Parallel algorithms for algebraic problems

Reviewer: Richard J. Fateman

Theoretically fast algorithms are described for solving problems in algebraic symbol manipulation, including greatest common divisor (gcd) of univariate polynomials, factoring of polynomials over finite fields, and related problems. Other than the Euclidean algorithm, the others are probabilistic, but have very low failure probability. For all the algorithms the number of processors is polynomial in n, where n is a degree bound on the input polynomials. Von zur Gathen shows that the extended Euclidean algorithm can be executed (although he ignores the need to deal with very long integer coefficients in intermediate results) in time O(log 2 n), rather than time O( n 3), by reduction to algorithms for solving linear systems rapidly in parallel. The paper consists of theoretical analysis, but provides no clues as to the impracticality of the algorithms using current technology. By examination of the principal reference [1], one learns that the number of processors required for these speed-ups is substantial: between n 3.5 and n 15. Since gcd calculations of polynomials of low degree are trivial, and gcd calculations of polynomials of high degree are more likely to benefit from exploitation of sparseness than by use of this technique, this paper must be considered primarily of theoretical interest in the parallel-computation reduction hierarchy. It will not change the way the algebraic computations are actually performed using current sequential or parallel computers.