International Association for Cryptologic Research

International Association
for Cryptologic Research


Daniel Günther


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 $\sim5 n\log n$ and $\sim4.75 n\log n$, and today's most efficient construction of Liu et al. (CRYPTO'21) has size $\sim3n\log n$. 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 \rightarrow \omega)-Lookup Tables (LUTs) that map $\rho$ input bits to $\omega$ 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 1.12$\times$ - $2.18\times$ 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 $1.45\times$.
Efficient and Scalable Universal Circuits
A universal circuit (UC) can be programmed to simulate any circuit up to a given size  n by specifying its program inputs. It provides elegant solutions in various application scenarios, e.g., for private function evaluation (PFE) and for improving the flexibility of attribute-based encryption schemes. The asymptotic lower bound for the size of a UC is $$\Omega (n\log n)$$ Ω ( n log n ) , and Valiant (STOC’76) provided two theoretical constructions, the so-called 2-way and 4-way UCs (i.e., recursive constructions with 2 and 4 substructures), with asymptotic sizes $${\sim }\,5n\log _2n$$ ∼ 5 n log 2 n and $${\sim }\,4.75n\log _2n$$ ∼ 4.75 n log 2 n , respectively. In this article, we present and extend our results published in (Kiss and Schneider EUROCRYPT’16) and (Günther et al. ASIACRYPT’17). We validate the practicality of Valiant’s UCs by realizing the 2-way and 4-way UCs in our modular open-source implementation. We also provide an example implementation for PFE using these size-optimized UCs. We propose a 2/4-hybrid approach that combines the 2-way and the 4-way UCs in order to minimize the size of the resulting UC. We realize that the bottleneck in universal circuit generation and programming becomes the memory consumption of the program since the whole structure of size $${\mathcal {O}}(n\log n)$$ O ( n log n ) is handled by the algorithms in memory. In this work, we overcome this by designing novel scalable algorithms for the UC generation and programming. Both algorithms use only $${\mathcal {O}}(n)$$ O ( n ) memory at any point in time. We prove the practicality of our scalable design with a scalable proof-of-concept implementation for generating Valiant’s 4-way UC. We note that this can be extended to work with optimized building blocks analogously. Moreover, we substantially improve the size of our UCs by including and implementing the recent optimization of Zhao et al. (ASIACRYPT’19) that reduces the asymptotic size of the 4-way UC to  $${\sim }\,4.5n\log _2n$$ ∼ 4.5 n log 2 n . Furthermore, we include their optimization in the implementation of our 2/4-hybrid UC which yields the smallest UC construction known so far.