Fourier Sparsity of Delta Functions and Matching Vector PIRs

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.

💡 Research Summary

This paper investigates a fundamental question at the intersection of Fourier analysis, combinatorics, and cryptography: how sparse can the Fourier representation of a “delta function” on the Boolean hypercube be? Specifically, for a finite abelian group G (typically Z_m^r) and the subset B = {0,1}^r, a delta function f: G → F satisfies f(0)=1 and f(x)=0 for all other x in B. The Fourier sparsity of f is the number of non-zero coefficients in its Fourier expansion over the characters of G.

The primary motivation stems from Private Information Retrieval (PIR) schemes, particularly the Matching Vector (MV) based constructions, which are the only known way to achieve subpolynomial communication complexity with a constant number of servers. A key component in optimizing these schemes is finding a sparse “S-decoding polynomial.” The paper establishes a direct equivalence: the problem of finding a sparse S-decoding polynomial for the canonical set S (arising from the product of primes m=p1…pr) is precisely the problem of finding a delta function on {0,1}^r within the group G = Z_{p1} × … × Z_{pr} with low Fourier sparsity.

The main technical contributions are non-trivial upper and lower bounds on this Fourier sparsity:

• Lower Bounds: Any delta function on {0,1}^r within Z_m^r must have Fourier sparsity at least (m/(m-1))^r. For the PIR-relevant case where m is a product of distinct primes, a stronger, linear lower bound of at least r+1 is proven.
• Upper Bounds: Explicit constructions show that sparsity O(m^(r/(m-1))) is achievable. Notably, when m > r, sparsity as low as r+1 is possible and optimal.
• Implication for PIR: The r+1 lower bound for the distinct-prime case directly implies that any S-decoding polynomial for the canonical set S must have at least r+1 monomials. This places a fundamental limitation on improving MV-PIR schemes via better decoding polynomials alone. It proves that using the known matching vector families (from BBR/Grolmusz), no S-decoding polynomial can achieve polylogarithmic communication complexity with a constant number of servers; the communication remains at best exponential in (log n)^(1/t).
• Contrast with {-1,0,1}^r: The paper also studies delta functions on the larger set {-1,0,1}^r and proves a much stronger, m-independent lower bound of 2^r, highlighting a significant structural difference between the two settings.

The proofs are elementary, relying on linear algebra and combinatorial arguments. The paper concludes by highlighting compelling open questions, such as tightening the exponential gap for fixed m, and resolving the conjectured 2^r lower bound for delta functions over products of cyclic groups of distinct prime orders when using complex-valued Fourier analysis. This work underscores the richness of analyzing Boolean functions within larger algebraic domains and its concrete implications for coding and cryptography.