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Open AccessArticle

The Symmetric Key Equation for Reed–Solomon Codes and a New Perspective on the Berlekamp–Massey Algorithm

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Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Av. Països Catalans 26, 43007 Tarragona, Catalonia
2
Department of Mathematics and Statistics, San Diego State University, 5500 Campanile Drive, San Diego, CA 92182-7720, USA
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Author to whom correspondence should be addressed.
Symmetry 2019, 11(11), 1357; https://doi.org/10.3390/sym11111357
Received: 19 September 2019 / Revised: 23 October 2019 / Accepted: 25 October 2019 / Published: 2 November 2019
(This article belongs to the Special Issue Interactions between Group Theory, Symmetry and Cryptology)
This paper presents a new way to view the key equation for decoding Reed–Solomon codes that unites the two algorithms used in solving it—the Berlekamp–Massey algorithm and the Euclidean algorithm. A new key equation for Reed–Solomon codes is derived for simultaneous errors and erasures decoding using the symmetry between polynomials and their reciprocals as well as the symmetries between dual and primal codes. The new key equation is simpler since it involves only degree bounds rather than modular computations. We show how to solve it using the Euclidean algorithm. We then show that by reorganizing the Euclidean algorithm applied to the new key equation we obtain the Berlekamp–Massey algorithm. View Full-Text
Keywords: Reed–Solomon codes; key equation; Berlekamp–Massey algorithm; Sugiyama et al. algorithm; euclidean algorithm Reed–Solomon codes; key equation; Berlekamp–Massey algorithm; Sugiyama et al. algorithm; euclidean algorithm
MDPI and ACS Style

Bras-Amorós, M.; O’Sullivan, M.E. The Symmetric Key Equation for Reed–Solomon Codes and a New Perspective on the Berlekamp–Massey Algorithm. Symmetry 2019, 11, 1357.

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