ZK proof that two short solutions are equal Let $A_1$ and $A_2$ two $m \times n$ matrices defining SIS problems.
Does there exist a zero knowledge proof that two short solutions are the same, i.e.
$$y_1 = A_1 x $$
$$y_2= A_2 x $$
$$ \Vert x \Vert &lt; \beta$$
This paper Improved Zero-Knowledge Proofs of Knowledge for the ISIS Problem, and Applications is a good place to start researching. The paper appeared at PKC 2013 so have been peer-reviewed. In addition, there is an upcoming paper at CRYPTO 2016 which looks very related How to prove knowledge of small secrets.