Pseudo-random Walk on Ideals: Practical Speed-Up in Relation Collection for Class Group Computation

Document Type

Conference Article

Publication Title

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

Abstract

We introduce a technique to obtain practical speed up for relation collection in class group computations. The idea is to perform a pseudo-random walk over the ideals. The ideals visited by the walk are used in the manner exactly as in the previous algorithm due to Gélin (2018). Under the heuristic assumption that the ideals visited by the walk behave as the ideals randomly generated in Gélin’s algorithm, the asymptotic complexity of the new algorithm remains the same as that of Gélin’s algorithm. The main advantage of the new method over Gélin’s method is that the pseudo-random walk requires a single ideal multiplication to generate the next ideal in the walk, whereas Gélin’s algorithm requires a number of ideal multiplications to generate each ideal to be tested. We have made Magma implementations of both the new algorithm and Gélin’s algorithm. Timing results confirm that there is indeed a substantial practical speed-up in relation collection by the new algorithm over Gélin’s algorithm.

First Page

18

Last Page

31

DOI

10.1007/978-3-031-34671-2_2

Publication Date

1-1-2023

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