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Linear hash functions

Authors:

Martin Dietzfelbinger,

Peter Bro Miltersen,

Gábor TardosAuthors Info & Claims

Journal of the ACM (JACM), Volume 46, Issue 5

Pages 667 - 683

https://doi.org/10.1145/324133.324179

Published: 01 September 1999 Publication History

Abstract

Consider the set ℋ of all linear (or affine) transformations between two vector spaces over a finite field F. We study how good ℋ is as a class of hash functions, namely we consider hashing a set S of size n into a range having the same cardinality n by a randomly chosen function from ℋ and look at the expected size of the largest hash bucket. ℋ is a universal class of hash functions for any finite field, but with respect to our measure different fields behave differently.

If the finite field F has n elements, then there is a bad set S ⊂ F² of size n with expected maximal bucket size Ω(n^1/3). If n is a perfect square, then there is even a bad set with largest bucket size always at least √n. (This is worst possible, since with respect to a universal class of hash functions every set of size n has expected largest bucket size below √ + 1/2.)

If, however, we consider the field of two elements, then we get much better bounds. The best previously known upper bound on the expected size of the largest bucket for this class was O(2^{√ log n}). We reduce this upper bound to O(log n log logn). Note that this is not far from the guarantee for a random function. There, the average largest bucket would be Θ (log n/ log log n).

In the course of our proof we develop a tool which may be of independent interest. Suppose we have a subset S of a vector space D over Z₂, and consider a random linear mapping of D to a smaller vector space R. If the cardinality of S is larger than c_ε|R|log|R|, then with probability 1 - ϵ, the image of S will cover all elements in the range.

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Cited By

Dhar MDvir Z(2022) Linear Hashing with ℓ ∞ guarantees and two-sided Kakeya bounds 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS54457.2022.00047(419-428)Online publication date: Oct-2022
https://doi.org/10.1109/FOCS54457.2022.00047
Apple J(2021)HalftimeHash: Modern Hashing Without 64-Bit Multipliers or Finite FieldsAlgorithms and Data Structures10.1007/978-3-030-83508-8_8(101-114)Online publication date: 31-Jul-2021
https://doi.org/10.1007/978-3-030-83508-8_8
Kaslasi IRothblum RVasudevanr P(2021)Public-Coin Statistical Zero-Knowledge Batch Verification Against Malicious VerifiersAdvances in Cryptology – EUROCRYPT 202110.1007/978-3-030-77883-5_8(219-246)Online publication date: 17-Oct-2021
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Index Terms

Linear hash functions
1. Information systems
  1. Information storage systems
    1. Record storage systems
      1. Record storage alternatives
        Hashed file organization
2. Theory of computation
  1. Design and analysis of algorithms
    1. Data structures design and analysis
      1. Sorting and searching

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Published In

cover image Journal of the ACM

Journal of the ACM Volume 46, Issue 5

Sept. 1999

210 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/324133

Issue’s Table of Contents

Copyright © 1999 ACM.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 September 1999

Published in JACM Volume 46, Issue 5

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Cited By

Dhar MDvir Z(2022) Linear Hashing with ℓ ∞ guarantees and two-sided Kakeya bounds 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS54457.2022.00047(419-428)Online publication date: Oct-2022
https://doi.org/10.1109/FOCS54457.2022.00047
Apple J(2021)HalftimeHash: Modern Hashing Without 64-Bit Multipliers or Finite FieldsAlgorithms and Data Structures10.1007/978-3-030-83508-8_8(101-114)Online publication date: 31-Jul-2021
https://doi.org/10.1007/978-3-030-83508-8_8
Kaslasi IRothblum RVasudevanr P(2021)Public-Coin Statistical Zero-Knowledge Batch Verification Against Malicious VerifiersAdvances in Cryptology – EUROCRYPT 202110.1007/978-3-030-77883-5_8(219-246)Online publication date: 17-Oct-2021
https://dl.acm.org/doi/10.1007/978-3-030-77883-5_8
Chattopadhyay EGoodman JGoyal VKumar ALi XMeka RZuckerman D(2020)Extractors and Secret Sharing Against Bounded Collusion Protocols2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS46700.2020.00117(1226-1242)Online publication date: Nov-2020
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Chen XChan T(2019)Derandomized balanced allocationProceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3310435.3310589(2513-2526)Online publication date: 6-Jan-2019
https://dl.acm.org/doi/10.5555/3310435.3310589
Knudsen M(2019)Linear Hashing Is AwesomeSIAM Journal on Computing10.1137/17M112680148:2(736-741)Online publication date: 30-Apr-2019
https://doi.org/10.1137/17M1126801
Berman IDegwekar ARothblum RVasudevan P(2018)Multi-Collision Resistant Hash Functions and Their ApplicationsAdvances in Cryptology – EUROCRYPT 201810.1007/978-3-319-78375-8_5(133-161)Online publication date: 31-Mar-2018
https://doi.org/10.1007/978-3-319-78375-8_5
Knudsen M(2016)Linear Hashing Is Awesome2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS.2016.45(345-352)Online publication date: Oct-2016
https://doi.org/10.1109/FOCS.2016.45
Pǎtraşcu MThorup M(2015)On the k-Independence Required by Linear Probing and Minwise IndependenceACM Transactions on Algorithms10.1145/271631712:1(1-27)Online publication date: 16-Nov-2015
https://dl.acm.org/doi/10.1145/2716317
Meka RReingold ORothblum GRothblum R(2014)Fast Pseudorandomness for Independence and Load BalancingAutomata, Languages, and Programming10.1007/978-3-662-43948-7_71(859-870)Online publication date: 2014
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