Improved lower bounds for decomposable randomized encoding

2026-09-15

AI Summary

研究解决了DRE大小的最优值问题，特别是对于n位OR函数及非周期对称函数，通过证明首个超线性和Ω(nlogn)下界。

Abstract

A decomposable randomized encoding (DRE) for a function $f$ allows $n$ parties, using shared randomness, to encode their individual inputs locally so that the collection of encodings reveals $f(x_1,\ldots,x_n)$ and nothing else. DREs are widely used in efficient multiparty computation. Their main complexity measure is size, the total bit length of the local encodings. Yet the optimal DRE size remains poorly understood even for the $n$-bit OR function. We prove the first superlinear lower bound for OR and, more generally, for every non-periodic symmetric function. Under an additional symmetry assumption, we prove a sharp $Ω(n\log n)$ lower bound for OR, matching the classic construction of Feige, Kilian, and Naor (STOC 1994). We also prove the first $Ω(n^2)$ lower bound on DRE size for non-explicit Boolean functions.

Problem

• decomposable randomized encoding
• lower bound
• OR function
• symmetric function

Innovation

• superlinear lower bound
• decomposable randomized encoding (DRE)
• non-periodic symmetric function
• Ω(n log n) lower bound
• Ω(n^2) lower bound