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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 ${\bf x,y}$.

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  • $\begingroup$ 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. $\endgroup$ Jul 23 at 1:19

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