On the Complexity of the Montes Ideal Factorization Algorithm

Let p be a rational prime and let Φ(X) be a monic irreducible polynomial in Z[X], with nΦ = deg Φ and δΦ = vp(disc Φ). In [13] Montes describes an algorithm for the decomposition of the ideal pOK 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(n3+ Φ δ2+ Φ ). We now give a refined analysis that yields an improved estimate of O(n3+ Φ δΦ+n2+ Φ δ2+ Φ ) 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.

• Type: Conference
• Year: 2010
• Where: ANTS
• Authors: David Ford, Olga Veres
• Keywords: Algorithms, ANTS 2010, Bit Operations, Monic Irreducible Polynomial, Worst Case