In 1994, Moss Sweedler’s dog proposed a cryptosystem, known as Barkee’s Cryptosystem, and the related cryptanalysis. Its explicit aim was to dispel the proposal of using the urban legend that “Gröbner bases are hard to compute”, in order to devise a public key cryptography scheme. Therefore he claimed that “no scheme using Gröbner bases will ever work”. Later, further variations of Barkee’s Cryptosystem were proposed on the basis of another urban legend, related to the infiniteness (and consequent uncomputability) of non-commutative Gröbner bases; unfortunately Pritchard’s algorithm for computing (finite) non-commutative Gröbner bases was already available at that time and was sufficient to crash the system proposed by Ackermann and Kreuzer. The proposal by Rai, where the private key is a principal ideal and the public key is a bunch of polynomials within this principal ideal, is surely immune to Pritchard’s attack but not to Davenport’s factorization algorithm. It was recently adapted specializing and extending Stickel’s Diffie–Hellman protocols in the setting of Ore extension. We here propose a further generalization and show that such protocols can be broken simply via polynomial division and Buchberger reduction.

中文翻译：

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为什么你甚至不能希望在密码学中使用 Gröbner 基：失败的永恒金色辫子

1994 年，Moss Sweedler 的狗提出了一种密码系统，称为 Barkee's Cryptosystem，以及相关的密码分析。其明确目的是消除使用“格罗布纳基数难以计算”的都市传说的提议，以设计公钥密码方案。因此，他声称“使用 Gröbner 基的任何方案都不会奏效”。后来，根据另一个城市传说，提出了 Barkee 密码系统的进一步变体，与非交换 Gröbner 基的无限性（以及随之而来的不可计算性）有关；不幸的是，当时用于计算（有限）非交换 Gröbner 基的 Pritchard 算法已经可用，足以使 Ackermann 和 Kreuzer 提出的系统崩溃。赖的提议，其中私钥是一个主要理想，而公钥是这个主要理想内的一堆多项式，肯定不受 Pritchard 的攻击，但不受 Davenport 的因式分解算法的影响。它最近在 Ore 扩展的设置中专门用于和扩展 Stickel 的 Diffie-Hellman 协议。我们在这里提出了进一步的概括，并表明可以通过多项式除法和 Buchberger 约简来简单地破坏此类协议。