Linear Hashing with Tight ℓ∞ Guarantees and Two-Sided Kakeya Bounds

Core Concepts

A randomly chosen linear map over a finite field gives a good hash function in the ℓ∞ sense, with the output being close to uniform in the ℓ∞ norm. This complements the widely-used Leftover Hash Lemma which provides guarantees in the ℓ1 norm.

Abstract

The main result of this paper is to show that a randomly chosen linear map over a finite field gives a good hash function in the ℓ∞ sense. Specifically:

Consider a set S ⊂ Fₙ_q and a randomly chosen linear map L: Fₙ_q → Fₜ_q, where qᵗ is sufficiently smaller than |S|.
The main theorem shows that, with high probability over the choice of L, the random variable L(Uₛ) is close to uniform in the ℓ∞ norm. This means that every element in the range Fₜ_q has about the same number of elements in S mapped to it.
This complements the Leftover Hash Lemma (LHL) which proves an analog statement under the ℓ₁ distance for a richer class of functions.
The proof leverages a connection between linear hashing and the finite field Kakeya problem, and extends some of the tools developed in this area, in particular the polynomial method.
The results are shown to be tight, demonstrating that linear functions hash as well as truly random functions up to a constant factor in the entropy loss.

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Key Insights Distilled From

Linear Hashing with $\ell_\infty$ guarantees and two-sided Kakeya bounds

by Manik Dhar,Z... at arxiv.org 04-03-2024

https://arxiv.org/pdf/2204.01665.pdf

Deeper Inquiries

How can the techniques developed in this paper be applied to other families of universal hash functions beyond linear maps

The techniques developed in the paper can be extended to other families of universal hash functions beyond linear maps by considering different types of functions that satisfy the properties required for ℓ∞ guarantees. For example, one could explore the use of high-degree polynomials over finite fields or other algebraic structures as hash functions. By adapting the proof techniques and analysis to these different families of functions, it may be possible to establish similar guarantees for ℓ∞ hashing in those contexts. Additionally, exploring the connections between linear hashing and other mathematical structures could lead to the development of new families of hash functions with ℓ∞ guarantees.

What are the implications of these ℓ∞ hashing guarantees for applications in cryptography and theoretical computer science

The implications of ℓ∞ hashing guarantees for applications in cryptography and theoretical computer science are significant. In cryptography, having ℓ∞ guarantees for hash functions ensures a high level of security and unpredictability in cryptographic protocols. These guarantees can be utilized in key generation for digital signatures, batch verifying zero-knowledge proofs, and uniformly generating solutions to NP-search problems. The ability to rely on linear maps or other simple functions for hashing while maintaining ℓ∞ guarantees simplifies the implementation of cryptographic schemes and reduces the computational complexity involved in generating secure keys.
In theoretical computer science, ℓ∞ hashing guarantees open up new possibilities for designing efficient algorithms and data structures. By ensuring that hash functions behave similarly to truly random functions in terms of ℓ∞ distance, researchers can develop more robust and reliable systems for data storage, retrieval, and processing. These guarantees can enhance the performance and security of various computational tasks, leading to advancements in areas such as load balancing, data integrity verification, and randomized algorithms.

Can the connection between linear hashing and finite field Kakeya problems be further explored to yield insights in other areas

The connection between linear hashing and finite field Kakeya problems presents a rich area for further exploration and insights in various fields. By delving deeper into this connection, researchers can potentially uncover new results and applications in mathematics, computer science, and cryptography. One avenue for exploration could involve investigating the relationship between linear hashing guarantees and the structure of Kakeya sets in finite fields, leading to a better understanding of the underlying principles governing both areas.
Furthermore, the insights gained from studying this connection could be applied to problems in combinatorics, algebra, and geometry, providing novel perspectives and solutions to longstanding mathematical challenges. By leveraging the tools and techniques developed in the context of linear hashing and finite field Kakeya problems, researchers may discover unexpected connections and applications in diverse areas of mathematics and computer science.