Fast Reduction of Ternary Quadratic Forms

We show that a positive de nite integral ternary form can be reduced with O(M(s) log s) bit operations, where s is the binary encoding length of the form and M(s) is the bit-complexity of s-bit integer multiplication. This result is achieved in two steps. First we prove that the the classical Gaussian algorithm for ternary form reduction, in the variant of Lagarias, has this worst case running time. Then we show that, given a ternary form which is reduced in the Gaussian sense, it takes only a constant number of arithmetic operations and a constant number of binary-form reductions to fully reduce the form. Finally we describe how this algorithm can be generalized to higher dimensions. Lattice basis reduction and shortest vector computation in xed dimension d can be done with O(M(s) logd−1 s) bit-operations.