Let $p = a_{0} + a_{1}x +...+a_{k-1}x^{k-1}$ be a polinomial of degree $k-1$ with coefficients $a_i\in \mathbb{Z_{p}}$ . Let $A$ be a set of k points, say ${1, 2, 3, ...,k}$ and let $y_{i} = p(i), i\in {1, 2,..., k}$. I define partial interpolation(just like normal interpolation, only that the polynomial is evaluated at 0) with the following formula : $$a_0 = \sum_{i\in A} y_{i} \prod_{j\in A, j\neq i}\frac{j}{j-i} $$
Finding the modulo inverse of a number is a somewhat "expansive" task,especially when say p is a very large prime number, computationally speaking. Hence, one would want, when evaluating $a_0$ with the above formula to compute as few modular inverses as possible.
A naive approach would see as computing $k(k-1)$ modulo inverses, which is bad. A better approach is computing only $k$ inverses(computing the numerator and denominator in the inner product and do an inverse for each i). My question is this: can we perform only one modulo inverse to evaluate $a_0$ ?