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Abstract The implicit boundary integral method (IBIM) provides a framework to construct quadrature rules on regular lattices for integrals over irregular domain boundaries. This work provides a systematic error analysis for IBIMs on uniform Cartesian grids for boundaries with different degrees of regularity. First, it is shown that the quadrature error gains an additional order of$$\frac{d-1}{2}$$ $\frac{d-1}{2}$from the curvature for a strongly convex smooth boundary due to the “randomness” in the signed distances. This gain is discounted for degenerated convex surfaces. Then the extension of error estimate to general boundaries under some special circumstances is considered, including how quadrature error depends on the boundary’s local geometry relative to the underlying grid. Bounds on the variance of the quadrature error under random shifts and rotations of the lattices are also derived. 

Broadcast is a fundamental primitive that plays an important role in secure Multi-Party Computation (MPC) area. In this work, we revisit the broadcast with selective abort (hereafter, short for broadcast) proposed by Goldwasser and Lindell (DISC 2002; JoC 2005) and study the round complexity of broadcast under different setup assumptions. Our findings are summarized as follows: - We formally prove that 1-round broadcast is impossible under various widely-used setup assumptions (e.g., plain model, random oracle model, and common reference string model, etc.), even if we consider the static security and the stand-alone framework. More concretely, we formalize a notion called consistent oracle to capture these setups, and prove that our impossibility holds under the consistent oracle. Our impossibility holds in both honest majority setting and dishonest majority setting. - We show that 1-round broadcast protocol is possible in the Universal Composition (UC) framework, by assuming stateful trusted hardwares. Our protocol can be proven secure against all-but-one adaptive and malicious corruptions. We bypass our impossibility result since our stateful trusted hardwares do not satisfy the definition of consistent oracle. - We provide an application of 1-round broadcast: we construct the first 1-round multiple-verifier zero-knowledge (which is a special case of MPC) protocol, without assuming the broadcast hybrid world.