Evaluation properties of invariant polynomials

AbstractA polynomial invariant under the action of a finite group can be rewritten using generators of the invariant ring. We investigate the complexity aspects of this rewriting process; we show that evaluation techniques enable one to reach a polynomial cost

Characterizing NTRU-Variants Using Group Ring and Evaluating their Lattice Security

The encryption scheme NTRU is designed over a quotient ring of a polynomial ring. Basically, if the ring is changed to any other ring, NTRU-like cryptosystem is constructible. In this paper, we propose a variant of NTRU using group ring, which is called GR-NTRU. GR-NTRU includes NTRU as a special case. Moreover, we analyze and compare the security of GR-NTRU for several concrete groups. It is easy to investigate the algebraic structure of group ring by using group representation theory. We apply this fact to the security analysis of GR-NTRU. We show that the original NTRU and multivariate NTRU are most secure among several GR-NTRUs which we investigated

From Single-Molecule Interactions to Population-Level Dynamics: Understanding the Complex Organization of RNA Pol II in the Nucleus of Living Cells

Transcription involves a complex exchange within a reservoir of proteins in the nucleoplasm, and the specific recruitment of individual proteins at specific gene loci. However, understanding the spatial distribution of individual proteins and the temporal behavior in the nucleus of living cells remains challenging. Using 3D super-resolution fluorescence microscopy and cluster analysis, we observe that the distribution of RNA Polymerase II (Pol II) cluster sizes, measured as the number of polymerases per cluster, follows a −3/2 power law. Radial dependent analysis of the spatial distribution of Pol II also shows scale-invariance, consistent with a so-called self-organized criticality in a fractal geometry of dimension ∼2.7. These results suggest a diffusion-based mechanism whereby, via transient interactions, massive recruitment and dismissal of pol II molecules can occur at specific loci in the nucleoplasm. Kinetic measurements using single-molecule detection in live cells reveal Pol II binding dynamics within minutes. Serum-induced transcription increased Pol II binding kinetics in live cells by an order of magnitude. Together, these results provide a comprehensive view of the spatio-temporal organization of Pol II in the nucleus: from the global population distribution, to single molecule recruitment at specific loci in live cells. This comprehensive single-cell approach can be adopted for other proteins beside RNA Pol II, for real-time quantification of protein organization in vivo, with single-molecule sensitivity