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Goppa code
About: Goppa code is a research topic. Over the lifetime, 361 publications have been published within this topic receiving 11058 citations.
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30 Apr 1991
TL;DR: In this article, an AG-Codes and their parameters and constructions are compared with those of binary codes from AG-codes, as well as asymptotic bounds.
Abstract: 1 Codes- 11 Codes and their parameters- 12 Examples and constructions- 13 Asymptotic problems- 2 Curves- 21 Algebraic curves- 22 Riemann-Roch theorem- 23 Rational points- 24 Elliptic curves- 25 Singular curves- 26 Reductions and schemes- 3 AG-Codes- 31 Constructions and properties- 32 Examples- 33 Decoding- 34 Asymptotic results- 4 Modular Codes- 41 Codes on classical modular curves- 42 Codes on Drinfeld curves- 43 Polynomiality- 5 Sphere Packings- 51 Definitions and examples- 52 Asymptotically dense packings- 53 Number fields- 54 Analogues of AG-codes- Appendix Summary of results and tables- A1 Codes of finite length- A11 Bounds- A12 Parameters of certain codes- A13 Parameters of certain constructions- A14 Binary codes from AG-codes- A2 Asymptotic bounds- A21 List of bounds- A22 Diagrams of comparison- A23 Behaviour at the ends- A24 Numerical values- A3 Additional bounds- A31 Constant weight codes- A32 Self-dual codes- A4 Sphere packings- A41 Small dimensions- A42 Certain families- A43 Asymptotic results- Author index- List of symbols
560 citations
537 citations
TL;DR: It is shown that the key equation for decoding Goppa codes can be solved using Euclid's algorithm, and the error locator polynomial is proved the multiplierPolynomial for the syndrome poynomial multiplied by an appropriate scalar factor.
Abstract: In this paper we show that the key equation for decoding Goppa codes can be solved using Euclid's algorithm. The division for computing the greatest common divisor of the Goppa polynomial g(z) of degree 2t and the syndrome polynomial is stopped when the degree of the remainder polynomial is less than or equal to t − 1. The error locator polynomial is proved the multiplier polynomial for the syndrome polynomial multiplied by an appropriate scalar factor. The error evaluator polynomial is proved the remainder polynomial multiplied by an appropriate scalar factor. We will show that the Euclid's algorithm can be modified to eliminate multiplicative inversion, and we will evaluate the complexity of the inversionless algorithm by the number of memories and the number of multiplications of elements in GF(qm). The complexity of the method for solving the key equation for decoding Goppa codes is a few times as much as that of the Berlekamp—Massey algorithm for BCH codes modified by Burton. However the method is straightforward and can be applied for solving the key equation for any Goppa polynomial.
417 citations
01 Apr 1988
TL;DR: A systematic method of checking is suggested, and a generalized version of the cryptanalytic attack which reduces the work factor sigdicantly is described, which can be viewed as generalized probabilistic decoding algorithms for any linear error correcting codes.
Abstract: The best known cryptanalytic attack on McEliece's public-key cryptosystem based on algebraic coding theory is to repeatedly select k bits at random from an n-bit ciphertext vector, which is corrupted by at most t errors, in hope that none of the selected k bits are in error until the cryptanalyst recovers the correct message. The method of determining whether the recovered message is the correct one has not been throughly investigated. In this paper, we suggest a systematic method of checking, and describe a generalized version of the cryptanalytic attack which reduces the work factor sigdicantly (factor of 211 for the commonly used example of n=1024 Goppa code case). Some more improvements are also given. We also note that these cryptanalytic algorithms can be viewed as generalized probabilistic decoding algorithms for any linear error correcting codes.
321 citations