X-Nico

Bruun's FFT algorithm

However, one can combine these remainders recursively to reduce the cost, using the following trick: if we want to evaluate $x(z)$ modulo two polynomials $U(z)$ and $V(z)$, we can first take the remainder modulo their product $U(z)$ $V(z)$, which reduces the degree of the polynomial $x(z)$ and makes subsequent modulo operations less computationally expensive.

George Blakley

In contrast, Shamir's secret sharing scheme represents the secret as the y-intercept of an n-degree polynomial, and shares correspond to points on the polynomial.

see also