In lattice cryptography, people often work with q-ary lattices so that we can use the hardness of short integer solution (SIS) and learning with errors (LWE). I saw in some notes that sometimes we want $q$ to be prime or a prime power. However there wasn't any explanation to why that is the case. So I have a couple of questions about the choice of $q$:
- Is there any issues with taking q to be any positive integer? Or are there any values that have shown to be problematic?
- What's the advantage of choosing it to be prime or a prime power? And are there any primes that are better suited?