We study the problem of density of polynomials in the de Branges spaces ℋ($E$) of entire functions and obtain conditions (in terms of the distribution of the zeros of the generating function$E$) ensuring that the polynomials belong to the space ℋ($E$) or are dense in this space. We discuss the relation of these results with the recent paper of V. P. Havin and J. Mashreghi on majorants for the shift-coinvariant subspaces. Also, it is shown that the density of polynomials implies the hypercyclicity of translation operators in ℋ($E$).

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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.