Experimental Mathematics

Extended GCD and Hermite normal form algorithms via lattice basis reduction

George Havas, Bohdan S. Majewski, and Keith R. Matthews

Abstract

Extended gcd calculation has a long history and plays an important role in computational number theory and linear algebra. Recent results have shown that finding optimal multipliers in extended gcd calculations is difficult. We present an algorithm which uses lattice basis reduction to produce small integer multipliers $x_1,\dots,x_m$ for the equation $s=\gcd{(s_1,\dots,s_m)}=x_1s_1+\cdots+x_ms_m$, where $s_1,\dots,s_m$ are given integers. The method generalises to produce small unimodular transformation matrices for computing the Hermite normal form of an integer matrix.

Article information

Source
Experiment. Math., Volume 7, Issue 2 (1998), 125-136.

Dates
First available in Project Euclid: 24 March 2003

Permanent link to this document
https://projecteuclid.org/euclid.em/1048515660

Mathematical Reviews number (MathSciNet)
MR1700579

Zentralblatt MATH identifier
0922.11112

Subjects
Primary: 11Y16: Algorithms; complexity [See also 68Q25]
Secondary: 11H06: Lattices and convex bodies [See also 11P21, 52C05, 52C07]

Citation

Havas, George; Majewski, Bohdan S.; Matthews, Keith R. Extended GCD and Hermite normal form algorithms via lattice basis reduction. Experiment. Math. 7 (1998), no. 2, 125--136. https://projecteuclid.org/euclid.em/1048515660


Export citation

Corrections

  • See also authors' correction: Addenda and errata: Extended GCD and Hermite normal form algorithms via lattice basis reduction. Experiment. Math. Vol. 8, iss. 2 (1999), p. 205.