Breaking the Size Barrier: Universal Circuits Meet Lookup Tables

A Universal Circuit (UC) is a Boolean circuit of size Theta(n log n) that can simulate any Boolean function up to a certain size n. Valiant (STOC'76) provided the first two UC constructions of asymptotic sizes similar to 5n log n and similar to 4.75n log n, and today's most efficient construction of Liu et al. (CRYPTO'21) has size similar to 3n log n. Evaluating a public UC with a secure Multi-Party Computation (MPC) protocol allows efficient Private Function Evaluation (PFE), where a private function is evaluated on private data. Previously, most UC constructions have only been developed for circuits consisting of 2-input gates. In this work, we generalize UCs to simulate circuits consisting of (rho -> omega)-Lookup Tables (LUTs) that map. input bits to. output bits. Our LUT-based UC (LUC) construction has an asymptotic size of 1.5 rho omega n log omega n and improves the size of the UC over the best previous UC construction of Liu et al. (CRYPTO'21) by factors 1.12x-2.18x for common functions. Our results show that the greatest size improvement is achieved for rho = 3 inputs, and it decreases for rho > 3. Furthermore, we introduce Varying Universal Circuits (VUCs), which reduce circuit size at the expense of leaking the number of inputs rho and outputs omega of each LUT. Our benchmarks demonstrate that VUCs can improve over the size of the LUC construction by a factor of up to 1.45x.