与密码学相关的PPP-完备性
PPP-Completeness with Connections to Cryptography
摘要：
多项式鸽子洞原理（Polynomial Pigeonhole Principle，PPP）是TFNP的一个重要子类，与基本密码原语的复杂性有着深刻的联系：抗冲突哈希函数和单向排列。与TFNP的大多数其他子类相比，PPP没有完整的问题。我们的工作确定了第一个PPP完全问题，没有任何电路或图灵机明确给出的输入，因此，我们回答了一个长期悬而未决的问题[Papadimitriou 1994]。具体来说，我们表明，约束SIS（cSIS），一个广义的版本，著名的短解问题（SIS）从基于格的密码学，是PPP完全的。为了给我们的约束SIS减少背后的直觉，我们确定了另一个PPP完全问题的输入电路，但密切相关的格问题。我们把这个问题称为BLICHFELDT，它是与格理论中的Blichfeldt基本定理相关的计算问题。基于PPP与抗碰撞哈希函数的内在联系，我们使用我们的完整性结果来构建第一个自然哈希函数族，该函数族在最坏情况下捕获所有抗碰撞哈希函数的硬度，即在最坏情况下它是自然和通用的。我们的散列函数家族与SIS非常相似，这使我们得到了第一个候选的抗碰撞散列函数，它在平均情况下既自然又通用。最后，我们的研究结果丰富了我们对PPP，格问题和其他具体的密码学假设，如一般群上的离散对数问题之间的联系的理解。
英文摘要：
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, and thus we answer a longstanding open question from [Papadimitriou1994]. Specifically, we show that constrained-SIS (cSIS), a generalized version of the well-known Short Integer Solution problem (SIS) from lattice-based cryptography, is PPP-complete. In order to give 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. We call this problem BLICHFELDT and it is 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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