Mosaics of combinatorial designs for privacy amplification

Abstract

We introduce, motivate and study ε-almost collision-flat (ACFU) universal hash functions f : X ×S → A. Their main property is that the number of collisions in any given value is bounded. Each ε-ACFU hash function is an ε-almost universal (AU) hash function, and every ε-almost strongly universal (ASU) hash function is an ε-ACFU hash function. We study how the size of the seed set S depends on ε, |X | and |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 S or X , it is possible to obtain an ε-ACFU hash function from an ε-AU hash function or an ε-ASU hash function, generalizing the construction of mosaics of designs from a given resolvable design (Gnilke, Greferath, Pavčević, 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. * A short and preliminary version of this paper is [31]. It mainly contains: two of the lower bounds on the size of an ACFU hash function (Theorem 3.4) without proof; a less general version of Theorem 4.1, which shows how to construct an ACFU hash function from an almost universal hash function; all examples without the discussion of their design-theoretic properties; a discussion of the application of ACFU hash functions to privacy amplification which is more detailed than here (Section 5) and gives a better bound on the attained security level.