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Factorization of univariate integer polynomials by diophantine approximation and an improved basis reduction algorithm

  • Arnold Schönhage
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 172)

Abstract

All steps described in the preceding section for fixed m are certainly covered by the rather crude bound of \(O(m(n^{5 + \varepsilon } + n^3 (log|f|)^{2 + \varepsilon } ))\) bit operations. For a factor p of degree \(k \leqslant \frac{n}{2}\) this bound applies for m=2,4,... until some m with m/2<k≤m is reached. The sum of the corresponding time bounds is therefore \(O(k(n^{5 + \varepsilon } + n^3 (log|f|)^{2 + \varepsilon } ))\). Further factors are found in the very same way, dealing with f/p, etc. There is at most one factor of degree k>n/2 (possibly f itself), thus the final time bound \(O(n^{6 + \varepsilon } + n^4 (log|f|)^{2 + \varepsilon } ))\) is obtained.

It should be observed that the distinctions between the real and complex case in Lemma 6.1 and in Lemma 6.2 nicely match such that in both cases log B∼6mn+2(m+n) log|f|. Due to the estimation (6.1) sometimes shortcuts in the reduction process may be possible. As soon as |b m * |2 becomes greater than 3·22m-1|f|2, bm can be eliminated from the reduction, etc.

It is conceivable that further (mainly theoretical) improvements of our algorithm are possible, for instance by exploiting fast matrix multiplication, or by iterating the block reduction technique.

Keywords

Size Reduction Arithmetic Operation Reduction Algorithm Minimal Polynomial Basis Reduction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    E. Kaltofen: On the complexity of finding short vectors in integer lattices. Proc. EUROCAL '83, Lecture Notes Comp. Sci.Google Scholar
  2. [2]
    A.K. Lenstra: Polynomial factorization by root approximation. Preprint, mathem. centrum, Amsterdam 1983.Google Scholar
  3. [3]
    A.K. Lenstra, H.W. Lenstra, Jr., L. Lovász: Factoring polynomials with rational coefficients. Math. Ann. 261 (1982), 515–534.Google Scholar
  4. [4]
    A. Schönhage: The fundamental theorem of algebra in terms of computational complexity. Preliminary Report, Math. Inst. Univ. Tübingen, 1982.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • Arnold Schönhage
    • 1
  1. 1.Mathematisches Institutder Universität TübingenTübingenW-Germany

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