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引用次数: 0
摘要
离散对数问题及相关问题在公钥密码学中非常重要;然而,这些问题可以简化为无差别群的隐子群问题(HSP),并存在高效的量子算法。本文更广泛地将这些问题视为不同模块上的半群作用问题(SAP)。结果证明,如果模块上的作用是注入式的,或者隐藏子群的最小生成集的心数小于或等于环的最小生成集的心数,那么模块上相应的 SAP 可以在多项式时间内还原为一个无常群的 HSP;因此,大多数基于模块上 SAP 的密码系统都无法抵御量子密码分析。这些结果适用于群上的离散算术问题和矩阵作用问题,以及模块上的其他 SAP。对于半模块上的 SAP,则找不到这样的还原。基于某些半模块上的 SAP 的密码系统有可能抵御量子攻击。
Reduction of the semigroup-action problem on a module to the hidden-subgroup problem
The discrete-logarithm problem and related problems are important in public-key cryptography; however, these problems can be reduced to the hidden-subgroup problem (HSP) of an abelian group, for which efficient quantum algorithms exist. This paper more broadly regards these problems as semigroup-action problems (SAPs) on different modules. The results prove that if the action on a module is injective or the cardinality of the hidden subgroup’s least generating set is less than or equal to that of the ring’s least generating set, the corresponding SAP on the module can be reduced to the HSP of an abelian group in polynomial time; therefore, most cryptosystems based on the SAP on a module cannot resist quantum cryptanalysis. The results are applicable to the discrete-logarithm problem and matrix-action problem on a group, along with other SAPs on a module. Such reduction is not be found for the SAP on the semi-module. The cryptographic systems based on SAPs on some semi-modules are likely to resist quantum attacks.
期刊介绍:
Quantum Information Processing is a high-impact, international journal publishing cutting-edge experimental and theoretical research in all areas of Quantum Information Science. Topics of interest include quantum cryptography and communications, entanglement and discord, quantum algorithms, quantum error correction and fault tolerance, quantum computer science, quantum imaging and sensing, and experimental platforms for quantum information. Quantum Information Processing supports and inspires research by providing a comprehensive peer review process, and broadcasting high quality results in a range of formats. These include original papers, letters, broadly focused perspectives, comprehensive review articles, book reviews, and special topical issues. The journal is particularly interested in papers detailing and demonstrating quantum information protocols for cryptography, communications, computation, and sensing.