Paper 2021/792

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

Madhurima Mukhopadhyay and Palash Sarkar

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.