Discrete logarithm computations over finite fields using Reed-Solomon codes

Cheng and Wan have related the decoding of Reed-Solomon codes to the computation of discrete logarithms over finite fields, with the aim of proving the hardness of their decoding. In this work, we experiment with solving the discrete logarithm over GF(q^h) using Reed-Solomon decoding. For fixed h and q going to infinity, we introduce an algorithm (RSDL) needing O (h! q^2) operations over GF(q), operating on a q x q matrix with (h+2) q non-zero coefficients. We give faster variants including an incremental version and another one that uses auxiliary finite fields that need not be subfields of GF(q^h); this variant is very practical for moderate values of q and h. We include some numerical results of our first implementations

Complexity Analysis of Reed-Solomon Decoding over GF(2^m) Without Using Syndromes

For the majority of the applications of Reed-Solomon (RS) codes, hard decision decoding is based on syndromes. Recently, there has been renewed interest in decoding RS codes without using syndromes. In this paper, we investigate the complexity of syndromeless decoding for RS codes, and compare it to that of syndrome-based decoding. Aiming to provide guidelines to practical applications, our complexity analysis differs in several aspects from existing asymptotic complexity analysis, which is typically based on multiplicative fast Fourier transform (FFT) techniques and is usually in big O notation. First, we focus on RS codes over characteristic-2 fields, over which some multiplicative FFT techniques are not applicable. Secondly, due to moderate block lengths of RS codes in practice, our analysis is complete since all terms in the complexities are accounted for. Finally, in addition to fast implementation using additive FFT techniques, we also consider direct implementation, which is still relevant for RS codes with moderate lengths. Comparing the complexities of both syndromeless and syndrome-based decoding algorithms based on direct and fast implementations, we show that syndromeless decoding algorithms have higher complexities than syndrome-based ones for high rate RS codes regardless of the implementation. Both errors-only and errors-and-erasures decoding are considered in this paper. We also derive tighter bounds on the complexities of fast polynomial multiplications based on Cantor's approach and the fast extended Euclidean algorithm.Comment: 11 pages, submitted to EURASIP Journal on Wireless Communications and Networkin

Founsure 1.0: An erasure code library with efficient repair and update features

Founsure is an open-source software library that implements a multi-dimensional graph-based erasure coding entirely based on fast exclusive OR (XOR) logic. Its implementation utilizes compiler optimizations and multi-threading to generate the right assembly code for the given multi-core CPU architecture with vector processing capabilities. Founsure possesses important features that shall find various applications in modern data storage, communication, and networked computer systems, in which the data needs protection against device, hardware, and node failures. As data size reached unprecedented levels, these systems have become hungry for network bandwidth, computational resources, and average consumed power. To address that, the proposed library provides a three-dimensional design space that trades off the computational complexity, coding overhead, and data/node repair bandwidth to meet different requirements of modern distributed data storage and processing systems. Founsure library enables efficient encoding, decoding, repairs/rebuilds, and updates while all the required data storage and computations are distributed across the network nodes.Turkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK) Grant Number : 115C111 - 119E235WOS:000656825700019Scopus - Affiliation ID: 60105072Science Citation Index ExpandedQ3ArticleUluslararası işbirliği ile yapılmayan - HAYIRJanuary2021YÖK - 2020-2