Independence of Tabulation-Based Hash Classes

TL;DR

This paper investigates the independence properties of tabulation-based hash functions, demonstrating a new variant with higher independence and establishing lower bounds on the number of derived characters needed for k-wise independence.

Contribution

It introduces a variant of tabulation hashing achieving (2d-1)-wise independence and provides a lower bound on the number of derived characters for k-wise independence.

Findings

A new variant achieves (2d-1)-wise independence.

A lower bound on derived characters for k-wise independence.

Analysis based on algebraic and geometric properties.

Abstract

A tabulation-based hash function maps a key into d derived characters indexing random values in tables that are then combined with bitwise xor operations to give the hash. Thorup and Zhang (2004) presented d-wise independent tabulation-based hash classes that use linear maps over finite fields to map a key, considered as a vector (a,b), to derived characters. We show that a variant where the derived characters are a+b*i for i=0,..., q-1 (using integer arithmetic) yielding (2d-1)-wise independence. Our analysis is based on an algebraic property that characterizes k-wise independence of tabulation-based hashing schemes, and combines this characterization with a geometric argument. We also prove a non-trivial lower bound on the number of derived characters necessary for k-wise independence with our and related hash classes.