Existence of injective one-way real functions (Levin framework)

Determine whether injective one-way real functions exist in Levin’s framework; specifically, ascertain whether there is a partial computable injection f: dom(f) ⊆ 2^ω -> 2^ω that is random-preserving and has no partial computable probabilistic inversion with positive probability. In particular, since such functions cannot be total computable, resolve the existence of partial computable injective one-way real functions.

Background

The paper studies computable real functions in the sense of Levin, where a one-way function is partial computable, random-preserving, and lacks a partial computable probabilistic inversion. The authors construct collision-resistant one-way functions that are nowhere injective.

They highlight that while injective one-way maps (permutations) are important in classical complexity, the existence of injective one-way functions on the reals is unknown; moreover, prior results imply such functions cannot be total computable. Later in Section 5, they reiterate the unresolved status more precisely: they do not know whether partial computable injective one-way functions exist, and provide upper bounds that show such injections would be easier to invert than total many-to-one maps.