## Experimental Mathematics

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

### 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

#### 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.