Online Linear Extractors for Independent Sources

In this work, we characterize online linear extractors. In other words, given a matrix $A \in \mathbb{F}_2^{n \times n}$, we study the convergence of the iterated process $\mathbf{S} \leftarrow A\mathbf{S} \oplus \mathbf{X}$, where $\mathbf{X} \sim D$ is repeatedly sampled independently from some fixed (but unknown) distribution $D$ with (min)-entropy at least $k$. Here, we think of $\mathbf{S} \in \{0,1\}^n$ as the state of an online extractor, and $\mathbf{X} \in \{0,1\}^n$ as its input. As our main result, we show that the state $\mathbf{S}$ converges to the uniform distribution for all input distributions $D$ with entropy $k > 0$ if and only if the matrix $A$ has no non-trivial invariant subspace (i.e., a non-zero subspace $V \subsetneq \mathbb{F}_2^n$ such that $AV \subseteq V$). In other words, a matrix $A$ yields an online linear extractor if and only if $A$ has no non-trivial invariant subspace. For example, the linear transformation corresponding to multiplication by a generator of the field $\mathbb{F}_{2^n}$ yields a good online linear extractor. Furthermore, for any such matrix convergence takes at most $\widetilde{O}(n^2(k+1)/k^2)$ steps. We also study the more general notion of condensing---that is, we ask when this process converges to a distribution with entropy at least $\ell$, when the input distribution has entropy greater than $k$. (Extractors corresponding to the special case when $\ell = n$.) We show that a matrix gives a good condenser if there are relatively few vectors $\mathbf{w} \in \mathbb{F}_2^n$ such that $\mathbf{w}, A^T\mathbf{w}, \ldots, (A^T)^{n-k-1} \mathbf{w}$ are linearly dependent. As an application, we show that the very simple cyclic rotation transformation $A(x_1,\ldots, x_n) = (x_n,x_1,\ldots, x_{n-1})$ condenses to $\ell = n-1$ bits for any $k > 1$ if $n$ is a prime satisfying a certain simple number-theoretic condition. Our proofs are Fourier-analytic and rely on a novel lemma, which gives a tight bound on the product of certain Fourier coefficients of any entropic distribution

No Time to Hash: On Super Efficient Entropy Accumulation

Real-world random number generators (RNGs) cannot afford to use (slow) cryptographic hashing every time they refresh their state $R$ with a new entropic input $X$. Instead, they use ``superefficient\u27\u27 simple entropy-accumulation procedures, such as $$R \leftarrow \mathsf{rot}_{\alpha, n}(R) \oplus X,$$ where $\mathsf{rot}_{\alpha,n}$ rotates an $n$-bit state $R$ by some fixed number $\alpha$. For example, Microsoft\u27s RNG uses $\alpha=5$ for $n=32$ and $\alpha=19$ for $n=64$. Where do these numbers come from? Are they good choices? Should rotation be replaced by a better permutation $\pi$ of the input bits? In this work we initiate a rigorous study of these pragmatic questions, by modeling the sequence of successive entropic inputs $X_1,X_2,\ldots$ as independent (but otherwise adversarial) samples from some natural distribution family ${\mathcal D}$. Our contribution is as follows. * We define $2$-monotone distributions as a rich family ${\mathcal D}$ that includes relevant real-world distributions (Gaussian, exponential, etc.), but avoids trivial impossibility results. * For any $\alpha$ with $\gcd(\alpha,n)=1$, we show that rotation accumulates $\Omega(n)$ bits of entropy from $n$ independent samples $X_1,\ldots,X_n$ from any (unknown) $2$-monotone distribution with entropy $k > 1$. * However, we also show that some choices of $\alpha$ perform much better than others for a given $n$. E.g., we show $\alpha=19$ is one of the best choices for $n=64$; in contrast, $\alpha=5$ is good, but generally worse than $\alpha=7$, for $n=32$. * More generally, given a permutation $\pi$ and $k\ge 1$, we define a simple parameter, the covering number $C_{\pi,k}$, and show that it characterizes the number of steps before the rule $$(R_1,\ldots,R_n)\leftarrow (R_{\pi(1)},\ldots, R_{\pi(n)})\oplus X$$ accumulates nearly $n$ bits of entropy from independent, $2$-monotone samples of min-entropy $k$ each. * We build a simple permutation $\pi^*$, which achieves nearly optimal $C_{\pi^*,k}\approx n/k$ for all values of $k$ simultaneously, and experimentally validate that it compares favorably with all rotations $\mathsf{rot}_{\alpha,n}$

