Two efficient algorithms for the computation of ideal sums in quadratic orders

Weilert, André

doi:10.1090/S0025-5718-05-01799-0

Two efficient algorithms for the computation of ideal sums in quadratic orders
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Math. Comp. 75 (2006), 941-981 Request permission

Abstract:

This paper deals with two different asymptotically fast algorithms for the computation of ideal sums in quadratic orders. If the class number of the quadratic number field is equal to 1, these algorithms can be used to calculate the GCD in the quadratic order. We show that the calculation of an ideal sum in a fixed quadratic order can be done as fast as in $\mathbf {Z}$ up to a constant factor, i.e., in $O(\mu (n) \log n),$ where $n$ bounds the size of the operands and $\mu (n)$ denotes an upper bound for the multiplication time of $n$-bit integers. Using Schönhage–Strassen’s asymptotically fast multiplication for $n$-bit integers, we achieve 𝜇(𝑛)=𝑂(𝑛log𝑛loglog𝑛).

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