More Efficient Universal Circuit Constructions

Abstract

A universal circuit (UC) can be programmed to simulate any circuit up to a given size n by specifying its program bits. UCs have several applications, including private function evaluation (PFE). The asymptotical lower bound for the size of a UC is proven to be Ω(n log n). In fact, Valiant (STOC'76) provided two theoretical UC constructions using so-called 2-way and 4-way constructions, with sizes 5n log 2 n and 4.75n log 2 n, respectively. The 2-way UC has recently been brought into practice in concurrent and independent results by Kiss and Schneider (EUROCRYPT'16) and Lipmaa et al. (Eprint 2016/017). Moreover, the latter work generalized Valiant's construction to any k-way UC. In this paper, we revisit Valiant's UC constructions and the recent results, and provide a modular and generic embedding algorithm for any k-way UC. Furthermore, we discuss the possibility for a more efficient UC based on a 3-way recursive strategy. We show with a counterexample that even though it is a promising approach, the 3-way UC does not yield an asymptotically better result than the 4-way UC. We propose a hybrid approach that combines the 2-way with the 4-way UC in order to minimize the size of the resulting UC. We elaborate on the concrete size of all discussed UC constructions and show that our hybrid UC yields on average 3.65% improvement in size over the 2-way UC. We implement the 4-way UC in a modular manner based on our proposed embedding algorithm, and show that our methods for programming the UC can be generalized for any k-way construction.

Efficient and Scalable Universal Circuits

“…This construction results in UCs of smaller size ∼ 4.75n log 2 n but has a more complicated structure and programming algorithm. We have studied and implemented this universal circuit in [31] and recapitulate our results here and in Sect. 7.…”

“…In this section, we describe Valiant's UC constructions [66,68] that can be programmed to evaluate any function of size n. We explain the general idea behind Valiant's UC construction [66] in Sects. 3.1 and 3.2, and the 2-way and 4-way UCs along with improvements of [31,45,46,72] in Sects. 3.3 and 3.4, respectively.…”

“…This journal article is a significantly extended and improved version of the conference publications [45] and [31]. Our added contributions are as follows.…”

“…5a. As before, we provide so-called head H (4) and tail T (4) blocks that occur at the top and bottom of a skeleton in [31,], respectively. The blocks are connected such that the 4 top (resp.…”

“…We designed the detailed embedding algorithm and the open-source UC implementation of [45] specifically for the 2-way UC, dealing with the whole UC skeleton as one block. In contrast, based on the modular design of [46], we modularized the edge-embedding task into multiple subtasks and described how they can be performed separately in [31]. In this section, we detail this modular approach for edge-embedding a graph into Valiant's -way EUG, where = 2 or = 4: The edge-embedding can be split into two parts, which are then combined.…”