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.