We use the signal function from RLWE key exchange to derive an efficient zero knowledge authentication protocol to validate an RLWE key p=as+e with secret s and error e in the Random Oracle Model (ROM). With this protocol, a verifier can validate that a key presented to him by a prover P is of the form p=as+e with s,e small and that the prover knows s. We accompany the description of the protocol with proof to show that it has negligible soundness and completeness error. The soundness of our protocol relies directly on the hardness of the RLWE problem. The protocol is applicable for both LWE and RLWE but we focus on the RLWE based protocol for efficiency and practicality. We also present a variant of the main protocol with a commitment scheme to avoid using the ROM.

How to properly specify boundary conditions for pressure is a longstanding problem for the incompressible Navier–Stokes equations with no-slip boundary conditions. An analytical resolution of this issue stems from a recently developed formula for the pressure in terms of the commutator of the Laplacian and Leray projection operators. Here we make use of this formula to (a) improve the accuracy of computing pressure in two kinds of existing time-discrete projection methods implicit in viscosity only, and (b) devise new higher-order accurate time-discrete projection methods that extend a slip-correction idea behind the well-known finite-difference scheme of Kim and Moin. We test these schemes for stability and accuracy using various combinations of C0 finite elements. For all three kinds of time discretization, one can obtain third-order accuracy for both pressure and velocity without a time-step stability restriction of diffusive type. Furthermore, two kinds of projection methods are found stable using piecewise-linear elements for both velocity and pressure.

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