Lattice problem such as NTRU problem and LWE problem is widely used as the security base of post-quantum cryptosystems. And currently doing lattice reduction by BKZ algorithm is the most efficient way to solve it. In this paper, we give several further improvements on BKZ algorithm, which can be used for different SVP subroutines base on both enumeration and sieving. These improvements in combination provide a speed up of 2^{3~4} in total. It is significant in concrete attacks. Using these new techniques, we solved the 656 dimensional ideal lattice challenge in only 380 thread hours (also with a enumeration based SVP subroutine), much less than the previous records (which costs 4600 thread hours in total). With these improvements enabled, we can still simulate the new BKZ algorithm easily. One can also use this simulator to find the blocksize strategy (and the corresponding cost) to make Pot of the basis (defined in section 5.2) decrease as fast as possible, which means the length of the first basis vector decrease the fastest if we accept the GSA assumption. It is useful for analyzing concrete attacks on lattice-based cryptography.

We introduce a class of M×M sample covariance matrices Q which subsumes and generalizes several previous models. The associated population covariance matrix Σ=EQ is assumed to differ from the identity by a matrix of bounded rank. All quantities except the rank of Σ−IM may depend on M in an arbitrary fashion. We investigate the principal components, i.e.\ the top eigenvalues and eigenvectors, of Q. We derive precise large deviation estimates on the generalized components ⟨w,ξi⟩ of the outlier and non-outlier eigenvectors ξi. Our results also hold near the so-called BBP transition, where outliers are created or annihilated, and for degenerate or near-degenerate outliers. We believe the obtained rates of convergence to be optimal. In addition, we derive the asymptotic distribution of the generalized components of the non-outlier eigenvectors. A novel observation arising from our results is that, unlike the eigenvalues, the eigenvectors of the principal components contain information about the \emph{subcritical} spikes of Σ. The proofs use several results on the eigenvalues and eigenvectors of the uncorrelated matrix Q, satisfying EQ=IM, as input: the isotropic local Marchenko-Pastur law established in [9], level repulsion, and quantum unique ergodicity of the eigenvectors. The latter is a special case of a new universality result for the joint eigenvalue-eigenvector distribution.

We explore the higher-form generalized global symmetries and higher anomalies based on a generalized cobordism theory. Our cobordism calculation guides us to classify higher anomalies and topological terms for Yang-Mills (YM) gauge theories and sigma models in various dimensions. Some of YM gauge theories can be obtained from dynamically gauging the SU(N) time-reversal symmetric cobordism invariants (SU(N)-generalized topological superconductors/insulators [arXiv:1711.11587]). We elaborate the cases of YM theory with a compact Lie gauge group (such as SU(N) with N=2, 3 and others) particularly in a 4d spacetime. We provide the relevant homotopy and cobordism group calculations of higher classifying spaces, based on mathematical tools of algebraic topology, to support the physics stories.

For solving time-dependent convection-dominated partial differential equations (PDEs), which arise frequently in computational physics, high order numerical methods, including finite difference, finite volume, finite element and spectral methods, have been undergoing rapid developments over the past decades. In this article we give a brief survey of two selected classes of high order methods, namely the weighted essentially non-oscillatory (WENO) finite difference and finite volume schemes and discontinuous Galerkin (DG) finite element methods, emphasizing several of their recent developments: bound-preserving limiters for DG, finite volume and finite difference schemes, which address issues in robustness and accuracy; WENO limiters for DG methods, which address issues in non-oscillatory performance when there are strong shocks, and inverse Lax-Wendroff type boundary treatments for finite difference schemes, which address issues in solving complex geometry problems using Cartesian meshes.

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