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

Abstract

We introduce, motivate and study $$\varepsilon $$ ε -almost collision-flat universal (ACFU) hash functions $$f:\mathcal X\times \mathcal S\rightarrow \mathcal A$$ f : X × S → A . Their main property is that the number of collisions in any given value is bounded. Each $$\varepsilon $$ ε -ACFU hash function is an $$\varepsilon $$ ε -almost universal (AU) hash function, and every $$\varepsilon $$ ε -almost strongly universal (ASU) hash function is an $$\varepsilon $$ ε -ACFU hash function. We study how the size of the seed set $$\mathcal S$$ S depends on $$\varepsilon ,|\mathcal X |$$ ε , | X | and $$|\mathcal A |$$ | 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 $$\mathcal S$$ S or $$\mathcal X$$ X , it is possible to obtain an $$\varepsilon $$ ε -ACFU hash function from an $$\varepsilon $$ ε -AU hash function or an $$\varepsilon $$ ε -ASU hash function, generalizing the construction of mosaics of designs from a given resolvable design (Gnilke et al. in Des. Codes Cryptogr. 86(1):85–95, 2017). 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.