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.

We prove that the algebra of singular cochains on a smooth manifold, equipped with the cup product, is equivalent to the A1 structure on the Lagrangian Floer cochain group associated to the zero section in the cotangent bundle. More generally, given embeddings with isomorphic normal bundles of a closed manifold B into manifolds Q1 and Q2, we construct a differential graded category from the singular cochains of these spaces, and prove that it is equivalent to the A1 category obtained by considering exact Lagrangian embeddings of Q1 and Q2 which intersect cleanly along B.

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