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Journal ArticleDOI

Improved decoding of Reed-Solomon and algebraic-geometry codes

01 Sep 1999-IEEE Transactions on Information Theory (IEEE)-Vol. 45, Iss: 6, pp 1757-1767
TL;DR: An improved list decoding algorithm for decoding Reed-Solomon codes and alternant codes and algebraic-geometry codes is presented and a solution to a weighted curve-fitting problem is presented, which may be of use in soft-decision decoding algorithms for Reed- Solomon codes.
Abstract: Given an error-correcting code over strings of length n and an arbitrary input string also of length n, the list decoding problem is that of finding all codewords within a specified Hamming distance from the input string. We present an improved list decoding algorithm for decoding Reed-Solomon codes. The list decoding problem for Reed-Solomon codes reduces to the following "curve-fitting" problem over a field F: given n points ((x/sub i//spl middot/y/sub i/))/sub i=1//sup n/, x/sub i/, y/sub i//spl isin/F, and a degree parameter k and error parameter e, find all univariate polynomials p of degree at most k such that y/sub i/=p(x/sub i/) for all but at most e values of i/spl isin/(1,...,n). We give an algorithm that solves this problem for e 1/3, where the result yields the first asymptotic improvement in four decades. The algorithm generalizes to solve the list decoding problem for other algebraic codes, specifically alternant codes (a class of codes including BCH codes) and algebraic-geometry codes. In both cases, we obtain a list decoding algorithm that corrects up to n-/spl radic/(n(n-d')) errors, where n is the block length and d' is the designed distance of the code. The improvement for the case of algebraic-geometry codes extends the methods of Shokrollahi and Wasserman (see in Proc. 29th Annu. ACM Symp. Theory of Computing, p.241-48, 1998) and improves upon their bound for every choice of n and d'. We also present some other consequences of our algorithm including a solution to a weighted curve-fitting problem, which may be of use in soft-decision decoding algorithms for Reed-Solomon codes.
Citations
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Book ChapterDOI
02 May 2004
TL;DR: This work provides formal definitions and efficient secure techniques for turning biometric information into keys usable for any cryptographic application, and reliably and securely authenticating biometric data.
Abstract: We provide formal definitions and efficient secure techniques for turning biometric information into keys usable for any cryptographic application, and reliably and securely authenticating biometric data.

1,914 citations

Journal ArticleDOI
TL;DR: In this article, the authors provide formal definitions and efficient secure techniques for turning noisy information into keys usable for any cryptographic application, and, in particular, reliably and securely authenticating biometric data.
Abstract: We provide formal definitions and efficient secure techniques for turning noisy information into keys usable for any cryptographic application, and, in particular, reliably and securely authenticating biometric data. Our techniques apply not just to biometric information, but to any keying material that, unlike traditional cryptographic keys, is (1) not reproducible precisely and (2) not distributed uniformly. We propose two primitives: a fuzzy extractor reliably extracts nearly uniform randomness $R$ from its input; the extraction is error-tolerant in the sense that $R$ will be the same even if the input changes, as long as it remains reasonably close to the original. Thus, $R$ can be used as a key in a cryptographic application. A secure sketch produces public information about its input $w$ that does not reveal $w$ and yet allows exact recovery of $w$ given another value that is close to $w$. Thus, it can be used to reliably reproduce error-prone biometric inputs without incurring the security risk inherent in storing them. We define the primitives to be both formally secure and versatile, generalizing much prior work. In addition, we provide nearly optimal constructions of both primitives for various measures of “closeness” of input data, such as Hamming distance, edit distance, and set difference.

1,279 citations

Journal ArticleDOI
26 Oct 2000
TL;DR: A polynomial-time soft-decision decoding algorithm for Reed-Solomon codes is developed and it is shown that the asymptotic performance can be approached as closely as desired with a list size that does not depend on the length of the code.
Abstract: A polynomial-time soft-decision decoding algorithm for Reed-Solomon codes is developed. This list-decoding algorithm is algebraic in nature and builds upon the interpolation procedure proposed by Guruswami and Sudan(see ibid., vol.45, p.1757-67, Sept. 1999) for hard-decision decoding. Algebraic soft-decision decoding is achieved by means of converting the probabilistic reliability information into a set of interpolation points, along with their multiplicities. The proposed conversion procedure is shown to be asymptotically optimal for a certain probabilistic model. The resulting soft-decoding algorithm significantly outperforms both the Guruswami-Sudan decoding and the generalized minimum distance (GMD) decoding of Reed-Solomon codes, while maintaining a complexity that is polynomial in the length of the code. Asymptotic analysis for alarge number of interpolation points is presented, leading to a geo- metric characterization of the decoding regions of the proposed algorithm. It is then shown that the asymptotic performance can be approached as closely as desired with a list size that does not depend on the length of the code.

672 citations


Cites background or methods from "Improved decoding of Reed-Solomon a..."

  • ...Once this is done, we appeal to the interpolation-based techniques of [14]....

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  • ...The algorithm of Guruswami and Sudan [14] corrects any fraction of erroneous positions for a Reed–Solomon code of rate ....

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  • ...The next section contains a brief overview of the Guruswami-Sudan list-decoding algorithm [14]....

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  • ...The proposed algorithm is based on the algebraic interpolation techniques developed by Guruswami and Sudan [14], [27]....

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  • ...GURUSWAMI and Sudan [27], [14] have recently achieved a breakthrough in algebraic decoding of Reed–Solomon codes....

