Efficient arithmetic Hash function Modern hash functions are considered to be efficient to calculate by boolean circuits (i.e their implementation are using bits- operations).

I'm looking for a cryptographic hash function, that can be efficient calculate by arithmetic circuits (i.e they use OR and AND gates (or $+$ and $\cdot$ gates) and work over $\operatorname{GF}(q)$.

After bit of search, I found MASH-1 and MASH-2 constructions, but they seem to include the $\text{xor}$ function that I'm trying to eliminate.
You could try Ajtai's hash function:

What is Ajtai's hash function?
Given a matrix $A \hookleftarrow U(\mathbb{Z}_q^{n \times m})$ and a column vector $\vec{m} \in \mathbb{Z}_d^m$, the hash of the message $\vec{m}$ is given by
$H(\vec{m}) = A\vec{m} \mod q$

If you're not comfortable with linear algebra (and perhaps lattices too) it might be more challenging to implement than something like SHA256.
As for the efficiency, that depends on your definition of &quot;efficient&quot;. I doubt it will compete with something like SHA256/SHA512/BLAKE2 for speed; But to be fair, few algorithms will beat BLAKE2.
Note
You will need to take extra care to ensure that the implementation functions in constant time. With bitwise operations, this is significantly easier to accomplish, which is one motivation to use them.