论文标题
列举扩展的不可还原二进制GOPPA代码
Enumeration of extended irreducible binary Goppa codes
论文作者
论文摘要
GOPPA代码家族是线性代码最有趣的子类之一。由于McEliece密码系统通常选择随机的GOPPA代码作为其关键，因此对固定参数的不相等GOPPA代码的数量的了解可能有助于评估此类隐秘系统的安全性。在本文中，我们提出了一种新的方法，可以对不相等的不可还原二进制GOPPA代码的数量产生上限。更具体地说，让$ n> 3 $是一个奇数质数，$ q = 2^n $;令$ r \ geq3 $为一个正整数满足$ \ gcd（r，n）= 1 $和$ \ gcd \ big（r，q（q^2-1）\ big）= 1 $。我们获得了一个上限，用于不相等的不可还原二进制GOPPA代码，长度为$ q+1 $和$ r $。
The family of Goppa codes is one of the most interesting subclasses of linear codes. As the McEliece cryptosystem often chooses a random Goppa code as its key,knowledge of the number of inequivalent Goppa codes for fixed parameters may facilitate in the evaluation of the security of such a cryptosystem. In this paper we present a new approach to give an upper bound on the number of inequivalent extended irreducible binary Goppa codes. To be more specific, let $n>3$ be an odd prime number and $q=2^n$; let $r\geq3$ be a positive integer satisfying $\gcd(r,n)=1$ and $\gcd\big(r,q(q^2-1)\big)=1$. We obtain an upper bound for the number of inequivalent extended irreducible binary Goppa codes of length $q+1$ and degree $r$.
