Fourier Sparsity of Delta Functions and Matching Vector PIRs

2025-12-05

AI Summary

This paper investigates the Fourier sparsity lower bound of the delta function over the Boolean cube—i.e., $f(0)=1$ and $f(x)=0$ for all nonzero $xin{0,1}^r$—when extended to $mathbb{Z}_m^r$. Leveraging a synthesis of Fourier analysis, Boolean function theory, and algebraic combinatorics, we establish the first tight asymptotic bounds on the size of its Fourier support. Our results demonstrate that, under any constant number of servers, private information retrieval (PIR) schemes based on existing matching vector families cannot achieve subpolynomial communication complexity: specifically, their $S$-decoding polynomials necessarily incur polynomial—rather than polylogarithmic—communication cost. This lower bound exposes a fundamental theoretical barrier inherent to this PIR construction paradigm, thereby establishing a definitive ceiling on its efficiency and providing a critical criterion for evaluating future breakthrough constructions.

Technology Category

• Machine Learning: Privacy
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

In this paper we study a basic and natural question about Fourier analysis of Boolean functions, which has applications to the study of Matching Vector based Private Information Retrieval (PIR) schemes. For integers m and r, define a delta function on {0,1}^r to be a function f: Z_m^r -> C with f(0) = 1 and f(x) = 0 for all nonzero Boolean x. The basic question we study is how small the Fourier sparsity of a delta function can be; namely how sparse such an f can be in the Fourier basis? In addition to being intrinsically interesting and natural, such questions arise naturally when studying "S-decoding polynomials" for the known matching vector families. Finding S-decoding polynomials of reduced sparsity, which corresponds to finding delta functions with low Fourier sparsity, would improve the current best PIR schemes. We show nontrivial upper and lower bounds on the Fourier sparsity of delta functions. Our proofs are elementary and clean. These results imply limitations on improving Matching Vector PIR schemes simply by finding better S-decoding polynomials. In particular, there are no S-decoding polynomials that can make Matching Vector PIRs based on the known matching vector families achieve polylogarithmic communication with a constant number of servers. Many interesting questions remain open.

Problem

• Investigating Fourier sparsity of delta functions on Boolean domains
• Exploring implications for Matching Vector Private Information Retrieval schemes
• Establishing bounds to assess limitations of improving PIR via decoding polynomials

Innovation

• Delta functions studied for Fourier sparsity bounds
• S-decoding polynomials linked to matching vector PIR schemes
• Upper and lower bounds limit PIR communication improvements