The Power of Hashing with Mersenne Primes

TL;DR

This paper explores advantages of using Mersenne primes in hash functions, introducing a 'Two for one' hashing technique that improves efficiency and maintains strong theoretical properties, demonstrated in Count Sketch for second-moment estimation.

Contribution

The paper introduces the 'Two for one' hashing method leveraging Mersenne primes, providing a new fast division/modulus algorithm, and applying these to improve hash-based algorithms.

Findings

Enhanced hash functions with Mersenne primes preserve strong theoretical properties.

The 'Two for one' hashing technique allows multiple hash functions from a single computation.

New fast branch-free code for division and modulus with Mersenne primes and Pseudo-Mersenne primes.

Abstract

The classic way of computing a $k$-universal hash function is to use a random degree-$(k-1)$ polynomial over a prime field $\mathbb Z_p$. For a fast computation of the polynomial, the prime $p$ is often chosen as a Mersenne prime $p=2^b-1$. In this paper, we show that there are other nice advantages to using Mersenne primes. Our view is that the hash function's output is a $b$-bit integer that is uniformly distributed in $\{0, \dots, 2^b-1\}$, except that $p$ (the all \texttt1s value in binary) is missing. Uniform bit strings have many nice properties, such as splitting into substrings which gives us two or more hash functions for the cost of one, while preserving strong theoretical qualities. We call this trick "Two for one" hashing, and we demonstrate it on 4-universal hashing in the classic Count Sketch algorithm for second-moment estimation. We also provide a new fast branch-free code for division and modulus with Mersenne primes. Contrasting our analytic work, this code generalizes to any Pseudo-Mersenne primes $p=2^b-c$ for small $c$.