  We introduce, motivate and study $\varepsilon$-almost collision-flat (ACFU)
universal hash functions $f:\mathcal X\times\mathcal S\to\mathcal A$. Their
main property is that the number of collisions in any given value is bounded.
Each $\varepsilon$-ACFU hash function is an $\varepsilon$-almost universal (AU)
hash function, and every $\varepsilon$-almost strongly universal (ASU) hash
function is an $\varepsilon$-ACFU hash function. We study how the size of the
seed set $\mathcal S$ depends on $\varepsilon,|\mathcal X|$ and $|\mathcal A|$.
Depending on how these parameters are interrelated, seed-minimizing ACFU hash
functions are equivalent to mosaics of balanced incomplete block designs
(BIBDs) or to duals of mosaics of quasi-symmetric block designs; in a third
case, mosaics of transversal designs and nets yield seed-optimal ACFU hash
functions, but a full characterization is missing. By either extending
$\mathcal S$ or $\mathcal X$, it is possible to obtain an $\varepsilon$-ACFU
hash function from an $\varepsilon$-AU hash function or an $\varepsilon$-ASU
hash function, generalizing the construction of mosaics of designs from a given
resolvable design (Gnilke, Greferath, Pav{\v c}evi\'c, Des. Codes Cryptogr.
86(1)). The concatenation of an ASU and an ACFU hash function again yields an
ACFU hash function. Finally, we motivate ACFU hash functions by their
applicability in privacy amplification.
