MMH* with arbitrary modulus is always almost-universal

TL;DR

This paper generalizes the MMH* universal hash family to arbitrary moduli, proving that the new GMMH* family is almost-universal with a bound depending on the smallest prime divisor of the modulus.

Contribution

It introduces GMMH*, extending MMH* to arbitrary moduli, and establishes tight bounds on its almost-universality based on prime divisors.

Findings

GMMH* is $rac{1}{p}$-almost-$ riangle$-universal for modulus n

The bound on universality is tight

Applicable to arbitrary integer moduli in hash functions

Abstract

Universal hash functions, discovered by Carter and Wegman in 1979, are of great importance in computer science with many applications. MMH$^*$ is a well-known $\triangle$-universal hash function family, based on the evaluation of a dot product modulo a prime. In this paper, we introduce a generalization of MMH$^*$, that we call GMMH$^*$, using the same construction as MMH$^*$ but with an arbitrary integer modulus $n>1$, and show that GMMH$^*$ is $\frac{1}{p}$-almost-$\triangle$-universal, where $p$ is the smallest prime divisor of $n$. This bound is tight.