Frobenius inner product polynomial rings

Question

I'm trying to implement the zero-knowledge proof presented in this paper. The proof has a rejection step (page 14), which can be computed as follows:

Where B and Z are in $R^{m \times n}$ for some ring. Although I understand how it works for the ring $R=\mathbb{Z}$, I don't get the point of how could work when $R=\mathbb{Z}[x]/(x^{n}+1)$. If I am not misunderstanding something, the Frobenius product between two matrices would output an element in the ring, and thus, the previous algorithm could only operate over integers.

What I am missing? Thanks in advance for your help.

Answer 1

Like $||B||^2$ is defined in Section 2.1 to be the norm of the vector of the integer coefficients comprising the elements of $B$, $<Z,B>$ is the inner product of these two integer vectors. Basically, flatten Z and B into integer vectors, and take the inner product. Sorry, it should have been defined. And also, in Figure 1, where this rejection sampling step is used, B=SC.