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I'm interested in exploring the construction of injective pseudorandom generators (PRGs), i.e., PRGs that are also 1-1 functions. We know that we can construct regular PRGs from one-way functions, as shown by Håstad et al.

While it is known that injective PRGs can be built from one-way permutations (as discussed in this post), one-way functions do not imply the existence of one-way permutations, as demonstrated by Rudich. This raises the question: are there any known constructions of injective PRGs that rely directly on one-way functions, without assuming the existence of one-way permutations?

Mahesh S R
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1 Answers1

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There is no black-box construction of injective OWFs from OWFs. This is a folklore result that I don't think is formally stated anywhere, but follows from (the same analysis as) existing black-box separations between OWFs and OWPs. This is for example mentioned explicitly in a footnote on the first page of this paper, where it is credited as a result "implicit" in those three papers (I did not personally check whether it's true, but I believe the authors on this topic). It follows that there is no black-box construction of injective PRG from OWFs.

On the positive side, if you start from an exponentially secure OWF, then it's possible to construct (in a black-box way) an almost 1-1 family of OWFs, via the construction of this paper. It looks highly plausible to me that plugging this "almost 1-1 OWF family" into the construction of injective PRG from OWP would yield an "almost injective PRG family" (i.e., a family of PRGs where, with overwhelming probability over the sampling of a random PRG from the family, the sampled PRG is injective). I did, however, not thoroughly check this claim and one would have to carefully prove it, I don't think it has been formally proven anywhere.

Geoffroy Couteau
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