The decoding of BCH codes readily reduces to the solution of a certain key equation. An iterative algorithm is presented for solving this equation over any field. Following a heuristic derivation of the algorithm, a complete statement of the algorithm and proofs of its principal properties are given. The relationship of this algorithm to the classical matrix methods and the simplification which the algorithm takes in the special case of binary codes is then discussed. The generalization of the algorithm to BCH codes with a slightly different definition, the generalization of the algorithm to decode erasures as well as errors, and the extension of the algorithm to decode more thanterrors in certain eases are also presented.