求解有限交换环上代数方程组及其应用
Solving Systems of Algebraic Equations Over Finite Commutative Rings and Applications
AI总结：
本文结合Gröbner基方法求解有限交换环上的代数方程组，并利用斜多项式与Plücker坐标将有限域上的秩解码和MinRank代数方法推广到有限主理想环。
摘要：
有限环上的代数几何与编码理论中的若干问题可建模为代数方程组。其中，秩解码问题被用于公钥密码体制的构造。2004年，Nechaev和Mikhailov提出了两种求解有限链环上多项式方程组的方法，这些方法利用剩余域上的解逐步构造出全部解。然而，对于某些类型的代数方程，人们只需要部分解。本文结合两种已有方法，展示了有限链环上的Gröbner基如何用于求解有限交换环上的代数方程组。随后，我们利用斜多项式和Plücker坐标证明，一些用于求解有限域上秩解码问题和MinRank问题的代数方法可以推广到有限主理想环。
英文摘要：
Several problems in algebraic geometry and coding theory over finite rings are modeled by systems of algebraic equations. Among these problems, we have the rank decoding problem, which is used in the construction of public-key cryptography. In 2004, Nechaev and Mikhailov proposed two methods for solving systems of polynomial equations over finite chain rings. These methods used solutions over the residual field to construct all solutions step by step. However, for some types of algebraic equations, one simply needs partial solutions. In this paper, we combine two existing approaches to show how Gröbner bases over finite chain rings can be used to solve systems of algebraic equations over finite commutative rings. Then, we use skew polynomials and Plücker coordinates to show that some algebraic approaches used to solve the rank decoding problem and the MinRank problem over finite fields can be extended to finite principal ideal rings.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
