  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 $c$.
