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Let $X$ be an abstract not necessarily compact orientable CR manifold of dimension $2n-1$, $n\geqslant2$, and let $L^k$ be the $k$-th tensor power of a CR complex line bundle $L$ over $X$. Given $q\in\set{0,1,\ldots,n-1}$, let $\Box^{(q)}_{b,k}$ be the Gaffney extension of Kohn Laplacian for $(0,q)$ forms with values in $L^k$. For $\lambda\geq0$, let $\Pi^{(q)}_{k,\leq\lambda}:=E((-\infty,\lambda])$, where $E$ denotes the spectral measure of $\Box^{(q)}_{b,k}$. In this work, we prove that $\Pi^{(q)}_{k,\leq k^{-N_0}}F^*_k$, $F_k\Pi^{(q)}_{k,\leq k^{-N_0}}F^*_k$, $N_0\geq1$, admit asymptotic expansions with respect to $k$ on the non-degenerate part of the characteristic manifold of $\Box^{(q)}_{b,k}$, where $F_k$ is some kind of microlocal cut-off function. Moreover, we show that $F_k\Pi^{(q)}_{k,\leq0}F^*_k$ admits a full asymptotic expansion with respect to $k$ if $\Box^{(q)}_{b,k}$ has small spectral gap property with respect to $F_k$ and $\Pi^{(q)}_{k,\leq0}$ is $k$-negligible away the diagonal with respect to $F_k$. By using these asymptotics, we establish almost Kodaira embedding theorems on CR manifolds and Kodaira embedding theorems on CR manifolds with transversal CR $S^1$ action.

In this paper, we introduce a new provably secure ephemeral-only RLWE+Rounding-based key exchange protocol and a proper approach to more accurately estimate the security level of the RLWE problem with only one sample. Since our scheme is an ephemeral-only key exchange, it generates only one RLWE sample from protocol execution. We carefully analyze how to estimate the practical security of the RLWE problem with only one sample, which we call the ONE-sample RLWE problem. Our approach is different from existing approaches that are based on estimation with multiple RLWE samples. Though our analysis is based on some recently developed techniques in Darmstadt, our type of practical security estimate was never done before and it produces security estimates substantial different from the estimates before based on multiple RLWE samples. We show that the new design improves the security and reduce the communication cost of the protocol simultaneously by using one RLWE+Rounding sample technique. We also present two parameter choices ensuring 2^{−60} key exchange failure probability which cover security of AES-128/192/256 with concrete security analysis and implementation. We believe that our construction is secure, simple, efficient and elegant with wide application prospects.

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Durhuus and Jonsson (1995) introduced the class of “locally constructible” (LC) 3-spheres and showed that there are only exponentially many combinatorial types of simplicial LC 3-spheres. Such upper bounds are crucial for the convergence of models for 3D quantum gravity.