We introduce Grotto, a framework and C++ library for space- and time-efficient (2+1)-party piecewise polynomial (i.e., spline) evaluation on secrets additively shared over ℤ2n. Grotto improves on the state-of-the-art approaches based on distributed comparison functions (DCFs) in almost every metric, offering asymptotically superior communication and computation costs with the same or lower round complexity. At the heart of Grotto is a novel observation about the structure of the ''tree'' representation underlying the most efficient distributed point functions (DPFs) from the literature, alongside an efficient algorithm that leverages this structure to do with a lightweight DPF what state-of-the-art approaches require comparatively heavyweight DCFs to do. Our open-source Grotto implementation supports dozens of useful functions out of the box, including trigonometric and hyperbolic functions with their inverses; various logarithms; roots, reciprocals, and reciprocal roots; sign testing and bit counting; and over two dozen of the most common univariate activation functions from the deep-learning literature.

We present Sabre, a family of sender-anonymous messaging protocols with instances supporting both Twitter-like anonymous bulletin boards and Secure Drop-like mailboxes. Both variants provide strong anonymity while potentially scaling to anonymity sets in the tens of millions and beyond. Sabre protocols inherit their basic structure from Riposte and Express while replacing those schemes’ linear-cost audit protocols with new logarithmic-cost ones. Sabre also incorporates a host of innovations that improve concrete performance by an order of magnitude or more under “ideal” circumstances, while providing asymptotic speedups in the face of resource exhaustion-style DoS attacks.