On Black-Box Separations among Injective One-Way Functions

A one-way permutation (OWP) is one of the most fundamental cryptographic primitives, and can be used as a building block for most of basic symmetric-key cryptographic primitives. However, despite its importance and usefulness, previous black-box separation results have shown that constructing a OWP from another primitive seems hopeless, unless building blocks already achieve “one-way” property and “permutation” property simultaneously. In this paper, in order to clarify more about the constructions of a OWP from other primitives, we study the construction of a OWP from primitives that are very close to a OWP. Concretely, as a negative result, we show that there is no fully black-box construction of a OWP from a length-increasing injective one-way function (OWF), even if the latter is just 1-bit-increasing and achieves strong form of one-wayness which we call adaptive one-wayness. As a corollary, we show that there is no fully black-box construction of a OWP from a regular OWF with regularity greater than 1. Since a permutation is length-preserving and injective, and is a regular OWF with regularity 1, our negative result indicates that to construct a OWP from another primitive is quite difficult, even if we use very close primitives to a OWP as building blocks. Moreover, we extend our separation result of a OWP from a length-increasing injective OWF, and show a certain restrictive form of black-box separations among injective OWFs in terms of how much a function stretches its input. This result shows a hierarchy among injective OWFs (including a OWP).