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Consider a permutation $f:\{0,1\}^*\rightarrow \{0,1\}^*$, which is not a one-way function, i.e. there exists an efficient probabilistic adversary $\mathcal{A}$ and some polynomial $q(n)$ such that for infinitely many $n$

\begin{equation} \mathrm{Pr}[\mathcal{A}(f(x)) = x] > \frac{1}{q(n)}, \end{equation} where the probability is over the internal randomness of $\mathcal{A}$ and $x$ drawn uniformly at random from $\{0,1\}^n$.

I would now like to prove that $f^{p(n)}$ is not a one way function, for any polynomial $p(n)$. Is this true for all permutations $f$ which are not one way?

My intuition was that this would be true, and that one could prove this by showing that $\tilde{\mathcal{A}} = \mathcal{A}^{p(n)}$ is a suitable adversary for $f^{p(n)}$. However, I can't get this to work. Is there a clever way to prove this, or is the statement I am trying to prove actually false?

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    $\begingroup$ Let's considet the identity function as f. $\endgroup$ – xagawa Nov 28 '19 at 9:13
  • $\begingroup$ I would like to understand whether for all permutations $f$ which are not one way, $f^{p(n)}$ is also not one way. The identity function is an example of a specific permutation $f$ which is not one way, and for which $f^{p(n)}$ is also not one way for all $p(n)$. As such, it doesn't provide a counter example to the conjecture, it just provides one example of a specific function which has this property - I would really like to understand if this holds for all permutations which are not one way. I have edited the question slightly to make this clear. $\endgroup$ – Ryan Nov 28 '19 at 12:01
  • $\begingroup$ I think this statement might be false because an adversary only needs to be successful for infinitely many $n$ and not all of them, so a non OWP f could be constructed as "return the first n-1 bits" if $n$ is even and "run this actual OWP $\pi$" if $n$ is odd, then this would be a non-OWF because for every second length there is an inverter but inverting twice is hard. $\endgroup$ – SEJPM♦ Nov 28 '19 at 13:34
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    $\begingroup$ I don't think this construction works as a counter-example, because for $n$ even we have that $f:\{0,1\}^n \rightarrow \{0,1\}^{n-1}$, and therefore $f$ is not a permutation for even $n$. $\endgroup$ – Ryan Nov 28 '19 at 14:11
  • $\begingroup$ You are right, I was not precise enough on the permutation part, at least the above is still an instructive counter-example for the more general non-OWF case even if it doesn't work for non-OWPs. Maybe there is still a way to change the length of the output if we design a clever permutation say for every two or three consecutive bitlengths. $\endgroup$ – SEJPM♦ Nov 28 '19 at 15:06
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If a permutation $f$ is not one way, we can not conclude about the one-wayness of $f^{p(n)}$. In fact, even $f^2$ could be one-way, if there are one-way length-preserving permutations that is.

Constructive proof: assume $g:\{0,1\}^*\rightarrow \{0,1\}^*$ is a length-preserving OWP, not invertible starting with rank $m$. Define $f$ from $g$ such that, for any bitstring $x$,

$$\begin{align}f(x\mathbin\|0)&=x\mathbin\|1\\f(x\mathbin\|1)&=g(x)\mathbin\|0\end{align}$$

$f$ is a permutation. It is trivially invertible for half its inputs of any fixed width, thus is not a OWP.

$f^2$ is also a permutation, with

$$\begin{align}f^2(x\mathbin\|0)&=g(x)\mathbin\|0\\f^2(x\mathbin\|1)&=g(x)\mathbin\|1\end{align}$$

An hypothetical algorithm that inverts $f^2$ starting at width $n>m$ could be used to invert $g$ at width $n-1\ge m$. With $g$ being a OWF, there is no such algorithm. Hence $f^2$ is one-way.

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  • $\begingroup$ Straight from The Book :) $\endgroup$ – Krystian Nov 29 '19 at 13:57
  • $\begingroup$ How would a permutation not be length preserving? $\endgroup$ – Maeher Dec 2 '19 at 8:18
  • $\begingroup$ @Maeher: how about the transformation $h:\{0,1\}^*\rightarrow \{0,1\}^*$ which leaves all bitstrings unchanged, except the all-zero bitstrings (including empty), with that of length $2k$ and $2k+1$ exchanged? $\endgroup$ – fgrieu♦ Dec 2 '19 at 8:41

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