Further results on Goppa codes and their applications to constructing efficient binary codes

Sugiyama, Y.; Kasahara, Masao; Hirasawa, Shigeichi; Namekawa, T.

doi:10.1109/tit.1976.1055610

Cited by 60 publications

(34 citation statements)

References 22 publications

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“…, α n , g). Sugiyama, Kasahara, Hirasawa, and Namekawa showed in [51] that . The Sugiyama-KasaharaHirasawa-Namekawa identity Γ q (.…”

Section: Application To Classical Goppa Codesmentioning

confidence: 97%

Simplified High-Speed High-Distance List Decoding for Alternant Codes

Bernstein

2011

Post-Quantum Cryptography

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Abstract. This paper presents a simplified list-decoding algorithm to correct any number w of errors in any alternant code of any length n with any designed distance t + 1 over any finite field F q ; in particular, in the classical Goppa codes used in the McEliece and Niederreiter public-key cryptosystems. The algorithm is efficient for w close to, and in many cases slightly beyond, the F q Johnson bound J = n − n (n − t − 1) where n = n(q − 1)/q, assuming t + 1 ≤ n . In the typical case that qn/t ∈ (lg n) O(1) and that the parent field has (lg n) O(1) bits, the algorithm uses n(lg n) O(1) bit operations for w ≤ J − n/(lg n) O(1) ; O(n 4.5 ) bit operations for w ≤ J + o((lg n)/ lg lg n); and n O(1) bit operations for w ≤ J + O((lg n)/ lg lg n).

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“…, α n , g). Sugiyama, Kasahara, Hirasawa, and Namekawa showed in [51] that . The Sugiyama-KasaharaHirasawa-Namekawa identity Γ q (.…”

Section: Application To Classical Goppa Codesmentioning

confidence: 97%

Simplified High-Speed High-Distance List Decoding for Alternant Codes

Bernstein

2011

Post-Quantum Cryptography

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“…Wild Goppa codes [14] are a subclass of Goppa codes and have been suggested for the Wild McEliece [15]. It has been shown in [14] that the wild Goppa codes Γ(L, b q ) and Γ(L, b q−1 ) are the same code of length |L| = n, dimension k ≥ n − rm and distance d ≥ q q−1 r + 1,…”

Section: Wild Goppa Codesmentioning

confidence: 99%

On Decoding and Applications of Interleaved Goppa Codes

Holzbaur

Liu

Puchinger

et al. 2019

2019 IEEE International Symposium on Information Theory (ISIT)

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Goppa Codes are a well-known class of codes with, among others, applications in code-based cryptography. In this paper, we present a collaborative decoding algorithm for interleaved Goppa codes (IGC). Collaborative decoding increases the decoding radius beyond half of the designed minimum distance. We consider wild Goppa codes and show that we can collaboratively correct more errors for binary Goppa codes than the Patterson decoder. We propose a modified version of the McEliece cryptosystem using wild IGC based on a recently proposed system by Elleuch et al., analyze attacks on the system and present some parameters with the corresponding key sizes.

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“…These symbols are also independent of each other extending the length of the code and, importantly, increasing the d min of the code. Sugiyama et al [9,10] described a construction technique mixing the standard Goppa code construction with the Goppa external parity check construction. We give below a simpler construction method applicable to BCH codes and to more general Goppa codes.…”

Section: Extended Bch Codes As Goppa Codesmentioning

confidence: 99%