A reduced strain gradient crystal plasticity theory which involves the gradient of a single scalar field is presented. Rate-dependent and rate-independent crystal plasticity settings are considered. The theory is then reformulated following first the micromorphic approach and second a Lagrange multiplier approach. The finite element implementation of the latter is detailed. Computational efficiency of the Lagrange multiplier approach is highlighted in an example involving regularization of strain localization. The numerical performance improvement is shown to reach up to two orders of magnitude in computation time speedup. Then, size effects predicted by micromorphic and Lagrange multiplier based formulations of strain gradient plasticity are assessed. First of all numerical comparisons are performed on single crystal wires in torsion. Saturation of the size effects induced by the micromorphic approach and absence of saturation with the Lagrange multiplier approach when sample size is decreased are demonstrated. The Lagrange multiplier based formulation is finally applied to characterize size effects predicted for the ductile growth of porous unit-cells at imposed stress triaxiality. Excellent agreement with micromorphic results is obtained.