Nowadays it's assumed (it can still turn out to be different) that elementary particles are "given" mass by interaction with the Higgs field.
I read in the only answer to this question:
If you know which string theory vacuum you have (which compactification, which fluxes etc) and you know how the symmetries are broken, then you could in principle compute the masses.
Is this a different mechanism or is the Higgs field somehow included in this answer? The Higgs-particle itself is also an elementary particle. That is, an excitation in the Higgs field.