Computing the ideal class group and the regulator of a number field is of both number theoretic and cryptographic interest. There exist a lot of similarities between algorithms for number fields and for Jacobians of algebraic curves, even if finding reductions between these problems is still an open problem.

After an introduction on number fields and the features of the ideal class group, we will present methods based on the quadratic sieve to enhance the subexponential algorithms for class group and regulator computation of quadratic number fields. We will also see how improvements to linear algebra algorithms can lead to significant speed-ups, and provide comparisons between our algorithms and those available in Magma.