General factoring and one-way functions

Question

Let a function $f$ be one-way, if there exists a probabilistic polynomial time algorithm to find the preimage of $y = f(x)$ for uniformly chosen $x$ with non-negligible probability.

Define the function as $f(x,y) = x\times y$, where the values of $x$ and $y$ range across the integers in $[2^{n-1}, 2^n-1]$ (basically, they are $n$-bit numbers with the left-most bit set). Is $f$ one-way??

(Although with probability 0.75 the received $y$ is even, $2$ and $y/2$ cannot be output as the factors, since I want the preimages to lie in that specific range)

Answer 1

What you are wondering about is the difference between weak one-way function, and strong one-way function. Google should give you the definitions (e.g. here is the first link I found). The factoring function for general inputs is not a strong one-way function, since there are outputs for which it is easy to invert. But it is a weak OWF, because no polytime adversary can have more than a probability $1-1/\mathsf{poly}$ of inverting it (as far as we know).

Furthermore, even though a weak OWF is not a strong OWF in general, we know a generic way of building strong OWFs from weak OWFs - in other words, both assumptions are formally equivalent. See e.g. this online course, that deals specifically with the product function which you care about.