# Efficient proof for Cartesian product

I am trying to find some efficient zero-knowledge arguments that could prove the vector $${\bf v}$$ is the Cartesian product of two vectors $${\bf x}$$ and $${\bf y}$$. I know there are efficient inner product arguments, but are there any efficient arguments for Cartesian products?

For example, given three (vector) commitments $$com({\bf x})$$, $$com({\bf y})$$, and $$com({\bf v})$$ to $${\bf x,y}$$ and $${\bf v}$$, respectively, how to prove the knowledge of the opening to the three commitments and $${\bf v}$$ is the Cartesian product of $${\bf x,y}$$.

• You probably should spell out what you mean by "zk", and make it more clear how this is a crypto problem and not a math one. Jul 23 at 1:19