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In this paper, we develop an adaptive Generalized Multiscale Discontinuous Galerkin Method (GMs-DGM) for a class of high-contrast flow problems, and derive a-priori and a-posteriori error estimates for the method. Based on the a-posteriori error estimator, we develop an adaptive enrichment algorithm for our GMsDGM and prove its convergence. The adaptive enrichment algorithm gives an automatic way to enrich the approximation space in regions where the solution requires more basis functions, which are shown to perform well compared with a uniform enrichment. We also discuss an approach that adaptively selects multiscale basis functions by correlating the residual to multiscale basis functions (cf.[4]). The proposed error indicators are L2-based and can be inexpensively computed which makes our approach efficient. Numerical results are presented that demonstrate the robustness of the proposed error indicators.

Multivariate Public Key Cryptosystems (MPKCs) are often touted as future-proofing against Quantum Computers. It also has been known for efficiency compared to “traditional” alternatives. However, this advantage seems to erode with the increase of arithmetic resources in modern CPUs and improved algorithms, especially with respect to Elliptic Curve Cryptography (ECC). In this paper, we show that hardware advances do not just favor ECC. Modern commodity CPUs also have many small integer arithmetic/logic resources, embodied by SSE2 or other vector instruction sets, that are useful for MPKCs. In particular, Intel’s SSSE3 instructions can speed up both public and private maps over prior software implementations of Rainbow-type systems up to 4×. Furthermore, MPKCs over fields of relatively small odd prime characteristics can exploit SSE2 instructions, supported by most modern 64-bit Intel and AMD CPUs. For example, Rainbow over F_{31} can be up to 2× faster than prior implementations of similarly-sized systems over F_{16}. Here a key advance is in using Wiedemann (as opposed to Gauss) solvers to invert the small linear systems in the central maps. We explain the techniques and design choices in implementing our chosen MPKC instances over fields such as F_{31}, F_{16} and F_{256}. We believe that our results can easily carry over to modern FPGAs, which often contain a large number of small multipliers, usable by odd-field MPKCs.