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Algebraic soft-decision (ASD) decoding algorithm of Reed-Solomon (RS) codes can achieve better performance-complexity tradeoff than other soft-decision decoding algorithms. The interpolation is a major step of ASD algorithms. In the case that multiple test vectors are involved, the interpolation needs to be carried out for each vector and leads to very high hardware complexity. To enable the sharing of computation results in the interpolation for different vectors, a backward interpolation scheme was developed previously to eliminate points from a given interpolation result, which is a Grobner basis. However, this scheme can only eliminate all points in the same code position when their multiplicities are all one and the number of points is one less than the number of polynomials in the Grobner basis. Larger multiplicities are required to achieve better error-correcting performance. Moreover, for general ASD algorithms, the points in the same code position may have different multiplicities. In this paper, a generalized backward interpolation algorithm is proposed through constructing equivalent Grobner basis. It is capable of reducing the multiplicity of each point in the same code position by one at a time until zero, and the multiplicity of each point can be different. As an example, the proposed scheme is applied to a Chase-type decoding that has multiplicity two in the flipping points, and efficient hardware architectures are developed. For a (255, 239) RS code with eight test vectors, employing the proposed backward interpolation leads to 18% area reduction and 23% speedup compared to repeating the interpolation over the flipping points for each test vector.