relatively high rates of meiotic recombination (hotspots). We show that a Ty element adjacent to the HIS4 recombination hotspot substantially reduces its activity, eliminating local DSB formation. We demonstrate that the Ty has a closed (nuclease-insensitive) chromatin configuration that is also imposed on the flanking DNA sequences. The compact chromatin structure is determined by sequences at the N terminus of the Ty. Increased binding of the Rap1 protein to the hotspot restores both open chromatin conformation and DSB formation. The chromatin configuration of Ty elements precludes initiation of recombination, thus preventing potentially lethal exchanges between repeated sequences