On the Complexity of the Montes Ideal Factorization Algorithm

  • David Ford
  • Olga Veres
Conference paper

DOI: 10.1007/978-3-642-14518-6_16

Part of the Lecture Notes in Computer Science book series (LNCS, volume 6197)
Cite this paper as:
Ford D., Veres O. (2010) On the Complexity of the Montes Ideal Factorization Algorithm. In: Hanrot G., Morain F., Thomé E. (eds) Algorithmic Number Theory. ANTS 2010. Lecture Notes in Computer Science, vol 6197. Springer, Berlin, Heidelberg

Abstract

Let p be a rational prime and let Φ(X) be a monic irreducible polynomial in Z[X], with nΦ = degΦ and δΦ = vp(discΦ). In [13] Montes describes an algorithm for the decomposition of the ideal \(p\mathcal{O}K\) in the algebraic number field K generated by a root of Φ. A simplified version of the Montes algorithm, merely testing Φ(X) for irreducibility over Qp, is given in [19], together with a full Maple implementation and a demonstration that in the worst case, when Φ(X) is irreducible over Qp, the expected number of bit operations for termination is O(nΦ3 + εδΦ2 + ε). We now give a refined analysis that yields an improved estimate of O(nΦ3 + εδΦ + nΦ2 + εδΦ2 + ε) bit operations. Since the worst case of the simplified algorithm coincides with the worst case of the original algorithm, this estimate applies as well to the complete Montes algorithm.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • David Ford
    • 1
  • Olga Veres
    • 1
  1. 1.Concordia UniversityMontréalCanada

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