Talk:Binary Goppa code Source: en.wikipedia.org/wiki/Talk:Binary_Goppa_code

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

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

Please change the line

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

into

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

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

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)[reply]

Hello! The GRS Niederreiter is certainly "faster" but it was broken by Sidelnikov&Shestakov (see ^[1]). 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) ^[2] 212.79.106.136 (talk) 08:57, 3 December 2019 (UTC)[reply]