Cryptography friends: How to prove that a hash function can produces a length-fixed output？

Question

Consider the following hash function based on RSA. $M_i < n$. The hash value of a message consisting of two blocks is calculated by

$H(M) = H(M_1,M_2) = ((M_1^e\ mod\ n)\ XOR\ M_2)^e mod\ n$

Q1.Does this hash function can produce fixed output size?

My answer: Because any output is limited to the modulus(n), the hash function can produce fixed output size?(I think of it for a long time, but I still wonder about it)

Q2.Is the hash function easy to calculate?

My answer: Because the hash function just contain encryption and XOR operations, it is easy to calculate.(Is it right or completed?)

Answer 1

It's a weird chaining mode of RSA, and the output can be seen as a string of the length equal to that of $n$, as we output a number modulo $n$. So yes to Q1, provided you output a fully padded string.

It's not easy to calculate, but that's a bit subjective. The cost is two RSA operations essentially (xor is very cheap on computers) and those are relatively expensive compared to real hash functions (like SHA256 or SHA3 and many others). It's easily doable, but relatively expensive.

It's very insecure though.