Talk:Binary Goppa code

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The parity check matrix $H$ is in the form $H=VD$. As currently written in the Wiki, matrix $V$ is $(t+1)\times n$, and $D$ is $n\times n$. Therefore, $H$ must be $(t+1)\times n$, whereas the text mentions that $H$ is a $t$-by-$n$ matrix. I believe matrix $V$ must be corrected. MSDousti (talk) 20:02, 16 July 2013 (UTC)

I have corrected the matrix by changing the highest powers from incorrect $t$ to correct $t-1$. Stefkar (talk) 18:39, 15 February 2018 (UTC)

Misleading use of $\mathbb {Z} _{n}$

Please change the line

$\forall i,j\in \mathbb {Z} _{n}:L_{i}\in GF(2^{m})\land L_{i}\neq L_{j}\land g(L_{i})\neq 0$

into

$\forall i,j\in \lbrace 1,...,n\rbrace :L_{i}\in GF(2^{m})\land L_{i}\neq L_{j}\land g(L_{i})\neq 0$

In the latter $i,j$ are just indizes running from $1$ through $n$, in the former they are elements of the coset ring $\mathbb {Z} _{n}=\mathbb {Z} /n\mathbb {Z}$, which has a lot more algebraic structure. (Addtion, multiplication and $x=x+kn$.) It was taking me nearly 15 minutes wondering what the purpose of the coset ring is and how it is related to $n$ before I understood that $\mathbb {Z} _{n}$ is meant to be an ugly short form for $\lbrace 1,...,n\rbrace$

Goppa Codes in Niederreiter's cryptosystem?

The text currently states: ... the binary Goppa codes are used in several post-quantum cryptosystems, notably McEliece cryptosystem and Niederreiter cryptosystem.

However, the Niederreiter cryptosystem proposed using Generalized Reed-Solomon (GRS) codes instead of Goppa codes in an attempt to make his cryptosystem more efficient and practical compared to McEliece's cryptosystem. - Markovisch (talk) 19:58, 19 April 2017 (UTC)

Hello! The GRS Niederreiter is certainly "faster" but it was broken by Sidelnikov&Shestakov (see ). As far as I know, binary Goppa codes and MDPCs are now the "simplest" codes useful in Niederreiter. Also see the thesis of Weger (2017) 212.79.106.136 (talk) 08:57, 3 December 2019 (UTC)

References