一种确定戈帕码真正最小距离的准则

中文摘要 AI 辅助

戈帕码因其高效的解码算法以及在梅利尔密码系统中作为底层私钥码的应用，在基于码的密码学中发挥着重要作用。确定戈帕码的真正最小距离是一个极其困难的问题。本文建立了一个戈帕码达到其设计距离的准则。考虑形如\(G(x)=U(x)H(x)+V(x)H'(x)\)的戈帕多项式，其中\(°(G)=t\)且\(H(x)\in\mathbb{F}_q[x]\)是次数为\(t + 1\)的首一不可约多项式，其根包含在支撑集\(L\)中。证明了相应的戈帕码\(\Gamma(L,G)\)包含重量为\(t + 1\)的码字当且仅当\(\frac{V(\alpha_{i_{t + 1}})}{V(\alpha_{i_j})}\in\mathbb{F}_q^*\)，\(1\leq j\leq t\)，其中\(\alpha_{i_1},\ldots,\alpha_{i_{t + 1}}\)是\(H(x)\)的根。基于此准则，通过开发基于插值的戈帕多项式构造，推导出一类达到设计距离的通用戈帕码族。通过考虑辅助多项式\(V(x)\)的单项式、二项式及其乘积来确定戈帕多项式，进一步得到了几类戈帕码族。通过取\(H(x)\)为不同的不可约二项式和三项式，得到了几个最小距离等于设计距离的显式戈帕码族。

英文摘要

Goppa codes play an important role in code-based cryptography due to their efficient decoding algorithms and their use as underlying private codes in the McEliece cryptosystem. To determine the true minimum distances of Goppa codes is a notoriously difficult problem. In this paper, we establish a criterion for a Goppa code to attain its designed distance. We consider Goppa polynomials of the form $G(x)=U(x)H(x)+V(x)H'(x)$, where $°(G)=t$ and $H(x)\in\mathbb{F}_q[x]$ is a monic irreducible polynomial of degree $t+1$ whose roots are contained in the support $L$. We prove that the corresponding Goppa code $Γ(L,G)$ contains a codeword of weight $t+1$ if and only if \[ \frac{V(α_{i_{t+1}})}{V(α_{i_j})}\in\mathbb{F}_q^*, \qquad 1\leq j\leq t, \] where $α_{i_1},\ldots,α_{i_{t+1}}$ are the roots of $H(x)$. Based on this criterion, we derive a general family of Goppa codes that attain their designed distance by developing an interpolation-based construction of Goppa polynomials. We further obtain families of Goppa codes whose Goppa polynomials are determined by considering monomial, binomial, and their product of the auxiliary polynomial $V(x)$. By taking $H(x)$ to be different irreducible binomials and trinomials, we obtain several explicit families of Goppa codes whose minimum distances are equal to designed distance.

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26 pages, 5 tables