A deterministic multivariate interpolation algorithm for small finite fields

We present a new multivariate interpolation algorithm over arbitrary fields which is primarily suited for small finite fields. Given function values at arbitrary t points, we show that it is possible to find an n-variable interpolating polynomial with at most t terms, using the number of field operations that is polynomial in t and n. The algorithm exploits the structure of the multivariate generalized Vandermonde matrix associated with the problem. Relative to the univariate interpolation, only the minimal degree selection of terms cannot be guaranteed and several term selection heuristics are investigated toward obtaining low-degree polynomials. The algorithms were applied to obtain Reed-Muller and related transforms for incompletely specified functions.