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In this paper, a high efficient decoding algorithm is developed here in order to correct both erasures and errors for Reed-Solomon (RS) codes based on the Euclidean algorithm together with the Berlekamp-Massey (BM) algorithm. The new decoding algorithm computes the errata locator polynomial and the errata evaluator polynomial simultaneously without performing polynomial divisions, and there is no need for the computation of the discrepancies and the field element inversions. Also, the separate computation of the Forney syndrome needed in the decoder is completely avoided. As a consequence, the complexity of this new decoding algorithm is dramatically reduced. Finally, the new algorithm has been verified through a software simulation using C++ language. An illustrative example of (255,239) RS code using this program shows that the speed of the decoding process is approximately three times faster than that of the inverse-free Berlekamp-Massey algorithm.