Let $A x = 0 \bmod q$ with $\Vert x \Vert < \beta$ as part of a lattice SIS problem. Does there exist an efficient zero knowledge proof of knowledge for such a solution?
My idea is to use it for an authentication protocol. But all ZK protocols I've seen so far are promise or gap problems. Due to this gap, I cannot see how to build an authentication protocol. Because an attacker which has a solution $x$ with $x > \beta$ but $x$ inside the gap could also convince the verifier.
If you are interested in lattice-based (..and non-interactive..) zero knowledge, the state-of-the-art is from a 2008 paper (yes, 2008..) found here: https://web.eecs.umich.edu/~cpeikert/pubs/latticeNISZK.pdf
Getting NIZK from lattices is a major open problem. (As far as we know, you might have to go all the way up to multilinear maps and IND obfuscation for P/poly to get a "lattice-like" form of NIZK.)
If you need more than the SZKs from that paper, you must use a non-SIS, non-LWE solution (or, you could go for the big bucks and build NIZK from LWE yourself!)
The statement "I know an $x$ so that $Ax = 0\,\text{mod}\,q$ and $\Vert x\Vert < \beta$" is plainly in NP, so any zkSNARK can give you such a proof, e.g. this paper. Though, this is an argument of knowledge (not a proof of knowledge) but there seems to be little practical difference between the two.
If you dislike non-falsifiable assumptions, you could start with a generic ZKP for your SIS statement (built from e.g. this nice recent paper), then compile it into a proof of knowledge using a standard transform.
