At each step of the process, the prover has two vectors, u and v. These are compressed by some compression factor λ (e.g. 4) to produce new vectors u and v with length λ times shorter.

At each step, the prover computes s polynomials p_i(x) and q_i(x), (where s = u.len() / λ). The polynomials p_i(x) are defined as the polynomials passing through particular coordinates of the vector u, and the polynomials q_i(x) are defined as the polynomials passing through particular coordinates of the vector v.

Now the prover generates a distributed zero knowledge proof. This proof is just a polynomial G(x) which is defined as the sum from i=0 to i=s of p_i(x) * q_i(x).

Rather than computing the coefficients of this polynomial G(x), the prover can just generate 2*λ - 1 points on this curve - as that uniquely defines the polynomial. We will always generate the y values of the points corresponding to the x-coordinates 0, 1, 2, ..., 2*λ - 2.

The first λ points can be found trivially by just multiplying the known values of p_i(x) and q_i(x). The next λ - 1 points require more computation. The prover can just use Lagrange interpolation to generate the next λ - 1 points on each polynomial p_i(x) and q_i(x), and then multiply these together and sum over i.

See a concrete example in this sheet: https://docs.google.com/spreadsheets/d/10tCSgiZyQbt4bMB3KqhAUOCfgfO8xIni83J2saipFCg/edit?usp=sharing
Tab: "All Lagrange Interpolation"