Algorithm for computing the reduced covariance -coupling kernels using HEALPix tools. We use the same notation as in the diagram of Fig. 2. The HEALPix functions require , with n_side the chosen resolution of maps W and Y_ℓm. As we choose the resolution n_side to be on the order of the multipoles indices ℓ,ℓ′, it is equivalent to say that they require O((ℓ + ℓ′)³). As operations are done O(ℓ + ℓ′) times, the whole operation of computing , is O((ℓ + ℓ′)⁴). Finally, it is clear in this diagram that the computing time of this kernel is at least doubled when multiple masks are used. Different masks would be used as inputs in the first line. As a result, the coupling coefficients would need to be computed for each of the masks, as shown in Eq. (D.3).