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I am studying a little bit about one-way functions from Dr Goyal's notes.

He constructs the following OWF:

Let $D=\Bbb Z_2^{n^3}$ and $R=\Bbb Z_2^{2 n}$. Given $x \in D$, $f$ interprets $x$ as a set $S$ of $n^2$ numbers, each of $n$ bits and uses deterministic PT primality test to find the first two 2 primes in $S$. If $f$ finds $p$ and $q$, then output $p q$. Otherwise, it outputs FAIL.

My Question here is what does it mean Mathematically the phrase $\color{blue}{\text{$f$ interprets $x$ as a set $S$ of $n^2$ numbers, each of $n$ bits}}$.

Can you please give a clarification together with an Example, in case mine is not correct?

For example, I took $n=2$ and $x=11101110\in \Bbb Z_2^{2^3}$, so $|x|=2^3$. Thus, we get the $n^2=2^2=4$, $n=2$-bit integers $11,10,11,10$, which correspond to $3$, $2$. Is this what he means?

Thanks!

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It is common to find this sort of high-level abstraction of how things are encoded. For this specific case, it uses the fact that the set $\Bbb Z_2^{n^3}$ has the same cardinality as $\Bbb Z_2^{n}\times \Bbb Z_2^{n}\times\dots\times \Bbb Z_2^{n}$ where the product is over $n^2$ copies of $\Bbb Z_2^{n}$. So there is a one-to-one mapping $\varphi$ that takes an element of $\Bbb Z_2^{n^3}$ and maps it to an element of $\Bbb Z_2^{n}\times \Bbb Z_2^{n}\times\dots\times \Bbb Z_2^{n}$. Note that formally speaking $S$ is a tuple of $n^2$ numbers and not a set since elements can repeat. It doesn't really matter what $\varphi$ is, only that it is one-to-one. Changing $\varphi$ would change the function $f$, but it is still one-way since you're only changing the encoding.

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  • $\begingroup$ Thank you for your answer, So, you say that $\Bbb Z_2^{n^3}$ is isomorphic to $\underbrace{\Bbb Z_2^{n}\times \Bbb Z_2^{n}\times\dots\times \Bbb Z_2^{n}}_{n^2}$, but as what structure? As rings? Also, what is precisely the map? In addition, it makes sense to me that $S$ is a collection of numbers; if my example is correct, then by taking the pairs of $11101110$ we get $11,10,11,10$, so in decimal $3,2$. So, the distinct elements of $S$ are only 2. But is it correct? $\endgroup$
    – Chris
    Oct 5, 2023 at 21:20
  • $\begingroup$ Isomorphic is not the right word, I meant that they have the same cardinality. I'll edit my answer. For your example, 11101110 is the binary representation of 356, so the function $\varphi$ would map 356 to the tuple (3,2,3,2) which is (11,10,11,10) in binary. $\endgroup$
    – lamontap
    Oct 6, 2023 at 12:40
  • $\begingroup$ So, $\varphi$ is in fact a bijection and my example is a correct interpretation. :) $\endgroup$
    – Chris
    Oct 6, 2023 at 20:09

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