A Stable Integer Relation Algorithm

Abstract

We study the following problem: given x 2 IR n either find a short integer relation m 2 ZZ n ; so that ! x; m ?= 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 this problem. We present a stable variation of the HJLS--algorithm that preserves lower bounds on (x) for infinitesimal changes of x: Given x 2 IR n and ff 2 IN this algorithm finds a nearby point x 0 and a short integer relation m for x 0 : The nearby point x 0 is 'good' in the sense that no very short relation exists for points x within half the x 0 --distance from x: On the other hand if x 0 = x then m is, up to a factor 2 n=2 ; a shortest integer relation for x: Our algorithm uses, for arbitrary real input x; at most O(n 4 (n + log ff)) many arithmetical operations on real numbers. If x is rational the algorithm operates on integers having at most O(n 5 + n 3 (log ff) 2 +..