We initiate the study of principled, automated methods for analyzing hardness assumptions in generic group models, following the approach of symbolic cryptography. We start by defining a broad class of generic and symbolic group models for different settings—symmetric or asymmetric (leveled) k-linear groups—and by proving “computational soundness” theorems for the symbolic models. Based on this result, we formulate a very general master theorem that formally relates the hardness of a (possibly interactive) assumption in these models to solving problems in polynomial algebra. Then, we systematically analyze these problems. We identify different classes of assumptions and obtain decidability and undecidability results. Next, we develop and implement automated procedures for verifying the conditions of master theorems, and thus the validity of hardness assumptions in generic group models. The concrete outcome of this work is an automated tool which takes as input the statement of an assumption and outputs either a proof of its generic hardness or shows an algebraic attack against the assumption.

中文翻译：

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通用组模型中密码假设的自动分析

我们遵循符号密码学的方法，开始研究用于分析通用组模型中的硬度假设的有原则的自动化方法。我们首先为不同的设置定义一大类通用和符号群模型——对称或非对称（水平）k 线性群——并通过证明符号模型的“计算合理性”定理。基于这个结果，我们制定了一个非常通用的主定理，该定理将这些模型中（可能是交互的）假设的难度与解决多项式代数问题正式联系起来。然后，我们系统地分析这些问题。我们识别不同类别的假设并获得可判定性和不可判定性结果。接下来，我们开发并实施用于验证主定理条件的自动化程序，以及通用组模型中硬度假设的有效性。这项工作的具体成果是一个自动化工具，它将假设的陈述作为输入，并输出其通用硬度的证明或显示对假设的代数攻击。