Abstract
We consider the following computational problem: we are given two coprime univariate polynomials f 0 and f 1 over a ring \(\mathcal{R}\) and want to find whether after a small perturbation we can achieve a large gcd. We solve this problem in polynomial time for two notions of “large” (and “small”): large degree (when \(\mathcal{R} = \mathbb{F}\) is an arbitrary field, in the generic case when f 0 and f 1 have a so-called normal degree sequence), and large height (when \(\mathcal{R} =\mathbb{Z}\)).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bini, D.A., Boito, P.: Structured Matrix-Based Methods for Polynomial ε-gcd: Analysis and Comparisons. In: Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation ISSAC 2007, Waterloo, Ontario, Canada, pp. 9–16 (2007)
von zur Gathen, J., Gerhard, J.: Modern Computer Algebra, 2nd edn. Cambridge University Press, Cambridge (2003), http://www-math.upb.de/~aggathen/mca/
Howgrave-Graham, N.: Approximate integer common divisor. In: Silverman, J.H. (ed.) Cryptography and Lattices: International Conference, CaLC 2001, Providence. LNCS, vol. 2146, pp. 51–66. Springer, Heidelberg (2001), http://www.springerlink.com/content/ak783wexe7ghp5db/
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
von zur Gathen, J., Shparlinski, I.E. (2008). Approximate Polynomial gcd: Small Degree and Small Height Perturbations. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds) LATIN 2008: Theoretical Informatics. LATIN 2008. Lecture Notes in Computer Science, vol 4957. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78773-0_24
Download citation
DOI: https://doi.org/10.1007/978-3-540-78773-0_24
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-78772-3
Online ISBN: 978-3-540-78773-0
eBook Packages: Computer ScienceComputer Science (R0)