A stable integer relation algorithm

  • We study the following problem: given x element Rn either find a short integer relation m element Zn, so that =0 holds for the inner product <.,.>, or prove that no short integer relation exists for x. Hastad, Just Lagarias and Schnorr (1989) give a polynomial time algorithm for the problem. We present a stable variation of the HJLS--algorithm that preserves lower bounds on lambda(x) for infinitesimal changes of x. Given x \in {\RR}^n and \alpha \in \NN this algorithm finds a nearby point x' and a short integer relation m for x'. The nearby point x' is 'good' in the sense that no very short relation exists for points \bar{x} within half the x'--distance from x. On the other hand if x'=x then m is, up to a factor 2^{n/2}, a shortest integer relation for \mbox{x.} Our algorithm uses, for arbitrary real input x, at most \mbox{O(n^4(n+\log \alpha))} many arithmetical operations on real numbers. If x is rational the algorithm operates on integers having at most \mbox{O(n^5+n^3 (\log \alpha)^2 + \log (\|q x\|^2))} many bits where q is the common denominator for x.

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Author:Claus Peter SchnorrGND, Carsten Rössner
Parent Title (German):Technical Report 94-016, ICSI, Berkeley, CA, 1994
Place of publication:Frankfurt am Main
Document Type:Report
Date of Publication (online):2005/07/13
Year of first Publication:1994
Publishing Institution:Universitätsbibliothek Johann Christian Senckenberg
Release Date:2005/07/13
Issue:April 1994
Page Number:13
Source:Technical Report 94-016, ICSI, Berkeley, CA, 1994 , http://www.mi.informatik.uni-frankfurt.de/research/papers.html
Institutes:Informatik und Mathematik / Mathematik
Informatik und Mathematik / Informatik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):License LogoDeutsches Urheberrecht