Computing the Greatest Common Divisor of Multivariate Polynomials over Finite Fields

Richard Zippel’s sparse modular GCD algorithm is widely used to compute the monic greatest common divisor (GCD) of two multivariate polynomials over Z. In this report, we present how this algorithm can be modified to solve the GCD problem for polynomials over finite fields of small cardinality. When the GCD is not monic, Zippel’s algorithm cannot be applied unless the normalization problem is resolved. In [6], Alan Wittkopf et al. developed the LINZIP algorithm for solving the normalization problem. Mahdi Javadi proposed a refinement to the LINZIP algorithm in [4]. We implemented his approach and will show that it is efficient and effective on polynomials over small finite fields. Zippel’s algorithm also uses properties of transposed Vandermonde systems to reduce the time and space complexity of his algorithm. We also investigated how this can be applied to our case.