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We introduce, motivate and study $\varepsilon$-almost collision-flat (ACFU)universal hash functions $f:\mathcal X\times\mathcal S\to\mathcal A$. Theirmain 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) hashfunction is an $\varepsilon$-ACFU hash function. We study how the size of theseed set $\mathcal S$ depends on $\varepsilon,|\mathcal X|$ and $|\mathcal A|$.Depending on how these parameters are interrelated, seed-minimizing ACFU hashfunctions are equivalent to mosaics of balanced incomplete block designs(BIBDs) or to duals of mosaics of quasi-symmetric block designs; in a thirdcase, mosaics of transversal designs and nets yield seed-optimal ACFU hashfunctions, but a full characterization is missing. By either extending$\mathcal S$ or $\mathcal X$, it is possible to obtain an $\varepsilon$-ACFUhash function from an $\varepsilon$-AU hash function or an $\varepsilon$-ASUhash function, generalizing the construction of mosaics of designs from a givenresolvable 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 anACFU hash function. Finally, we motivate ACFU hash functions by theirapplicability in privacy amplification.

$\varepsilon$-Almost collision-flat universal hash functions and mosaics of designs