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I am trying to prove that a function is a one-way function.

The function I am working on in particular is $f'(x,y)=f(x)||f(x \oplus y)$.

For what I have understood looking at similar solved solutions (e.g. 1c), the strategy is the following:

  1. Assume $f'$ is not one way, so exists an $A'$ that can invert $f'$ with non-negligible probability
  2. Use $A'$ to construct $A$ that can invert $f$.
  3. Show that $A$ inverts $A$ with non-negligible probability
  4. Contradiction.

Based on this knowledge, I am trying to prove the given

$$f'(x,y)=f(x)||f(x \oplus y)$$


So, here is my attempt:

  1. Assume $f'(x,y)$ is not one way function
  2. So, exists a $A'$ that can invert $f'$ probability
  3. Construct

    $A(z)$:

    • pick a $w \in f(x)$ (for any $x$)
    • $x,y <- A'(f(z, y)) = A'(f(x) || f(x \oplus w))$
    • we have now inverted the first half
    • ...
    • return $x,y$

However, although I understand the method I cannot find any valid way of making the construction:

  • How can you build such construction?
  • Is there something I am doing wrong? or am I in the right direction?
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The easiest way to construct a probabilistic $A$ is simply, given $z$:

  • Select a random $w$
  • Compute $A'(z || f(w))$.
  • If $A'$ succeeds, then it'll evaluate to an $(x, y)$ pair with $z = f(x)$ and $f(w) = f(x \oplus y)$.
  • Discard $y$, and return $x$ as $A(z)$.

We select a random $f(w)$, rather than just inserting another copy of $z$, because it is possible that $A'$ would always fail on an input of the form $(z || z)$. As I have written, if $A'$ succeeds on a nontrivial fraction of its inputs, then $A$ will succeed on the same nontrivial fraction of its inputs.

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  • $\begingroup$ this is not enough though, we have to show that A' is non-negligible for the contraddiction $\endgroup$ – graphtheory92 Feb 20 '16 at 21:56
  • $\begingroup$ @graphtheory92 The key observation is that if $x$ and $y$ are uniformly and independently distributed, then so are $x$ and $x \oplus y$, so since $w$ is chosen independently of $x$, $f(x)||f(w)$ is identically distributed to $f(x)||f(x\oplus y)$, and so the success probability of $A'$ is at least that of $A$ (because if $A$ succeeds, so does $A'$). $\endgroup$ – fkraiem Feb 21 '16 at 0:30

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