Analysis of the p-adic construction of multivariate correction coefficiencts in polynomial factorization: Iteration vs. recursion

Abstract

In a recently published paper [4] the author described a new algorithm for multivariate polynomial factorization over the integers. One important feature of this algorithm is the variable-by-variable p-adic construction of all factors in parallel. To perform such p-adic construction of the factors, a second p-adic construction for multivariate "correction coefficients" is used. This second p-adic construction is in an "inner loop" of the construction process. Therefore its efficiency is of central importance to the entire factoring algorithm. In [4], an iterative algorithm is used for this second p-adic construction. A timing analysis of this iterative algorithm is included. A new recursive method for the p-adic construction of the correction coefficients is presented and analyzed.

Timing comparisons, both theoretical, and empirical are made of these two approaches. The recursive algorithm is shown to be much more efficient.

Research supported in part by the National Science Foundation under Grant NSF MCS 78-02234 and in part by MIT Laboratory for Computer Science.