An elementary linear-algebraic proof without computer-aided arguments for the group law on elliptic curves
An elementary linear-algebraic proof without computer-aided arguments for the group law on elliptic curves
复制标题
椭圆曲线群律的基本线性代数证明，无需计算机辅助论证
DOI：
10.1142/s2661335221500015
复制
发表时间：
2021
影响因子：
0.2
通讯作者：
Koji Nuida
中科院分区：
文献类型：
--
作者：
Koji Nuida
椭圆曲线有理点上的群结构在数学中以及最近在密码学等其他领域中都有着重要的作用。然而，群性质的著名证明(特别是它的结合定律)需要一些高深的数学知识，因此非数学家不容易获得。另一方面，文献中也有给出初等证明的尝试，但这些证明中的某些部分依赖于计算机辅助计算。在这篇文章中，我们给出了这一运算的结合律的一个完备的证明，假设只有基本线性代数水平的数学知识，不需要计算机辅助论证。
The group structure on the rational points of elliptic curves plays several important roles, in mathematics and recently also in other areas such as cryptography. However, the famous proofs for the group property (in particular, for its associative law) require somewhat advanced mathematics and therefore are not easily accessible by non-mathematician. On the other hand, there have been attempts in the literature to give an elementary proof, but those rely on computer-aided calculation for some part in their proofs. In this paper, we give a self-contained proof of the associative law for this operation, assuming mathematical knowledge only at the level of basic linear algebra and not requiring computer-aided arguments.