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. *
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