On sparse interpolation over finite fields

Abstract

Extended Abstract of Poster Presented at ISSAC 2010Let f be a polynomial in n variables x 1 , x 2 , ..., x n of degree d with t non-zero terms over a finite field with characteristic p. It is well known that f can be interpolated from (1 + d) n points using Newton interpolation. But this is exponential in n. If t(1 + d) n , that is, f is sparse, we seek algorithms for interpolating f whose complexity is polynomial in d, t and n.The first such algorithm was engineered by Richard Zippel in 1979 (see [9]). Zippel's algorithm interpolates one variable at a time using O(ndt) points. Zippel's algorithm is probabilistic; it needs p ndt to succeed with high probability. Zippel developed his algorithm to interpolate a sparse polynomial GCD with integer coefficients. Zippel's algorithm is now used in Maple, Magma and Mathematica as the default GCD algorithm for multivariate GCD computation over the integers. Rayes, Wang and Weber attempted a parallel implementation in [8]. In [4] we extended it to compute GCDs over algebraic function fields.The second such algorithm was developed by Ben-Or and Tiwari in 1988 for interpolation in characteristic 0.