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A new fast matrix-vector multiplication scheme for the solution of the electric field integral equation is presented in this work. Similarly to other fast methods, our approach reduces the matrix-vector multiplication cost from O(N2) to O(N logN). Differently from other fast solvers, however, the effectiveness of EFIE preconditioning techniques such as quasi-Helmholtz decompositions or Calderón approaches is maintained by our method even for very high matrix compression rates. This is thanks to the fact that, in the scheme we are proposing, the contribution from the scalar potential when applied to or tested with solenoidal functions is always zero independent of the compression error. In addition, the new method will take advantage of the redundancies of the EFIE matrix in the low-frequency/dense discretization regime, and it will further decrease both the memory storage and the multiplication cost with respect to currently available fast solvers. Numerical results will show the effectiveness of our approach and its impact on the solution of realistic problems.