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Say that I have a random oracle function $H$. This function outputs a value in $\mathbb{F}_{p}$ where $p \approx 2^{256}$. $H$ can accept either one or two inputs (outputting a single value in both cases).

I can hash two elements $x$ and $y$ using either

case 1: $H(x, y)$

case 2: $H(x) + H(y)$ (using modular addition)

How does the security of these approaches differ?

In case 1 there must be collisions because we're mapping two elements to one element. If $H$ is a random oracle then we should have collision odds $1/p$.

Is there something I'm missing with case 2? I'm assuming we get security from Schwartz-Zippel, $H(x) + H(y)$ being a multivariate linear polynomial with both variables randomly distributed in $\mathbb{F}$. Is the security the same as that of $H$? Does this significantly change based on the actual implementation of $H$ (e.g. sha256 vs poseidon vs md5 vs etc).

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    $\begingroup$ Hint (lesser than the next one): what about first preimage resistance of 2 ? $\endgroup$
    – fgrieu
    Apr 9, 2023 at 8:36
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    $\begingroup$ HINT: There's a very easy second preimage attack $\endgroup$
    – Daniel S
    Apr 9, 2023 at 10:34
  • $\begingroup$ @DanielS sorry to ask this, but in which case is he talking about ? I don t understand the meaning of the first sentence besides I know what is a finite field. $\endgroup$ Jul 10, 2023 at 22:56
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    $\begingroup$ @user2284570 Are you referring to the comment below the answer? If so the approach to find a preimage for a target $z$ in case 2 is to make two lists of length about $2^{n/2}$ one of values $H(x)$ and one of values $z-H(y)$ for random $x$ and $y$. We then look for a value that appears in both lists. Our total work is then about $2^{n/2}$. $\endgroup$
    – Daniel S
    Jul 11, 2023 at 7:16
  • $\begingroup$ @DanielS I’m referring to the question itself. My case is the first one anyway. Does this question apply only to the hash result of x and y of different lengths ? $\endgroup$ Jul 11, 2023 at 9:52

1 Answer 1

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Ok thank you for the comments.

For an input $x$ and $y$ there's a simple second pre-image attack in case 2:

$H(x) + H(y) = H(y) + H(x)$

The same problem applies if the elements are combined with multiplication as well.

There's also a first pre-image resistance problem. If you want a hash $z$ all you need to do is find $H(x) = z / 2$, then provide $x$ as the input twice. It follows that given $H(x) + H(y) = z$ the pre-image for any hash $2(z - H(x))$ or $2(z-H(y))$ is known.

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  • $\begingroup$ That was not my idea about first preimage resistance for 2. Rather, it's that if $H$ has $n$ bits, there is a first preimage attack with expected cost like $2^{n/2}$ hashes somewhat like for the birthday problem. $\endgroup$
    – fgrieu
    Apr 9, 2023 at 17:29
  • $\begingroup$ There’s an even easier first pre-image attack. Consider the hash of $k$ copies of $x$. $\endgroup$
    – Daniel S
    Apr 10, 2023 at 5:22

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