Browsing by Author "Luo, Qinglong"
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- ItemOpen AccessComputing Class Groups of Cubic Orders Using Kleinjung’s Sieving Method(2024-06-13) Luo, Qinglong; Scheidler, Renate; Jacobson, Michael J., Jr.; Bauer, Mark; Tran, HaWe introduce two techniques to generate relations in the computation of class groups of orders of number fields using Buchmann’s index calculus method. In particular, we focus on primitive cubic orders. The first is an adaptation of the homogeneous method from the number field sieve to compute the class groups of a special class of orders known as rationally monogenic orders, which includes primitive cubic orders. The second is a generalisation of recent work by Kleinjung on the quadratic sieve to arbitrarily high degrees. We combine the two into one algorithm, implemented it, and test its performance against MAGMA. The results indicate that our method outperforms MAGMA if the given defining form is skew, and essentially tying MAGMA if the given form is flat.
- ItemOpen AccessOn the Performance of Kleinjung’s Quadratic Sieving Method(2020-01-17) Luo, Qinglong; Scheidler, Renate; Jacobson, Michael J.; Nguyen, Dang Khoa; Dimitrov, Vassil S.Kleinjung published a recent improvement in the quadratic sieving step of the self-initialising variant of the quadratic sieve method (SIQS) for computing class groups of orders of quadratic fields. To date, a comprehensive comparison between Kleinjung’s method and SIQS has not been done. Here, we implement a variation of both Kleinjung’s method and the original SIQS method, and run these two methods side by side over a wide range of test parameters to test their relative performance. We observed that Kleinjung’s method performs better than the SIQS method, especially when the number of primes used in constructing the self-initialising polynomial set is high. However, this performance advantage shrinks as the sieving interval increases.