论文标题
在最小代码中具有高代数免疫力的布尔功能的新颖应用
A Novel Application of Boolean Functions with High Algebraic Immunity in Minimal Codes
论文作者
论文摘要
在某些流密码中，具有高代数免疫力的布尔功能是重要的加密原始功能。在本文中，提出了两种用于构建二进制最小代码，布尔函数和矢量布尔函数的方法，具有高代数免疫。更准确地说，提出了使用Reed-Muller代码中包含的最小代码和没有非零低度歼灭器的集合中的最小代码的一般最小代码的一般结构。另一种结构使我们能够从芦苇毛刺代码的某些子代码和具有高代数免疫力的矢量布尔函数中产生最小的代码。通过这些一般结构，获得了最小二进制线性$ m $且长度小于或等于$ m（m+1）/2 $的无限族。另外，建立了提出的最小线性代码的最小距离的下限。还提出了猜想和开放问题。本文的结果表明，具有较高代数免疫力的布尔功能在几个领域中具有很好的应用，例如对称密码学，编码理论和秘密共享方案。
Boolean functions with high algebraic immunity are important cryptographic primitives in some stream ciphers. In this paper, two methodologies for constructing binary minimal codes from sets, Boolean functions and vectorial Boolean functions with high algebraic immunity are proposed. More precisely, a general construction of new minimal codes using minimal codes contained in Reed-Muller codes and sets without nonzero low degree annihilators is presented. The other construction allows us to yield minimal codes from certain subcodes of Reed-Muller codes and vectorial Boolean functions with high algebraic immunity. Via these general constructions, infinite families of minimal binary linear codes of dimension $m$ and length less than or equal to $m(m+1)/2$ are obtained. In addition, a lower bound on the minimum distance of the proposed minimal linear codes is established. Conjectures and open problems are also presented. The results of this paper show that Boolean functions with high algebraic immunity have nice applications in several fields such as symmetric cryptography, coding theory and secret sharing schemes.
