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.

翻译：本文引入、论证并研究了$\varepsilon$-几乎无碰撞全通（ACFU）通用哈希函数$f:\mathcal X\times\mathcal S\to\mathcal A$。其主要性质是任意给定值中的碰撞次数有界。每个$\varepsilon$-ACFU哈希函数均为$\varepsilon$-几乎通用（AU）哈希函数，而每个$\varepsilon$-几乎强通用（ASU）哈希函数也是$\varepsilon$-ACFU哈希函数。我们研究了种子集$\mathcal S$的大小如何依赖于$\varepsilon$、$|\mathcal X|$和$|\mathcal A|$。根据这些参数之间的相互关系，种子最小化的ACFU哈希函数等价于平衡不完全区组设计（BIBD）的镶嵌，或拟对称区组设计镶嵌的对偶；在第三种情形下，横截设计镶嵌与网格可产生种子最优的ACFU哈希函数，但完整的刻画尚待完善。通过扩展$\mathcal S$或$\mathcal X$，可以从$\varepsilon$-AU哈希函数或$\varepsilon$-ASU哈希函数构造出$\varepsilon$-ACFU哈希函数，这推广了基于给定可分解设计构建设计镶嵌的方法（参见Gnilke, Greferath, Pav{\v c}evi\'c, Des. Codes Cryptogr. 86(1)）。ASU与ACFU哈希函数的级联仍产生ACFU哈希函数。最后，我们通过ACFU哈希函数在隐私放大中的适用性论证了其应用价值。