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

55Mn NMR in Mn12 acetate: Hyperfine interaction and magnetic relaxation of cluster

The 55Mn NMR in oriented powder Mn12Ac has been investigated at 1.4-2.0 K in zero field and with external fields along the c-axis. Three kinds of 55Mn NMR composed of five-fold quadrupole-split lines for I=5/2 nuclei have been interpreted to arise from Mn4+ ion, and two crystallographically-inequivalent Mn3+ ions, respectively. It is found that the isotropic hyperfine field in the Mn4+ ion with 3d3 configuration indicates a large amount of reduction (26%) as compared with the theoretical evaluation. In the analysis for the hyperfine field of Mn3+ ions with 3d4 configuration, we have taken into account of the anisotropic dipolar contribution in addition to the Fermi-contact term in order to explain two kinds of 55Mn NMR frequencies in Mn3+ ions in inequivalent sites. By using the hyperfine coupling constants of twelve manganese ions in Mn12Ac, the total hyperfine interaction of the ferrimagnetic ground state of S=10 has been determined to amount to 0.3 cm-1 in magnitude at most, the magnitude of which corresponds to the nuclear hyperfine field he(0.32 kG seen by Mn12 cluster spin. The relaxation of the cluster magnetization was investigated by observing the recovery of the 55Mn spin-echo intensity in the fields of 0.20-1.90 T along the c-axis at 2.0 K. It was found that the magnetization of the cluster exhibits the (t-recovery in the short time regime. The relaxation time decreases with increasing external field following significant dips at every 0.45 T. This is interpreted to be due to the effects of thermally-assisted quantum tunneling between the spin states at magnetic level crossings.Comment: Text 25 pages, five figures and two table

Inferring pure context-free languages from positive data

We study the possibilities to infer pure context-free languages from positive data. We can show that while the whole class of pure context-free languages is not inferable from positive data, it has interesting subclasses which have the desired inference property. We study uniform pure languages, i.e., languages generated by pure grammars obeying restrictions on the length of the right hand sides of their productions, and pure languages generated by deterministic pure grammars