Efficient Root-Finding for Interpolation-Based Decoding of Elliptic and Hyperelliptic Codes

This paper proposes an efficient root-finding algorithm for the interpolation-based unique decoding of one-point elliptic and hyperelliptic codes. Instead of finding a message polynomial, it directly computes a codeword from the interpolation polynomial. It first determines the error positions through the error locator polynomial that is contained in the interpolation polynomial. Subsequently, the corresponding codeword symbols are determined based on the root-finding equation that is reformulated as a linear system of univariate polynomials. The proposed algorithm demonstrates superiority to the Roth-Ruckenstein (RR) algorithm, especially in the scenarios that the decoder is required to output a codeword and the re-encoding transform (ReT) technique is employed.