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I know that the normal construction for Ajtai hash functions is as follows:

Pick $n, m, q \in \mathbb{Z}^+$ such that $n \log q < m < \frac{q}{2n^4}$ and $q = O(n^c)$ for some $c>0$, and some random matrix $A \in {\mathbb{Z}_q}^{n \times m}$. Then the Ajtai hash function $f_A: \{ 0,1 \}^m \to \mathbb{Z}_q^n$ is defined as $$f_A(\vec{x}) = A \vec{x} \mod q.$$

I notice that $\mathsf{dom}(f_A) = \{ 0,1 \}^m$ but my question is what happens when $\mathsf{dom}(f_A) = D^m$ for any arbitrary $D \subset \mathbb{Z}_q$? What are the limitations on the kinds of $D$ I can pick such that the function retains all the useful properties of being a hash function, and how would the choice of $D$ impact the security constraints I mentioned above (i.e. $n \log q < m < \frac{q}{2n^4}$).

Like for example, what if instead of the input being a bitstring, I want it to be a byte array, so $D = \{ 0, 1, \dots, 255 \}$ (for some $q \gg 255$), or similar.

Is there any way to easily find these constraints and parameters? Rules of thumb and so on. Or any literature around this? Or is it a situation where you have to do mathematical analysis on a case-by-case basis?

And if you do find the new security constraints for a given $D$, and pick valid parameters $n, m, q$, how do you then go on to measure how many bits of security you get against collision attacks? (i.e 256 bits of security against collision attacks)

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Micciancio studies precisely this question (for an "algebraically structured" version of the Ajtai hash) in Generalized compact knapsacks, cyclic lattices, and efficient one-way functions from worst-case complexity assumptions. See Section 4 in particular.

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    $\begingroup$ Thankyou so much. Reading through this paper is like a blessing $\endgroup$ Commented Mar 22 at 5:47

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