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Journal ArticleDOI
01 Sep 1986
TL;DR: This chapter discusses algorithmics and modular computations, Theory of Codes and Cryptography (3), and the theory and practice of error control codes (3).
Abstract: algorithmics and modular computations, Theory of Codes and Cryptography (3).From an analytical 1. RE Blahut. Theory and practice of error control codes. eecs.uottawa.ca/∼yongacog/courses/coding/ (3) R.E. Blahut,Theory and Practice of Error Control Codes, Addison Wesley, 1983. QA 268. Cached. Download as a PDF 457, Theory and Practice of Error Control CodesBlahut 1984 (Show Context). Citation Context..ontinued fractions.

597 citations

Journal ArticleDOI
TL;DR: In this article, the traceback problem is reframed as a polynomial reconstruction problem and uses algebraic techniques from coding theory and learning theory to provide robust methods of transmission and reconstruction.
Abstract: We present a new solution to the problem of determining the path a packet traversed over the Internet (called the traceback problem) during a denial-of-service attack. This article reframes the traceback problem as a polynomial reconstruction problem and uses algebraic techniques from coding theory and learning theory to provide robust methods of transmission and reconstruction.

484 citations

References
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Book
01 Jan 1977
TL;DR: This book presents an introduction to BCH Codes and Finite Fields, and methods for Combining Codes, and discusses self-dual Codes and Invariant Theory, as well as nonlinear Codes, Hadamard Matrices, Designs and the Golay Code.
Abstract: Linear Codes. Nonlinear Codes, Hadamard Matrices, Designs and the Golay Code. An Introduction to BCH Codes and Finite Fields. Finite Fields. Dual Codes and Their Weight Distribution. Codes, Designs and Perfect Codes. Cyclic Codes. Cyclic Codes: Idempotents and Mattson-Solomon Polynomials. BCH Codes. Reed-Solomon and Justesen Codes. MDS Codes. Alternant, Goppa and Other Generalized BCH Codes. Reed-Muller Codes. First-Order Reed-Muller Codes. Second-Order Reed-Muller, Kerdock and Preparata Codes. Quadratic-Residue Codes. Bounds on the Size of a Code. Methods for Combining Codes. Self-dual Codes and Invariant Theory. The Golay Codes. Association Schemes. Appendix A. Tables of the Best Codes Known. Appendix B. Finite Geometries. Bibliography. Index.

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Book
01 Jan 1974
TL;DR: This text introduces the basic data structures and programming techniques often used in efficient algorithms, and covers use of lists, push-down stacks, queues, trees, and graphs.
Abstract: From the Publisher: With this text, you gain an understanding of the fundamental concepts of algorithms, the very heart of computer science. It introduces the basic data structures and programming techniques often used in efficient algorithms. Covers use of lists, push-down stacks, queues, trees, and graphs. Later chapters go into sorting, searching and graphing algorithms, the string-matching algorithms, and the Schonhage-Strassen integer-multiplication algorithm. Provides numerous graded exercises at the end of each chapter. 0201000296B04062001

9,262 citations

Book
01 Jan 2015
TL;DR: This is the revised edition of Berlekamp's famous book, "Algebraic Coding Theory," originally published in 1968, wherein he introduced several algorithms which have subsequently dominated engineering practice in this field.
Abstract: This is the revised edition of Berlekamp's famous book, "Algebraic Coding Theory," originally published in 1968, wherein he introduced several algorithms which have subsequently dominated engineering practice in this field. One of these is an algorithm for decoding Reed-Solomon and Bose–Chaudhuri–Hocquenghem codes that subsequently became known as the Berlekamp–Massey Algorithm. Another is the Berlekamp algorithm for factoring polynomials over finite fields, whose later extensions and embellishments became widely used in symbolic manipulation systems. Other novel algorithms improved the basic methods for doing various arithmetic operations in finite fields of characteristic two. Other major research contributions in this book included a new class of Lee metric codes, and precise asymptotic results on the number of information symbols in long binary BCH codes.Selected chapters of the book became a standard graduate textbook.Both practicing engineers and scholars will find this book to be of great value.

2,912 citations

Book
01 Jan 1993
TL;DR: The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods.
Abstract: A description of 148 algorithms fundamental to number-theoretic computations, in particular for computations related to algebraic number theory, elliptic curves, primality testing and factoring. The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods, including a detailed description of the number field sieve algorithm. The whole is rounded off with a description of available computer packages and some useful tables, backed by numerous exercises. Written by an authority in the field, and one with great practical and teaching experience, this is certain to become the standard and indispensable reference on the subject.

2,842 citations

Book
25 Jun 1993
TL;DR: This new edition, published in the series Graduate Texts in Mathematics, has been considerably expanded and contains numerous exercises that help the reader to understand the basic material.
Abstract: The theory of algebraic function fields has its origins in number theory, complex analysis (compact Riemann surfaces), and algebraic geometry. Since about 1980, function fields have found surprising applications in other branches of mathematics such as coding theory, cryptography, sphere packings and others. The main objective of this book is to provide a purely algebraic, self-contained and in-depth exposition of the theory of function fields. This new edition, published in the series Graduate Texts in Mathematics, has been considerably expanded. Moreover, the present edition contains numerous exercises. Some of them are fairly easy and help the reader to understand the basic material. Other exercises are more advanced and cover additional material which could not be included in the text. This volume is mainly addressed to graduate students in mathematics and theoretical computer science, cryptography, coding theory and electrical engineering.

2,041 citations