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Let $f\colon \{0,1\}^n \to \{0,1\}^n$ be a length-preserving (i.e., $|f(x)| = |x|$) one-way function. Then, for $k \in \mathbb{N}$ and $m = n^k$, the function $g\colon \{ 0,1\}^m \to \{0,1\}^n$ with $$g(x_1, \ldots, x_m) = f(x_1)$$ is also one-way. Otherwise, being given $f(x)$ and $1^n$ as well as the description of a PPT $A$ which inverts $g$, one could run $A(f(x), 1^m)$ to obtain a preimage $x'$ for $f(x)$. This procedure can be realized in polynomial time since $m = n^k$ is polynomial in $n$.

The above gives a one-way function $g$ for which $|g(x)| = |x|^{\frac{1}{k}}$ for constant $k$. My question is: Does there exist one-way functions whose value is even shorter than this? For example, is there a one-way $g$ for which $|g(x)| = \log |x|$?

Clearly, $|g(x)| \in O(1)$ is not possible since producing preimages for $g$ would then be possible with only constant information; hence my question about this apparent "gap" between $O(1)$ and $O(n^{\frac{1}{k}})$.

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    $\begingroup$ Remember why we give $1^n$ to the adversary in the first place. $\endgroup$
    – fkraiem
    May 10, 2019 at 12:55
  • $\begingroup$ @fkraiem So the complexity of inverting $f$ is related not only to $|f(x)|$, but also to the length $|x|$ of (some) preimage $x$. But why would this rule out $|f(x)| = \log |x|$? $\endgroup$
    – dkaeae
    May 10, 2019 at 13:02
  • $\begingroup$ What if $f$ is not length preserving? $\endgroup$
    – kelalaka
    May 10, 2019 at 14:06
  • $\begingroup$ @kelalaka Well, the construction of $g$ just presupposes there is such an $f$. The existence of a length-preserving OWF is directly implied by the existence of OWFs (by using padding; in a way it is similar to the construction here, but in "reverse"). $\endgroup$
    – dkaeae
    May 10, 2019 at 14:25

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