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

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

Additive Approximation Schemes for Load Balancing Problems

We formalize the concept of additive approximation schemes and apply it to load balancing problems on identical machines. Additive approximation schemes compute a solution with an absolute error in the objective of at most ? h for some suitable parameter h and any given ? > 0. We consider the problem of assigning jobs to identical machines with respect to common load balancing objectives like makespan minimization, the Santa Claus problem (on identical machines), and the envy-minimizing Santa Claus problem. For these settings we present additive approximation schemes for h = p_{max}, the maximum processing time of the jobs. Our technical contribution is two-fold. First, we introduce a new relaxation based on integrally assigning slots to machines and fractionally assigning jobs to the slots. We refer to this relaxation as the slot-MILP. While it has a linear number of integral variables, we identify structural properties of (near-)optimal solutions, which allow us to compute those in polynomial time. The second technical contribution is a local-search algorithm which rounds any given solution to the slot-MILP, introducing an additive error on the machine loads of at most ?? p_{max}

Information-Theoretically Secret Reed-Muller Identification with Affine Designs

We consider the problem of information-theoretic secrecy in identification schemes rather than transmission schemes. In identification, large identities are encoded into small challenges sent with the sole goal of allowing at the receiver reliable verification of whether the challenge could have been generated by a (possibly different) identity of his choice. One of the reasons to consider identification is that it trades decoding for an exponentially larger rate, however this may come with such encoding complexity and latency that it can render this advantage unusable. Identification still bears one unique advantage over transmission in that practical implementation of information-theoretic secrecy becomes possible, even considering that the information-theoretic secrecy definition needed in identification is that of semantic secrecy. Here, we implement a family of encryption schemes, recently shown to achieve semantic-secrecy capacity, and apply it to a recently-studied family of identification codes, confirming that, indeed, adding secrecy to identification comes at essentially no cost. While this is still within the one-way communication scenario, it is a necessary step into implementing semantic secrecy with two-way communication, where the information-theoretic assumptions are more realistic.Comment: 6 pages, 3 figures, accepted at European Wireless 202