Condition Assessment of Paper Insulation in Oil-Immersed Power Transformers Based on the Iterative Inversion of Resistivity

The resistivity of oil impregnated paper will decrease during its aging process. This paper takes paper resistivity as an assessment index to evaluate the insulation condition of oil impregnated paper in power transformer. The feasibility of this method are discussed in two aspects: reliability and sensitivity. Iterative inversion of paper resistivity was combined with finite element simulation. Both the bisection method and Newton’s method were used as iterative methods. After the analysis and comparison, Newton’s method was selected as the first option of paper resistivity iteration for its faster convergence. In order to consider the spatial distribution characteristic of paper aging and enhance the calculation accuracy, the resistivity calculation is expanded to a multivariate iteration based on Newton’s method, in order to consider the spatial distribution characteristic of paper aging and improve the calculation accuracy. This paper presents an exploratory research on condition assessment of oil impregnated paper insulation, and provides some reference to the security and economy operation of power transformers

Lateral Percolation and Its Effect on Shale Gas Accumulation on the Basis of Complex Tectonic Background

As a result of complex tectonic background, shale gas in China exhibits differential enrichment. Choosing a favorable exploration target accurately is a crucial problem to be solved. In this study, the tests show that there is a superior transportation pathway within shale layer. Gas in the shale layer percolates much more in the direction parallel to the plane. Therefore, the accumulation pattern of shale gas indicates a complex tectonic background. Gas in the lower part of the structure diffuses and percolates in the vertical direction into the surrounding rock. Most gas percolates towards the high part of the structure in the direction parallel to the plane. When the shale was exposed, gas percolated along the parallel direction into the air. In the case of fracture development, if there is a reverse fault, gas would be enriched in the footwall. However, if there is an unsealed fault, it would become a pathway for gas migration. The above accumulation pattern was proved in several Areas. Also, this research presented a basis of evaluation units division. According to the buried depth, fractures, and structural position, Xiuwu Basin was divided into five evaluation units and Unit A3 is the most favorable exploration target

Distinct genomic features between osteosarcomas firstly metastasing to bone and to lung

Background: Osteosarcoma initially metastasing to bone only shows distinct biological features compared to osteosarcoma that firstly metastasizes to the lung, which suggests us underlying different genomic pathogenetic mechanism. Methods: We analyzed whole-exome sequencing (WES) data for 38 osteosarcoma with paired samples in different relapse patterns. We also sought to redefine disease subclassifications for osteosarcoma based on genetic alterations and correlate these genetic profiles with clinical treatment courses to elucidate potential evolving cladograms. Results: We investigated WES of 12/38 patients with high-grade osteosarcoma (31.6%) with initial bone metastasis (group A) and 26/38 (68.4%) with initial pulmonary metastasis (group B), of whom 15/38 (39.5%) had paired samples of primary lesions and metastatic lesions. We found that osteosarcoma in group A mainly carries single-nucleotide variations displaying higher tumor mutation burden and neoantigen load and more tertiary lymphoid structures, while those in group B mainly exhibits structural variants. High conservation of reported genetic sequencing over time in their evolving cladograms. Conclusions: Osteosarcoma with mainly single-nucleotide variations other than structural variants might exhibit biological behavior predisposing toward bone metastases as well as better immunogenicity in tumor microenvironment