Computationally-efficient Sparse Polynomial Interpolation

We consider the problem of interpolating a high-degree sparse polynomial, where the sparsity is in the number of monomial terms with non-zero coefficients. We propose a probabilistic algorithm that requires only O(k) evaluations of a polynomial with complex coefficients, on the unit circle at specified points and has a complexity O(k log k), where k is the sparsity of the polynomial. Thus the evaluation complexity as well as the computational complexity are independent of the maximum degree n in contrast to existing algorithms in the literature. We extend our algorithm to polynomials defined over the finite field using fast algorithms in the literature to compute discrete logs for certain field sizes.