Linear Programming Relaxations for Goldreich's Generators over Non-Binary Alphabets

Goldreich suggested candidates of one-way functions and pseudorandom generators included in $\mathsf{NC}^0$. It is known that randomly generated Goldreich's generator using $(r-1)$-wise independent predicates with $n$ input variables and $m=C n^{r/2}$ output variables is not pseudorandom generator with high probability for sufficiently large constant $C$. Most of the previous works assume that the alphabet is binary and use techniques available only for the binary alphabet. In this paper, we deal with non-binary generalization of Goldreich's generator and derives the tight threshold for linear programming relaxation attack using local marginal polytope for randomly generated Goldreich's generators. We assume that $u(n)\in \omega(1)\cap o(n)$ input variables are known. In that case, we show that when $r\ge 3$, there is an exact threshold $\mu_\mathrm{c}(k,r):=\binom{k}{r}^{-1}\frac{(r-2)^{r-2}}{r(r-1)^{r-1}}$ such that for $m=\mu\frac{n^{r-1}}{u(n)^{r-2}}$, the LP relaxation can determine linearly many input variables of Goldreich's generator if $\mu>\mu_\mathrm{c}(k,r)$, and that the LP relaxation cannot determine $\frac1{r-2} u(n)$ input variables of Goldreich's generator if $\mu<\mu_\mathrm{c}(k,r)$. This paper uses characterization of LP solutions by combinatorial structures called stopping sets on a bipartite graph, which is related to a simple algorithm called peeling algorithm.Comment: 14 pages, 1 figur

Health Systems and Preferences for Places of Death : Poor Death vs Happy Death

Public interest for quality of death is increasing worldwide. This paper overviews previous studies analyzing some factors influenced on death using Discrete Choice Experiments (DCEs). Only a little research, however, has been conducted globally to quantify which types of death people consider as a good death. Last half of the paper, I overviewed the real situations for places of death in Japan

A Super Bubble Candidate in the Galactic Center and a Local Enhancement G359.77-0.09

A 20' \times 16' elliptical ring-like structure has been found near the Galactic center in the narrow energy band corresponding to the K$\alpha$ line from He-like sulfur. In the ring, two diffuse sources are found, a supernova remnant candidate G359.79-0.26 and an unidentified source G359.77-0.09. The X-ray spectrum of G359.77-0.09 is similar to that of G359.79-0.26, which can be explained by an absorbed thin thermal plasma model with temperatures of 0.7 and 1.0 keV. The absorption column densities of these two sources are large (N_H = 6.9 \times 10^{22} and 4.5 \times 10^{22} cm^{-2}) and are consistent with that of the Galactic center distance. The X-ray spectrum extracted from the ring-like structure is also represented by an absorbed thin thermal plasma model (kT_e \sim 0.9 keV). The thermal energy of the plasma filling the ring-like structure is estimated to be 1.0 \times 10^{51} erg. We therefore propose that the two sources comprise a single ring-like object, which is possibly a super bubble with a size of 49 pc \times 40 pc in the Galactic center region.Comment: 9 pages, 7 figures, accepted for publication in PAS