Efficient Reduction of Quadratic Forms

Abstract

The positive definite integer quadratic form, ax ² + bxy + cy ², is of some importance in number theory. For example such quadratic forms have been shown useful in factorization of large integers. For many applications it is important to be able to recognize when two quadratic forms are equivalent, so it is useful to be able to reduce these quadratic forms to a canonical representation.

For applications in factorization, the quadratic forms used have large coefficients, which must be represented as multiple computer words. This paper shows how to efficiently reduce such multi precision quadratic forms.

I am grateful to A. O. L. Atkin for his encouragement in this work. Most of this work was completed at the University of Illinois at Chicago in conjunction with Atkin’s project on the factorization of large integers.