Publication Details

Title: A Stable Integer Relation Algorithm
Author: C. Rössner and C. P. Schnorr
Group: ICSI Technical Reports
Date: April 1994
PDF: ftp://ftp.icsi.berkeley.edu/pub/techreports/1994/tr-94-016.pdf

Overview:
We study the following problem: given x ∈ 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 ar{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.

Bibliographic Information:
ICSI Technical Report TR-94-016

Bibliographic Reference:
C. Rössner and C. P. Schnorr. A Stable Integer Relation Algorithm. ICSI Technical Report TR-94-016, April 1994