Homomorphic message authentication codes (HMACs) allow users to authenticate data using a shared secret key, while supporting computation over authenticated data. Given data $\left(m_{1}, \ldots, m_{n}\right)$ and their tags $\left(\sigma_{1}, \ldots, \sigma_{n}\right)$, anyone can evaluate a circuit C on the data and tags to produce a succinct tag authenticating the output $C\left(m_{1}, \ldots, m_{n}\right)$. Importantly, tags remain succinct-of size polynomial in the security parameter $\lambda$-regardless of the size of C. This work introduces an enhanced variant of HMACs called algebraic HMAC (aHMAC), in which all tags (input and output) take the form $\vec{\Delta} \cdot m+\vec{K}$, as in standard information-theoretic MACs. We construct an aHMAC from group-based assumptions, including variants of the DDH and DCR assumptions, and use it to obtain group-based constructions of several cryptographic primitives:•Succinct CDS for circuits. For any $P:[N]^{k} \rightarrow[N]$ represented by circuit, we obtain a Conditional Disclosure of Secrets protocol with $\operatorname{poly}(\lambda, k, \log N)$ communication.•Succinct PSM for simple programs. For any $P:[N]^{k} \rightarrow[N]$ represented by a truth-table or shallow branching program, we obtain a Private Simultaneous Messages protocol or a garbling scheme with $\operatorname{poly}(\lambda, k, \log N)$ communication.•Constrained PRFs for circuits. We obtain the first groupbased constrained pseudorandom functions for general circuits, improving over a previous construction for $\mathrm{NC}^{1}$ circuits.Prior to our work, these applications could only be obtained from lattice assumptions or indistinguishability obfuscation.

This paper explores a stochastic approach for controlling swarms of independent robots toward a target distribution in a bounded domain. The robot swarm has no central controller, and individual robots lack both communication and localization capabilities. Robots can only measure a scalar field (e.g. concentration of a chemical) from the environment and from this deduce the desired local swarm density. Based on this value, each robot follows a simple control law that causes the swarm as a whole to diffuse toward the target distribution. Using a new holonomic drive robot, we present the first confirmation of this control law with physical experiment. Despite deviations from assumptions underpinning the theory, the swarm achieves the theorized convergence to the target distribution in both simulation and experiment. In fact, simulated and experimental performance agree with one another and with our hypothesis that the error from the target distribution is inversely proportional to the square root of the number of robots. This is evidence that the algorithm is both practical and easily scalable to large swarms.