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# On Hashing by (Random) Equations (Invited Talk)

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LIPIcs.ESA.2023.1.pdf
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## Cite As

Martin Dietzfelbinger. On Hashing by (Random) Equations (Invited Talk). In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, p. 1:1, Schloss Dagstuhl β Leibniz-Zentrum fΓΌr Informatik (2023)
https://doi.org/10.4230/LIPIcs.ESA.2023.1

## Abstract

The talk will consider aspects of the following setup: Assume for each (key) x from a set π° (the universe) a vector a_x = (a_{x,0},β¦ ,a_{x,{m-1}}) has been chosen. Given a list Z = (z_i)_{i β [m]} of vectors in {0,1}^r we obtain a mapping Ο_Z: π° β {0,1}^r, x β¦ β¨a_x,Zβ© := β¨_{i β [m]} a_{x,i} β z_i, where β¨ is bitwise XOR. The simplest way for creating a data structure for calculating Ο_Z is to store Z in an array Z[0..m-1] and answer a query for x by returning β¨ a_x,Zβ©. The length m of the array should be (1+Ξ΅)n for some small Ξ΅, and calculating this inner product should be fast. In the focus of the talk is the case where for all or for most of the sets S β π° of a certain size n the vectors a_x, x β S, are linearly independent. Choosing Z at random will lead to hash families of various degrees of independence. We will be mostly interested in the case where a_x, x β π° are chosen independently at random from {0,1}^m, according to some distribution π. We wish to construct (static) retrieval data structures, which means that S β π° and some mapping f: S β {0,1}^r are given, and the task is to find Z such that the restriction of Ο_Z to S is f. For creating such a data structure it is necessary to solve the linear system (a_x)_{x β S} β Z = (f(x))_{x β S} for Z. Two problems are central: Under what conditions on m and π can we expect the vectors a_x, x β S to be linearly independent, and how can we arrange things so that in this case the system can be solved fast, in particular in time close to linear (in n, assuming a reasonable machine model)? Solutions to these problems, some classical and others that have emerged only in recent years, will be discussed.

## Subject Classification

##### ACM Subject Classification
• Theory of computation β Sorting and searching
• Theory of computation β Randomness, geometry and discrete structures
##### Keywords
• Hashing
• Retrieval
• Linear equations
• Randomness

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