Abstract : In this paper, we exploit the symmetry properties of 4th-order cumulants to develop new blind channel identification algorithms that utilize the parallel factor (Parafac) decomposition of cumulant tensors by solving a single-step (SS) least squares (LS) problem. We first consider the case of single-input single-output (SISO) finite impulse response (FIR) channels and then we extend the results to multiple-input multiple-output (MIMO) instantaneous mixtures. Our approach is based on 4th-order output cumulants only and it is shown to hold for certain underdetermined mixtures, i.e. systems with more sources than sensors. A simplified approach using a reduced-order tensor is also discussed. Computer simulations are provided to assess the performance of the proposed algorithms in both SISO and MIMO cases, comparing them to other existing solutions. Initialization and convergence issues are also addressed.