The zero-check round polynomial is of degree $d+1$ where $d$ is the degree of the polynomial asserted to be zero over the hypercube (a "gate" in the context of GKR). Hence, we need evaluations of the polynomial points $g' := g(r_1, r_2, ..., r_{j-1}, X, b_{j+1},...,b_n)$ for $0 \le X\le d+1 $. It will most likely be cheaper to compute $g_j(d+1)$ from $g_j(X)$ values for $0\le X\le d$ using forward differencing, which saves some multiplications at the cost of $O(d^2)$ additions. (The number of multiplications saved is likely to be $O(d)$ if the user designs the circuit well.)
#1724 is mutually exclusive with this and must be preferred, as it reduces the degree of the round polynomial.
The zero-check round polynomial is of degree$d+1$ where $d$ is the degree of the polynomial asserted to be zero over the hypercube (a "gate" in the context of GKR). Hence, we need evaluations of the polynomial points $g' := g(r_1, r_2, ..., r_{j-1}, X, b_{j+1},...,b_n)$ for $0 \le X\le d+1 $ . It will most likely be cheaper to compute $g_j(d+1)$ from $g_j(X)$ values for $0\le X\le d$ using forward differencing, which saves some multiplications at the cost of $O(d^2)$ additions. (The number of multiplications saved is likely to be $O(d)$ if the user designs the circuit well.)
#1724 is mutually exclusive with this and must be preferred, as it reduces the degree of the round polynomial.