Computation of Highly Regular Nearby Points

We call a vector x 2 IRn highly regular if it satis es < x m >= 0 for some short, non{zero integer vector m where < : : > is the inner product. We present an algorithm which given x 2 IRn and 2 IN nds a highly regular nearby point x0 and a short integer relation m for x0 : The nearby point x0 is 'good' in the sense that no short relation m of length less than =2 exists for points x within half the x0 {distance from x: The integer relation m for x0 is for random x up to an average factor 2n=2 a shortest integer relation for x0 : Our algorithm uses, for arbitrary real input x at most O(n4(n + log )) many arithmetical operations on real numbers. If x is rational the algorithm operates on integers having at most O(n5 + n3(log )2 + log(kqxk2)) many bits where q is the common denominator for x.

• Type: Conference
• Year: 1995
• Where: ISTCS
• Authors: Carsten Rössner, Claus-Peter Schnorr
• Keywords: Applied Computing, Integer Relation, ISTCS 1995, Non{zero Integer Vector, Short Integer Relation