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 construct noncompact solutions to the affine normal flow of hypersurfaces, and show that all ancient solutions must be either ellipsoids (shrinking solitons) or paraboloids (translating solitons). We also provide a new proof of the existence of a hyperbolic affine sphere asymptotic to the boundary of a convex cone containing no lines, which is originally due to Cheng-Yau. The main techniques are local second-derivative estimates for a parabolic Monge-Amp`ere equation modeled on those of Ben Andrews and Guti´errez-Huang, a decay estimate for the cubic form under the affine normal flow due to Ben Andrews, and a hypersurface barrier due to Calabi.

We give existence results for simple closed curves with prescribed geodesic curvature on S2, which correspond to periodic orbits of a charge in a magnetic field.

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The endoscopic classification via the stable trace formula comparison provides certain character relations between irreducible cuspidal automorphic representations of classical groups and their global Arthur parameters, which are certain automorphic representations of general linear groups. It is a question of J. Arthur and W. Schmid that asks how to construct concrete modules for irreducible cuspidal automorphic representations of classical groups in term of their global Arthur parameters? In this paper, we formulate a general construction of concrete modules, using Bessel periods, for cuspidal automorphic representations of classical groups with generic global Arthur parameters. Then we establish the theory for orthogonal and unitary groups, based on certain well expected conjectures. Among the consequences of the theory in this paper is that the global Gan-Gross-Prasad conjecture for those classical groups is proved in full generality in one direction and with a global assumption in the other direction.