In the ε-Consensus-Halving problem, a fundamental problem in fair division, there are n agents with valuations over the interval [0, 1], and the goal is to divide the interval into pieces and assign a label "+" or "−" to each piece, such that every agent values the total amount of "+" and the total amount of "−" almost equally. The problem was recently proven by Filos-Ratsikas and Goldberg [18,19] to be the first "natural" complete problem for the computational class PPA, answering a decade-old open question.In this paper, we examine the extent to which the problem becomes easy to solve, if one restricts the class of valuation functions. To this end, we provide the following contributions. First, we obtain a strengthening of the PPA-hardness result of [19], to the case when agents have piecewise uniform valuations with only two blocks. We obtain this result via a new reduction, which is in fact conceptually much simpler than the corresponding one in [19]. Then, we consider the case of single-block (uniform) valuations and provide a parameterized polynomial time algorithm for solving ε-Consensus-Halving for any ε, as well as a polynomial-time algorithm for ε = 1/2; these are the first algorithmic results for the problem. Finally, an important application of our new techniques is the first hardness result for a generalization of Consensus-Halving, the Consensus-1/k-Division problem [33]. In particular, we prove that ε-Consensus-1/3-Division is PPAD-hard.

A. Hollender is supported by an EPSRC doctoral studentship (Reference 1892947).

Polynomial Pigeonhole Principle (PPP) is an important subclass of TFNP with profound connections to the complexity of the fundamental cryptographic primitives: collision-resistant hash functions and one-way permutations. In contrast to most of the other subclasses of TFNP, no complete problem is known for PPP. Our work identifies the first PPP-complete problem without any circuit or Turing Machine given explicitly in the input: constrained-SIS, and thus we answer a longstanding open question from [Pap94].constrained-SIS: a generalized version of the well-known Short Integer Solution problem (SIS) from lattice-based cryptography.In order to give some intuition behind our reduction for constrained-SIS, we identify another PPP-complete problem with a circuit in the input but closely related to lattice problems: BLICHFELDT.BLICHFELDT: the computational problem associated with Blichfeldt's fundamental theorem in the theory of lattices.Building on the inherent connection of PPP with collision-resistant hash functions, we use our completeness result to construct the first natural hash function family that captures the hardness of all collision-resistant hash functions in a worst-case sense, i.e. it is natural and universal in the worst-case. The close resemblance of our hash function family with SIS, leads us to the first candidate collision-resistant hash function that is both natural and universal in an average-case sense.Finally, our results enrich our understanding of the connections between PPP, lattice problems and other concrete cryptographic assumptions, such as the discrete logarithm problem over general groups. *

Background: Finding long matches in deoxyribonucleic acid (DNA) sequences in large aligned genetic sequences is a problem of great interest. A paradigmatic application is the identification of distant relatives via large common subsequences in DNA data. However, because of the sensitive nature of genomic data such computations without security consideration might compromise the privacy of the individuals involved. Methods: The secret sharing technique enables the computation of matches while respecting the privacy of the inputs of the parties involved. This method requires interaction that depends on the circuit depth needed for the computation. Results: We design a new depth-optimized algorithm for computing set-maximal matches between a database of aligned genetic sequences and the DNA of an individual while respecting the privacy of both the database owner and the individual. We then implement and evaluate our protocol. Conclusions: Using modern cryptographic techniques, difficult genomic computations are performed in a privacy-preserving way. We enrich this research area by proposing a privacy-preserving protocol for set-maximal matches.

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