Numerical schemes provably preserving the positivity of density and pressure are highly desirable for ideal magnetohydrodynamics (MHD), but the rigorous positivity-preserving (PP) analysis remains challenging. The difficulties mainly arise from the intrinsic complexity of the MHD equations as well as the indeterminate relation between the PP property and the divergence-free condition on the magnetic field. This paper presents the first rigorous PP analysis of conservative schemes with the Lax--Friedrichs (LF) flux for 1D and multidimensional ideal MHD. The significant innovation is the discovery of the theoretical connection between the PP property and a discrete divergence-free (DDF) condition. This connection is established through the generalized LF splitting properties, which are alternatives to the usually expected LF splitting property that does not hold for ideal MHD. The generalized LF splitting properties involve a number of admissible states strongly coupled by the DDF condition, making their derivation very difficult. We derive these properties via a novel equivalent form of the admissible state set and an important inequality, which is skillfully constructed by technical estimates. Rigorous PP analysis is then presented for finite volume and discontinuous Galerkin schemes with the LF flux on uniform Cartesian meshes. In the 1D case, the PP property is proved for the first-order scheme with proper numerical viscosity, and also for arbitrarily high-order schemes under conditions accessible by a PP limiter. In the 2D case, we show that the DDF condition is necessary and crucial for achieving the PP property. It is observed that even slightly violating the proposed DDF condition may cause failure to preserve the positivity of pressure. We prove that the 2D LF type scheme with proper numerical viscosity preserves both the positivity and the DDF condition. Sufficient conditions are derived for 2D PP high-order schemes, and extension to 3D is discussed. Numerical examples provided in the supplementary material further confirm the theoretical findings.

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Error reconciliation is an important technique for learning with error (LWE) and ring-LWE (RLWE)-based constructions. In this paper, we present a comparison analysis on two error reconciliation-based RLWE key exchange protocols: Ding et al. in 2012 (DING12) and Bos et al. in 2015 (BCNS15). We take them as examples to explain the core idea of error reconciliation, building key exchange over RLWE problem, implementation and real-world performance, and compare them comprehensively. We also analyse a LWE key exchange 'Frodo' that uses an improved error reconciliation mechanism in BCNS15. To the best of our knowledge, our work is the first to present at least 128-bit classic (80-bit quantum) and 256-bit classic (> 200-bit quantum) secure parameter choices for DING12 with efficient portable C/C++ implementations. Benchmark shows that our efficient implementation is 11× faster than BCNS15 and one key exchange execution only costs 0.07 ms on a four-year-old middle range CPU. Error reconciliation is 1.57× faster than BCNS15.

We introduce a curvature-dimension condition CD ($K$,$N$) for metric measure spaces. It is more restrictive than the curvature bound $\underline{{{\text{Curv}}}} {\left( {M,{\text{d}},m} \right)} > K$ (introduced in [Sturm K-T (2006) On the geometry of metric measure spaces. I. Acta Math 196:65–131]) which is recovered as the borderline case CD($K$, ∞). The additional real parameter$N$plays the role of a generalized upper bound for the dimension. For Riemannian manifolds, CD($K$,$N$) is equivalent to $${\text{Ric}}_{M} {\left( {\xi ,\xi } \right)} > K{\left| \xi \right|}^{2} $$ and dim($M$) ⩽$N$.